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+{"text":"\\section{Electronic Submission}\n\\label{submission}\n\nSubmission to ICML 2017 will be entirely electronic, via a web site\n(not email).  Information about the submission process and \\LaTeX\\ templates\nare available on the conference web site at:\n\\begin{center}\n\\textbf{\\texttt{http:\/\/icml.cc\/2017\/}}\n\\end{center}\nSend questions about submission and electronic templates to\n\\texttt{icml2017pc@gmail.com}.\n\nThe guidelines below will be enforced for initial submissions and\ncamera-ready copies.  Here is a brief summary:\n\\begin{itemize}\n\\item Submissions must be in PDF.\n\\item The maximum paper length is \\textbf{8 pages excluding references and acknowledgements, and 10 pages\n  including references and acknowledgements} (pages 9 and 10 must contain only references and acknowledgements).\n\\item Do \\textbf{not include author information or acknowledgements} in your initial\nsubmission.\n\\item Your paper should be in \\textbf{10 point Times font}.\n\\item Make sure your PDF file only uses Type-1 fonts.\n\\item Place figure captions {\\em under} the figure (and omit titles from inside\nthe graphic file itself).  Place table captions {\\em over} the table.\n\\item References must include page numbers whenever possible and be as complete\nas possible.  Place multiple citations in chronological order.  \n\\item Do not alter the style template; in particular, do not compress the paper\nformat by reducing the vertical spaces.\n\\item Keep your abstract brief and self-contained, one\n   paragraph and roughly 4--6 sentences.  Gross violations will require correction at the camera-ready phase.\n  Title should have content words capitalized.\n  \n\n\\end{itemize}\n\n\\subsection{Submitting Papers}\n\n{\\bf Paper Deadline:} The deadline for paper submission to ICML 2017\nis at \\textbf{23:59 Universal Time (3:59 p.m.\\ Pacific Standard Time) on February 24, 2017}.\nIf your full submission does not reach us by this time, it will \nnot be considered for publication. There is no separate abstract submission.\n\n{\\bf Anonymous Submission:} To facilitate blind review, no identifying\nauthor information should appear on the title page or in the paper\nitself.  Section~\\ref{author info} will explain the details of how to\nformat this.\n\n{\\bf Simultaneous Submission:} ICML will not accept any paper which,\nat the time of submission, is under review for another conference or\nhas already been published. This policy also applies to papers that\noverlap substantially in technical content with conference papers\nunder review or previously published. ICML submissions must not be\nsubmitted to other conferences during ICML's review period. Authors\nmay submit to ICML substantially different versions of journal papers\nthat are currently under review by the journal, but not yet accepted\nat the time of submission. Informal publications, such as technical\nreports or papers in workshop proceedings which do not appear in\nprint, do not fall under these restrictions.\n\n\\medskip\n\nTo ensure our ability to print submissions, authors must provide their\nmanuscripts in \\textbf{PDF} format.  Furthermore, please make sure\nthat files contain only Type-1 fonts (e.g.,~using the program {\\tt\n  pdffonts} in linux or using File\/DocumentProperties\/Fonts in\nAcrobat).  Other fonts (like Type-3) might come from graphics files\nimported into the document.\n\nAuthors using \\textbf{Word} must convert their document to PDF.  Most\nof the latest versions of Word have the facility to do this\nautomatically.  Submissions will not be accepted in Word format or any\nformat other than PDF. Really. We're not joking. Don't send Word.\n\nThose who use \\textbf{\\LaTeX} to format their accepted papers need to pay close\nattention to the typefaces used.  Specifically, when producing the PDF by first\nconverting the dvi output of \\LaTeX\\ to Postscript the default behavior is to\nuse non-scalable Type-3 PostScript bitmap fonts to represent the standard\n\\LaTeX\\ fonts. The resulting document is difficult to read in electronic form;\nthe type appears fuzzy. To avoid this problem, dvips must be instructed to use\nan alternative font map.  This can be achieved with the following two commands:\n\n{\\footnotesize\n\\begin{verbatim}\ndvips -Ppdf -tletter -G0 -o paper.ps paper.dvi\nps2pdf paper.ps\n\\end{verbatim}}\nNote that it is a zero following the ``-G''.  This tells dvips to use\nthe config.pdf file (and this file refers to a better font mapping).\n\nA better alternative is to use the \\textbf{pdflatex} program instead of\nstraight \\LaTeX. This program avoids the Type-3 font problem, however you must\nensure that all of the fonts are embedded (use {\\tt pdffonts}). If they are\nnot, you need to configure pdflatex to use a font map file that specifies that\nthe fonts be embedded. Also you should ensure that images are not downsampled\nor otherwise compressed in a lossy way.\n\nNote that the 2017 style files use the {\\tt hyperref} package to\nmake clickable links in documents.  If this causes problems for you,\nadd {\\tt nohyperref} as one of the options to the {\\tt icml2017}\nusepackage statement.\n\n\\subsection{Reacting to Reviews}\n\nWe will continue the ICML tradition in which the authors are given the\noption of providing a short reaction to the initial reviews. These\nreactions will be taken into account in the discussion among the\nreviewers and area chairs.\n\n\\subsection{Submitting Final Camera-Ready Copy}\n\nThe final versions of papers accepted for publication should follow the\nsame format and naming convention as initial submissions, except of\ncourse that the normal author information (names and affiliations)\nshould be given.  See Section~\\ref{final author} for details of how to\nformat this.\n\nThe footnote, ``Preliminary work.  Under review by the International\nConference on Machine Learning (ICML).  Do not distribute.'' must be\nmodified to ``\\textit{Proceedings of the\n$\\mathit{34}^{th}$ International Conference on Machine Learning},\nSydney, Australia, PMLR 70, 2017.\nCopyright 2017 by the author(s).'' \n\nFor those using the \\textbf{\\LaTeX} style file, this change (and others) is\nhandled automatically by simply changing\n$\\mathtt{\\backslash usepackage\\{icml2017\\}}$ to \n$$\\mathtt{\\backslash usepackage[accepted]\\{icml2017\\}}$$\nAuthors using \\textbf{Word} must edit the\nfootnote on the first page of the document themselves.\n\nCamera-ready copies should have the title of the paper as running head\non each page except the first one.  The running title consists of a\nsingle line centered above a horizontal rule which is $1$ point thick.\nThe running head should be centered, bold and in $9$ point type.  The\nrule should be $10$ points above the main text.  For those using the\n\\textbf{\\LaTeX} style file, the original title is automatically set as running\nhead using the {\\tt fancyhdr} package which is included in the ICML\n2017 style file package.  In case that the original title exceeds the\nsize restrictions, a shorter form can be supplied by using\n\n\\verb|\\icmltitlerunning{...}|\n\njust before $\\mathtt{\\backslash begin\\{document\\}}$.\nAuthors using \\textbf{Word} must edit the header of the document themselves.\n\n\\section{Format of the Paper} \n \nAll submissions must follow the same format to ensure the printer can\nreproduce them without problems and to let readers more easily find\nthe information that they desire.\n\n\\subsection{Length and Dimensions}\n\nPapers must not exceed eight (8) pages, including all figures, tables,\nand appendices, but excluding references and acknowledgements. When references and acknowledgements are included,\nthe paper must not exceed ten (10) pages.\nAcknowledgements should be limited to grants and people who contributed to the paper.\nAny submission that exceeds \nthis page limit or that diverges significantly from the format specified \nherein will be rejected without review.\n\nThe text of the paper should be formatted in two columns, with an\noverall width of 6.75 inches, height of 9.0 inches, and 0.25 inches\nbetween the columns. The left margin should be 0.75 inches and the top\nmargin 1.0 inch (2.54~cm). The right and bottom margins will depend on\nwhether you print on US letter or A4 paper, but all final versions\nmust be produced for US letter size.\n\nThe paper body should be set in 10~point type with a vertical spacing\nof 11~points. Please use Times typeface throughout the text.\n\n\\subsection{Title}\n\nThe paper title should be set in 14~point bold type and centered\nbetween two horizontal rules that are 1~point thick, with 1.0~inch\nbetween the top rule and the top edge of the page. Capitalize the\nfirst letter of content words and put the rest of the title in lower\ncase.\n\n\\subsection{Author Information for Submission}\n\\label{author info}\n\nTo facilitate blind review, author information must not appear.  If\nyou are using \\LaTeX\\\/ and the \\texttt{icml2017.sty} file, you may use\n\\verb+\\icmlauthor{...}+ to specify authors and \\verb+\\icmlaffiliation{...}+ to specify affiliations. (Read the TeX code used to produce this document for an example usage.)  The author information\nwill  not be printed unless {\\tt accepted} is passed as an argument to the\nstyle file. (Again, see the TeX code used to produce this PDF.) \nSubmissions that include the author information will not\nbe reviewed.\n\n\\subsubsection{Self-Citations}\n\nIf your are citing published papers for which you are an author, refer\nto yourself in the third person. In particular, do not use phrases\nthat reveal your identity (e.g., ``in previous work \\cite{langley00}, we \nhave shown \\ldots'').\n\nDo not anonymize citations in the reference section by removing or\nblacking out author names. The only exception are manuscripts that are\nnot yet published (e.g. under submission). If you choose to refer to\nsuch unpublished manuscripts \\cite{anonymous}, anonymized copies have \nto be submitted\nas Supplementary Material via CMT. However, keep in mind that an ICML\npaper should be self contained and should contain sufficient detail\nfor the reviewers to evaluate the work. In particular, reviewers are\nnot required to look a the Supplementary Material when writing their\nreview.\n\n\\subsubsection{Camera-Ready Author Information}\n\\label{final author}\n\nIf a paper is accepted, a final camera-ready copy must be prepared.\nFor camera-ready papers, author information should start 0.3~inches\nbelow the bottom rule surrounding the title. The authors' names should\nappear in 10~point bold type, in a row, separated by white space, and centered. \nAuthor names should not be broken across lines.\nUnbolded superscripted numbers, starting 1, should be used to refer to affiliations. \nAffiliations should be numbered in the order of appearance.  A single footnote block of text should be used to list all the affiliations. (Academic affiliations should list Department, University, City, State\/Region, Country. Similarly for industrial affiliations.)\nEach distinct affiliations should be listed once.  If an author has multiple affiliations, multiple superscripts should be placed after the name, separated by thin spaces.  If the authors would like to highlight equal contribution by multiple first authors, those authors should have an asterisk placed after their name in superscript, and the term ``\\textsuperscript{*}Equal contribution\" should be placed in the footnote block ahead of the list of affiliations.  A list of corresponding authors and their emails (in the format Full Name \\textless{}email@domain.com\\textgreater{}) can follow the list of affiliations. Ideally only one or two names should be listed.\n\n\nA sample file (in PDF) with author names is included in the ICML2017 \nstyle file package.  Turn on the \\texttt{[accepted]} option to the ICML stylefile to see the names rendered. \nAll of the guidelines above are automatically met by the \\LaTeX\\ style file.\n\n\\subsection{Abstract}\n\nThe paper abstract should begin in the left column, 0.4~inches below\nthe final address. The heading `Abstract' should be centered, bold,\nand in 11~point type. The abstract body should use 10~point type, with\na vertical spacing of 11~points, and should be indented 0.25~inches\nmore than normal on left-hand and right-hand margins. Insert\n0.4~inches of blank space after the body. Keep your abstract brief and \nself-contained,\nlimiting it to one paragraph and roughly 4--6 sentences.  Gross violations will require correction at the camera-ready phase.\n\n\\subsection{Partitioning the Text} \n\nYou should organize your paper into sections and paragraphs to help\nreaders place a structure on the material and understand its\ncontributions.\n\n\\subsubsection{Sections and Subsections}\n\nSection headings should be numbered, flush left, and set in 11~pt bold\ntype with the content words capitalized. Leave 0.25~inches of space\nbefore the heading and 0.15~inches after the heading.\n\nSimilarly, subsection headings should be numbered, flush left, and set\nin 10~pt bold type with the content words capitalized. Leave\n0.2~inches of space before the heading and 0.13~inches afterward.\n\nFinally, subsubsection headings should be numbered, flush left, and\nset in 10~pt small caps with the content words capitalized. Leave\n0.18~inches of space before the heading and 0.1~inches after the\nheading. \n\nPlease use no more than three levels of headings.\n\n\\subsubsection{Paragraphs and Footnotes}\n\nWithin each section or subsection, you should further partition the\npaper into paragraphs. Do not indent the first line of a given\nparagraph, but insert a blank line between succeeding ones.\n \nYou can use footnotes\\footnote{For the sake of readability, footnotes\nshould be complete sentences.} to provide readers with additional\ninformation about a topic without interrupting the flow of the paper. \nIndicate footnotes with a number in the text where the point is most\nrelevant. Place the footnote in 9~point type at the bottom of the\ncolumn in which it appears. Precede the first footnote in a column\nwith a horizontal rule of 0.8~inches.\\footnote{Multiple footnotes can\nappear in each column, in the same order as they appear in the text,\nbut spread them across columns and pages if possible.}\n\n\\begin{figure}[ht]\n\\vskip 0.2in\n\\begin{center}\n\\centerline{\\includegraphics[width=\\columnwidth]{icml_numpapers}}\n\\caption{Historical locations and number of accepted papers for International\n  Machine Learning Conferences (ICML 1993 -- ICML 2008) and\n  International Workshops on Machine Learning (ML 1988 -- ML\n  1992). At the time this figure was produced, the number of\n  accepted papers for ICML 2008 was unknown and instead estimated.}\n\\label{icml-historical}\n\\end{center}\n\\vskip -0.2in\n\\end{figure} \n\n\\subsection{Figures}\n \nYou may want to include figures in the paper to help readers visualize\nyour approach and your results. Such artwork should be centered,\nlegible, and separated from the text. Lines should be dark and at\nleast 0.5~points thick for purposes of reproduction, and text should\nnot appear on a gray background.\n\nLabel all distinct components of each figure. If the figure takes the\nform of a graph, then give a name for each axis and include a legend\nthat briefly describes each curve. Do not include a title inside the\nfigure; instead, the caption should serve this function.\n\nNumber figures sequentially, placing the figure number and caption\n{\\it after\\\/} the graphics, with at least 0.1~inches of space before\nthe caption and 0.1~inches after it, as in\nFigure~\\ref{icml-historical}.  The figure caption should be set in\n9~point type and centered unless it runs two or more lines, in which\ncase it should be flush left.  You may float figures to the top or\nbottom of a column, and you may set wide figures across both columns\n(use the environment {\\tt figure*} in \\LaTeX), but always place\ntwo-column figures at the top or bottom of the page.\n\n\\subsection{Algorithms}\n\nIf you are using \\LaTeX, please use the ``algorithm'' and ``algorithmic'' \nenvironments to format pseudocode. These require \nthe corresponding stylefiles, algorithm.sty and \nalgorithmic.sty, which are supplied with this package. \nAlgorithm~\\ref{alg:example} shows an example. \n\n\\begin{algorithm}[tb]\n   \\caption{Bubble Sort}\n   \\label{alg:example}\n\\begin{algorithmic}\n   \\STATE {\\bfseries Input:} data $x_i$, size $m$\n   \\REPEAT\n   \\STATE Initialize $noChange = true$.\n   \\FOR{$i=1$ {\\bfseries to} $m-1$}\n   \\IF{$x_i > x_{i+1}$} \n   \\STATE Swap $x_i$ and $x_{i+1}$\n   \\STATE $noChange = false$\n   \\ENDIF\n   \\ENDFOR\n   \\UNTIL{$noChange$ is $true$}\n\\end{algorithmic}\n\\end{algorithm}\n \n\\subsection{Tables} \n \nYou may also want to include tables that summarize material. Like \nfigures, these should be centered, legible, and numbered consecutively. \nHowever, place the title {\\it above\\\/} the table with at least \n0.1~inches of space before the title and the same after it, as in \nTable~\\ref{sample-table}. The table title should be set in 9~point \ntype and centered unless it runs two or more lines, in which case it\nshould be flush left.\n\n\n\\begin{table}[t]\n\\caption{Classification accuracies for naive Bayes and flexible \nBayes on various data sets.}\n\\label{sample-table}\n\\vskip 0.15in\n\\begin{center}\n\\begin{small}\n\\begin{sc}\n\\begin{tabular}{lcccr}\n\\hline\n\\abovespace\\belowspace\nData set & Naive & Flexible & Better? \\\\\n\\hline\n\\abovespace\nBreast    & 95.9$\\pm$ 0.2& 96.7$\\pm$ 0.2& $\\surd$ \\\\\nCleveland & 83.3$\\pm$ 0.6& 80.0$\\pm$ 0.6& $\\times$\\\\\nGlass2    & 61.9$\\pm$ 1.4& 83.8$\\pm$ 0.7& $\\surd$ \\\\\nCredit    & 74.8$\\pm$ 0.5& 78.3$\\pm$ 0.6&         \\\\\nHorse     & 73.3$\\pm$ 0.9& 69.7$\\pm$ 1.0& $\\times$\\\\\nMeta      & 67.1$\\pm$ 0.6& 76.5$\\pm$ 0.5& $\\surd$ \\\\\nPima      & 75.1$\\pm$ 0.6& 73.9$\\pm$ 0.5&         \\\\\n\\belowspace\nVehicle   & 44.9$\\pm$ 0.6& 61.5$\\pm$ 0.4& $\\surd$ \\\\\n\\hline\n\\end{tabular}\n\\end{sc}\n\\end{small}\n\\end{center}\n\\vskip -0.1in\n\\end{table}\n\nTables contain textual material that can be typeset, as contrasted \nwith figures, which contain graphical material that must be drawn. \nSpecify the contents of each row and column in the table's topmost\nrow. Again, you may float tables to a column's top or bottom, and set\nwide tables across both columns, but place two-column tables at the\ntop or bottom of the page.\n \n\\subsection{Citations and References} \n\nPlease use APA reference format regardless of your formatter\nor word processor. If you rely on the \\LaTeX\\\/ bibliographic \nfacility, use {\\tt natbib.sty} and {\\tt icml2017.bst} \nincluded in the style-file package to obtain this format.\n\nCitations within the text should include the authors' last names and\nyear. If the authors' names are included in the sentence, place only\nthe year in parentheses, for example when referencing Arthur Samuel's\npioneering work \\yrcite{Samuel59}. Otherwise place the entire\nreference in parentheses with the authors and year separated by a\ncomma \\cite{Samuel59}. List multiple references separated by\nsemicolons \\cite{kearns89,Samuel59,mitchell80}. Use the `et~al.'\nconstruct only for citations with three or more authors or after\nlisting all authors to a publication in an earlier reference \\cite{MachineLearningI}.\n\nAuthors should cite their own work in the third person\nin the initial version of their paper submitted for blind review.\nPlease refer to Section~\\ref{author info} for detailed instructions on how to\ncite your own papers.\n\nUse an unnumbered first-level section heading for the references, and \nuse a hanging indent style, with the first line of the reference flush\nagainst the left margin and subsequent lines indented by 10 points. \nThe references at the end of this document give examples for journal\narticles \\cite{Samuel59}, conference publications \\cite{langley00}, book chapters \\cite{Newell81}, books \\cite{DudaHart2nd}, edited volumes \\cite{MachineLearningI}, \ntechnical reports \\cite{mitchell80}, and dissertations \\cite{kearns89}. \n\nAlphabetize references by the surnames of the first authors, with\nsingle author entries preceding multiple author entries. Order\nreferences for the same authors by year of publication, with the\nearliest first. Make sure that each reference includes all relevant\ninformation (e.g., page numbers).\n\n\\subsection{Software and Data}\n\nWe strongly encourage the publication of software and data with the\ncamera-ready version of the paper whenever appropriate.  This can be\ndone by including a URL in the camera-ready copy.  However, do not\ninclude URLs that reveal your institution or identity in your\nsubmission for review.  Instead, provide an anonymous URL or upload\nthe material as ``Supplementary Material'' into the CMT reviewing\nsystem.  Note that reviewers are not required to look a this material\nwhen writing their review.\n\n\\section*{Acknowledgements} \n \n\\textbf{Do not} include acknowledgements in the initial version of\nthe paper submitted for blind review.\n\nIf a paper is accepted, the final camera-ready version can (and\nprobably should) include acknowledgements. In this case, please\nplace such acknowledgements in an unnumbered section at the\nend of the paper. Typically, this will include thanks to reviewers\nwho gave useful comments, to colleagues who contributed to the ideas, \nand to funding agencies and corporate sponsors that provided financial \nsupport.  \n\n\n\\nocite{langley00}\n\n\n\\section{Introduction}\n\\label{submission}\n\nAutomated driving development has radically changed during the past few years, driven by advances in Artificial Intelligence (AI), and specifically Deep Learning (DL). \nDeveloping an efficient and safe driving policy is in the heart of reaching high level of autonomy in a robot car. Traditional methods are driven with rule based approaches \\cite{le2006review}\\cite{pasquier2001fuzzylot}, while significant advancement in the field is driven by learning approaches \\cite{sallab2016end}\\cite{el2017deep} \\cite{bojarski2016end}. Reinforcement learning is the arm of AI which is concerned with solving the control problem based on learning while interacting with the environment.\n\nOur motivation is to take the work in \\cite{el2017deep} and \\cite{sallab2016end} a step further towards real car deployment. The constraints of such task are: 1) No damage caused by interaction with real environment (which was possible in the game engines world), 2) Sample efficiency, so learning time has to be reasonable 3) Generalization, where the learnt policy of the driving agent should be able to capture the basics of the intuitive driving when deployed in unseen environments.\n\nThere are several approaches to find an efficient driving policy. Optimal control methods based on traditional planning require knowledge of an environment model. While model based approaches are gaining more popularity \\cite{polydoros2017survey}, it is still hard to develop an efficient environment model especially for complex ones like urban and city scenarios. On the other hand, model free approaches \\cite{Watkins1989a} \\cite{sallab2016end} require the definition of a reward function, which only exist in simulated environments and game engines but not in real world.\nImitation learning is another successful approach, which might suffer the risk of de-generalization to unseen environments other than trained on. The most popular algorithms are based on data aggregation, which could be done through some hacks as in \\cite{bojarski2016end}, or formally as in \\cite{ross2011reduction}.\n\nIn this paper we propose a Meta learning framework for automated driving, to enforce transfer learning from one environment to another. Our hypothesis is that Meta learning will help capturing generic features without memorizing one specific environment. One proof of generalization is the visualization of which features the agent has learnt. If the learnt features are quite specific to the training environment, then it means that such agent will fail if deployed in another environment with different features. Under the Meta framework, we propose a novel algorithm; MetaDAgger, which is based on a Meta learning framework to aggregate data across different environments. To ensure generalization, we split the learning agent into two: 1) Meta learner, whose objective is to capture generic features not specific to an environment and 2) Low level learner, whose objective is to drive in a specific environment. The role of Meta learning is to smooth the learning performance across different environments. \n\nThe two learners are trained using Convolutional Neural Networks (ConvNets) on two different data sets. The meta learner is trained on a data set of environments; where  a set of environments are kept for training, while another independent set of environments are kept for testing and are not allowed to alter the learning process. One the other hand, the Low learner is trained on a data set aggregated from each specific environment, in the form of state-action pairs. \nThe link between both learners is done through continual lifelong learning, where the Low learner communicates its learning to the Meta learner whenever a switch from an environment to the other is undergone. In this way the learning is preserved across different environments, which enable the Meta learner to capture general features as proved by the features visualizations. It is worth noting that in \\cite{sallab2016end} an interesting conclusion is that continual learning highly improves the learning curve, which is in line with the proposed metal learning framework proposed in this work.\n\nThe experimental setup is based on The Open Race Car Simulator (TORCS) \\cite{wymann2000torcs}. The Meta dataset is formed of 19 tracks, with 10 training tracks and 9 testing tracks. Low learners are based on Convolutional Neural Networks (ConvNets) for policy learning. The results show significant advantage of MetaDAgger over DAgger on both training and testing tracks. MetaDAgger is not only improving generalization, but also significantly improving the sample efficiency and learning time over DAgger. The objective in each low level task is to keep the central lane. The demonstration in all tracks comes from a tradition PID controller, with access to the position of the ego car with respect to the left and right lanes. The control actions is applied to the steering wheel, with continuous actions output. The input states in all experiments are taken as the raw pixels information as provided by TORCS.\n\nThe rest of the paper is organized as follows; first we discuss the related work, then the Meta learning framework is described followed by the MetaDAgger algorithm. Then the experimental setup is described followed by the discussion of results and visualizations, and finally we conclude.\n\n\n\\section{Related Work}\n\\label{submission}\n\nThe problem of developing a driving policy is essentially formulated under the framework of optimal control. The solution of the optimization problem is easily found under the traditional planning framework only if an environment model exists, a requirement which is difficult to achieve in the real world of high way, urban or city driving. Model based approach is a wide topic that has been recently tackled in \\cite{polydoros2017survey}.\n\nOn the other hand, model free approaches have undergone a huge advancement in the area of human level control, the Deep Q Net is a famous example of which \\cite{MnihKavukcuogluSilverEtAl2015a} \\cite{MnihKavukcuogluSilverEtAl2013a}. The success of model free Reinforcement Learning \\cite{Sutton1988a} \\cite{Watkins1989a} has reached the area Automated Driving \\cite{Karavolos2013a} \\cite{el2017deep} \\cite{sallab2016end}. However, model free approaches require interacting with an environment through actions and rewards scheme. Such interaction usually requires a controlled environment such as simulators and game engines. The reward function is relatively easy acquired in such configuration of game engines. However, it is not clear how such reward function can be obtained in real world without causing too much damage. Moreover, model free approaches are known of their sample inefficiency, which is compensated by huge number of interactions with the simulated environment, which is again infeasible in real world. \n\nAnother learning approach based on human demonstrations has been successfully deployed in automated driving \\cite{bojarski2016end}.  Apprenticeship learning has been tackled in \\cite{AbbeelNg2004} \\cite{NgRussellothers2000a}, under the framework of Inverse Reinforcement Learning (IRL). IRL goal is to deduce the reward function the coach has been trying to achieve or maximize throughout the course of apprenticeship. Such hard goal is achieved on the expense of high complexity algorithm, involving two nested RL loops.\n\nIn order to tackle the problem of sample efficiency without knowing the environment models, supervised learning models are employed, where learning is based on demonstrated behavior to imitate (imitation learning). Learning from demonstrations has been approached as a supervised learning problem for automated driving in \\cite{bojarski2016end}. The risk of such an approach is the lack of generalization, where supervised learning schemes could converge to exact behavior cloning. The challenge of getting such approaches to work lies in the ability to enrich the training data by introducing new unseen states from the demonstration. One solution is data aggregation, where the agent is able to extend his actions to further states than the demonstrated ones.  Some hacks have been proposed in \\cite{bojarski2016end} to achieve data aggregation, where two cameras has been added in addition to the central camera to augment the training states with different situations supported with the correct action to take in each. A more formal approach has been formulated in the Dataset aggregation (DAgger) algorithm \\cite{ross2011reduction}. DAgger algorithm is also extensible to include safety constraints, as in SafeDAgger \\cite{zhang2016query}.\n\nTransfer of experience is an essential component of the human learning process. Transfer of knowledge could be in the form of external coaching, demonstrations or apprenticeship, or it could take the form of self accumulated experience through lifelong, continual learning \\cite{urgen1996simple}. Meta learning \\cite{thrun2012learning} \\cite{urgen1996simple} is another name of continual and transfer learning. Meta learning has also been used to search the best hyper parameters settings \\cite{hochreiter2001learning} \\cite{andrychowicz2016learning}. Continual learning fosters the transfer of experience across the life time of the agent. In addition, Meta learning provides a framework to transfer the knowledge acquired in one task to other tasks as well thanks to the captured information from task to task in the high level or Meta learner \\cite{thrun2012learning}.\n\n\\section{Meta Learning Framework for Automated Driving}\n\\label{submission}\n\nIn this section the Meta Learning framework is presented $Figure$ \\ref{Fig1}. The goal of the framework is to reach smooth and generalized performance over a set of environments\\{${E_{1},E_{2},..,E_{N}}$\\}. The performance is measured with the ability to take driving actions (e.g. steering) in a correct manner defined by the demonstrated behavior. \n\nTo ensure and test generalization, the set of environments are split into training and testing environments; $E_{train}$  and $E_{test}$. The learning algorithm is allowed to access the ground truth actions set of training environments $E_{train}$ , while it is not allowed to access the ground truth actions of the test environments $E_{test}$ . The learning is said to be generalized if it achieves the desired performance on the test environments $E_{test}$. The architecture is based on two learners:\n\n\\begin{figure*}\n\\centerline{\\includegraphics[width=120mm]{1}}\n\\caption{\\label{Fig1} Meta Learning Framework for Dataset Aggregation}\n\\label{icml-historical}\n\\end{figure*} \n\n\\subsection{Low Level Learner ($L$)}\n\nThe scope of the Low learner $L$ is to capture the specific features of each environment. The environment in operation $E$ is selected from a pool of training environments $E_{train}$. The low level learner is associated with a training data set $L_{train}$ which is formed due to interaction with an environment in operation $E$.  The training data set $L_{train}$  is formed of a list of state-action pairs \\{${s,a}$\\}$^t$, where t is the index of the interaction time stamp, with t=\\{$0,1,...,T$\\}, where $T$ is the end of interaction episode. An interaction episode is terminated with the $L$ reaching the goal, made a terminating mistake or after certain defined number of steps. The state $s$ is the measurement obtained by probing the environment $E$. The action $a$ is obtained from a demonstration, either from Human or from a reference algorithm.\n\nUnder the framework of Automated Driving, $L$ is trained using supervised learning imitating the demonstrated behavior.\n\n\n\\subsection{Meta Learner ($M$)}\n\nThe goal of the Meta Learner $M$ is to capture general features across all environments \\{${E_{1},E_{2},..,E_{N}}$\\}. With each interaction episode, $M$ communicates the initialization parameters to the Low learner $L$ to start interaction with the in operation environment $E$. After the Low learner finishes an episode, the learnt parameters of $L$ are communicated back to the $M$ to ensure continual learning.\n\nThe Meta Learner $M$ parameters are updated based on aggregated data over different interactions with the training environments $E_{train}$. The result of data set aggregation is called $M_{train}$. The members of $M_{train}$ have the same format of  state-action pairs  \\{${s,a}$\\}$^t$ as $L_{train}$. The training of $M$ is also following a supervised learning scheme using the training data in $M_{train}$.\n\nThe driving policy $\\pi^M$ is obtained as the parameters of the Meta Learner $M$, such that an action at time $t$ is obtained as  . In the context of Neural Networks, $\\pi^M$ is parameterized by the weights parameters of the network.\nThe performance of $\\pi^M$ is tested against a test data set $M_{test}$, which is formed from the set of test environments $E_{test}$. \n\n\n\\section{MetaDAgger Algorithm} \n \nIn this section the MetaDAgger algorithm is described as previous, in the light of the Meta learning framework described in $Figure$ \\ref{Fig1}.\n\n\\begin{algorithm}[tb]\n   \\caption{MetaDAgger}\n   \\label{alg:example}\n\\begin{algorithmic}\n   \\STATE {\\bfseries Loop on $E_{train}$:} \/\/ Data Collection Step\n    \\STATE  \\qquad Collect $L_{train}$ += \\{${s, a_{ref}}\\}^t$ \/\/ $a_{ref}$ is obtained \n    \\STATE  \\qquad from a demonstration or a reference algorithm \n    \\STATE  \\qquad Aggregate $M_{train}$ += $L_{train}$\n    \\STATE Fit a Meta model $M$ based on $M_{train}$\\\\\n    -------------------------------------------------------------------\n   \\STATE {\\bfseries Loop until $N$-iter is reached} \/\/ Data Aggregation Step\n   \\STATE \\qquad { \\textbf{for} each $E$ in $E_{train}$} \\textbf{do}\n   \\STATE \\qquad \\quad Initialize $L$ = $M$ \n   \\STATE \\qquad \\quad Loop for $N_{steps}$\n   \\STATE \\qquad \\qquad $L$ interacts with $E$ to measure current state $s$\n   \\STATE \\qquad \\qquad Execute $a_{L}$ = $\\pi^L$(s), where $\\pi^{L}$ is obtained us-\n   \\STATE \\qquad \\qquad ing the parameters of $L$.\n   \\STATE \\qquad \\qquad \\textbf{if}  {$a_{L} != a_{ref}$}: \/\/Aggregate incorrect actions\n   \\STATE  \\qquad \\qquad \\qquad Aggregate $L_{train}$ += \\{${s, a_{ref}}$\\}$^t$\n    \\STATE \\qquad \\qquad Fit a Low model $L$ based on $L_{train}$\n    \\STATE \\qquad \\quad Save back M=L\n \\STATE Return $M$, $\\pi^M$\n\n\\end{algorithmic}\n\\end{algorithm}\n\nThe algorithm is based on two main steps:\n\n\\subsection{Data Collection}\nDuring data collection the reference demonstration is interacting with the environment to collect reference data. The reference demonstration could be a human or a reference algorithm. The collected data take the shape of state-action \\{${s, a_{ref}}\\}^t$ pairs at each interaction time step $t$, where $a_{ref}$  is the action provided by the demonstration.\nThis data is aggregated into $M_{train}$  to train the initial model parameters $M$, which could be a ConvNet fitting the supervised data.\n\n\n\\subsection{Data Aggregation}\nData aggregation is repeated for a number of iterations $N$-iter $>$ $|E_{train}|$, which means that the aggregation is repeated more than once over all the training environments.\nEvery iteration, $L$ interacts with $E$, starting from the parameters captures in $M$, which ensures continual learning. The parameters of $L$ define the interaction policy $\\pi$ that will be executed. The interaction happens for N-steps, during which data aggregation takes place. The action to be executed is obtained from the policy parameters $\\pi^{L}$ based on the measured state st. The aggregated data $L_{train}$ is then used to retrain $L$. At the end of each episode, the model is saved back to $M$ to ensure lifelong and continual learning for the next environment.\n\nIn order to improve sample efficiency, only the incorrect actions are saved. Hence, only actions that are different from the reference actions are saved. The difference is taken within certain tolerance (say 40\\% error). This is important in two aspects: 1) the aggregated data set is focused on mistakes only, which need to be corrected when $L$ is re-trained, and 2) in real world, we have no access to the reference action $a_{ref}$, however, it is easier to ask a human supervisor to attend and correct only the algorithm mistakes, even before it causes damage, in which case the corrected state-action pair is aggregated and retraining happens to correct it.\n\n\n\n\n\\section{Experimental Setup} \n\n\\subsection{Environment}\nMetaDAgger is designed to be ready for deployment in real car. The algorithm steps ensure data sampling efficiency and zero damage to the experimental car. Moreover, aggregation of incorrect actions ensures easy human supervision requiring the least effort.\n\nAs a first step towards deploying MetaDAgger in real car, we perform our experiments under TORCS game engine. We divide the 19 tracks in TORCS into 10 training tracks ($E_{train}$), and 9 test tracks ($E_{test}$). We use the Gym-TORCS environment \\cite{GYM_TORCS}, with the visual image input \\cite{Visual_TORCS}. Hence, the input state is the raw image pixels (64x64) as provided by the visual TORCS client as shown in $Figure$ \\ref{Fig2}.\nThe actions provided by MetaDAgger are the continuous steering angle values. The supervision signal is obtained using a reference PID controller that can access the position of the car from the two side lanes, which is provided by the TORCS game engine. Although we could use this supervision signal to aggregate all encountered states in the data aggregation step, however, we keep aggregating only the incorrect ones. Since the actions are continuous, mapping the absolute action value to a steering angle is sensitive to small differences. Because of this, we relax the exact matching between the reference and algorithm actions to a certain tolerance, empirically set to 40\\%.\n\n\\begin{figure}\n\\vskip 0.2in\n\\begin{center}\n\\centerline{\\includegraphics[width=50mm]{2}}\n\\caption{\\label{Fig2}TORCS visual input state}\n\\label{icml-historical}\n\\end{center}\n\\vskip -0.2in\n\\end{figure}\n\n\n\\subsection{Network Architecture and Hyper Parameters}\n\nFor the Low learner $L$ and the Meta learner $M$ we fit the same ConvNet model shown in $Figure$ \\ref{Fig3} , which simplifies copying models back and forth between $M$ and $L$ for continual learning. The kernel sizes are kept to a small size (3x3) due to the small input image size (64x64). Batch normalization is found to significantly improve the learning time \\cite{ioffe2015batch}. Xaviar initialization \\cite{glorot2010understanding} is used to initialize all weights. Dropout of 0.25 is used in the convolution layers, followed by Dropout of 0.5 in the fully connected ones.\nReLU activation is used in all layers, except for the output, where linear activation to produce the continuous output. The loss criterion is Mean Squared Error (MSE) minimization, since our task is a regression one.\n\n\\begin{figure*}\n\\centerline{\\includegraphics[width=\\textwidth]{3}}\n\\caption{\\label{Fig3}ConvNet architecture for Low and Meta learners}\n\\label{icml-historical}\n\\end{figure*}\n\n\\subsection{Results}\nWe first evaluate the generalization performance of MetaDAgger. The evaluation is done over the 9 test tracks $M_{test}$. Results are shown in $Figure$ \\ref{Fig4}. The horizontal axis represents the number of data aggregation iterations ($N$-iter), while the vertical axis is the number of steps the car moved without exiting the lane. We should note that 1000 steps mean one complete lap.\n\n\\begin{figure}\n\\begin{center}\n\\centerline{\\includegraphics[width=\\columnwidth]{5}}\n\\caption{\\label{Fig4}MetaDAgger performance on test tracks. Most of the tracks are completed just after 0 or 1 iterations of data aggregation.}\n\\label{icml-historical}\n\\end{center}\n\\end{figure}\n\n\n\nWe compare the performance of MetaDAgger against traditional DAgger \\cite{ross2011reduction}. Results are shown in $Figure$ \\ref{Fig5}, where the vertical axis represents the number of steps without collision (one complete lap equals 1000 steps), and the horizontal axis represents the test track number. For some of the test tracks, MetaDAgger is able to complete one or two laps. For other tracks, performance is better than DAgger, although no laps are completed.\n\n\\begin{figure}\n\\begin{center}\n\\centerline{\\includegraphics[width=\\columnwidth]{4}}\n\\caption{\\label{Fig5}MetaDAgger vs. DAgger. Overall MetaDAgger is better than DAgger. For some tracks, it is still hard even with Meta learning to capture some hard turns.}\n\\label{icml-historical}\n\\end{center}\n\\end{figure}\n\nWe analyze the learnt features in case of DAgger and MetaDAgger. We use Grad CAM \\cite{selvaraju2016grad}  visualization technique of the gradient of the last neuron (linear activation for action output), projected back to the input image. The result is a heat map showing which part of the image contributes more to the output. In case of DAgger, the learnt features are more memorizing and capturing the details of the track it is trained on, as shown in $Figure$ \\ref{Fig6}. For example, we can see the most important part is the horizon or the place with the mountain; this is because the theme of this track is a desert one. It is then understood why it is hard for such model to generalize to other tracks with different themes.\n\n\\begin{figure}\n\\begin{center}\n\\centerline{\\includegraphics[width=\\columnwidth]{6}}\n\\caption{\\label{Fig6}DAgger visualization. Most of the features represent the horizon or the mountain features.}\n\\label{icml-historical}\n\\end{center}\n\\end{figure}\n\n\n\nOn the other hand, when we visualize the filters of MetaDAgger in $Figure$ \\ref{Fig7}, we see that the learnt features are more representing the relevant features to the driving task, like the positions of the lanes. This proves the generality of such model.\n\n\\begin{figure}\n\\begin{center}\n\\centerline{\\includegraphics[width=\\columnwidth]{7}}\n\\caption{\\label{Fig7}MetaDAgger filter visualization. Lane features are captured in many cases.}\n\\label{icml-historical}\n\\end{center}\n\\end{figure}\n\nWe further evaluate the sample efficiency of MetaDAgger versus DAgger in $Figure$ \\ref{Fig8}. Here we evaluate how many data aggregation iterations are needed in order for the algorithm to complete a lap. An iteration terminates after certain number of steps, a complete lap or a termination condition. In our experiments the number of steps are taken to be 1000 (complete lap), while the termination condition is met when the car is out of the track. In case of DAgger, it takes 4 iterations, while for MetaDAgger it takes only two. Moreover, just after the data collection and behavior cloning step, the algorithm is able to complete 80\\% of the track.\n\n\\begin{figure}\n\\begin{center}\n\\centerline{\\includegraphics[width=\\columnwidth]{8}}\n\\caption{\\label{Fig8}Sample efficiency of MetaDAgger vs. DAgger. After only data collection, MetaDAgger is able to complete 80\\% of the lap.}\n\\label{icml-historical}\n\\end{center}\n\\end{figure}\n\n\\section{Conclusion}\n\nIn this paper we presented a framework for generalized imitation learning, based on the principles of Meta learning and data set aggregation. The proposed algorithm MetaDAgger is shown to be able to generalize on unseen test tracks, achieving much less training time and better sample efficiency.  The results on TORCS show significant improvement on both training and testing tracks, supported by visualizations of the generic learnt features. The proposed algorithm is designed to be ready for real car deployment, where the data aggregation step is only limited to correcting the mistakes the algorithm makes in real world.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nSuppose \n$\\be$ is defined as a family of $C^*$-correspondences $\\{E_1, \\ldots,E_k\\},$ over a $C^*$-algebra $\\mathcal{B}$ along with the unitary\nisomorphisms $t_{i,j}: E_i \\ot E_j \\to E_j \\ot E_i$ ($i>j$) where $k\\in \\mathbb N.$ With the help of these unitary isomorphisms, we may identify $C^*$-correspondence  $\\be ({\\bf n})$ as $E_1^{\\ot^{ n_1}} \\ot \\cdots \\ot E_k^{\\ot^{n_k}},$ for all\n${\\bf n}=(n_1, \\cdots, n_k) \\in \\Nk.$  Define maps $t_{i,i} := \\id_{E_i \\ot E_i}$ and $t_{i,j} := t_{j,i}^{-1}$ when $i<j.$ \n In this case, we say that $\\be$ is a {\\it product system over $\\Nk$} (cf. \\cite{F02}). \n \n A {\\it completely bounded, covariant representation} (CB-representation) (cf. \\cite{S008}) of $\\be$ on a\n\t\tHilbert space $\\mathcal{H}_{V}$ is defined as a tuple $ V:=(\\sigma, V^{(1)}, \\ldots, V^{(k)})$, where $\\sigma$ is a\n\t\trepresentation of  $\\mathcal{B}$ on $B(\\mathcal{ H}_{V}),$ and  $V^{(i)}:E_i \\to B(\\mathcal{H}_{V})$ are  completely bounded linear  maps satisfying\n\t\t\\[ V^{(i)}(a \\xi_i b) = \\sigma(a) V^{(i)}(\\xi_i) \\sigma(b), \\;\\;\\; a,b \\in \\mathcal M, \\xi_i \\in E_i,\\]\n\t\tand satisfying the commutation relation\n\t\t\\begin{equation} \\label{rep} \\wV^{(i)} (I_{E_i} \\ot \\wV^{(j)}) = \\wV^{(j)} (I_{E_j} \\ot \\wV^{(i)}) (t_{i,j} \\ot I_{\\mathcal {H}_{V}})\\end{equation}\n\twhere $1\\leq i,j\\leq k.$ We shall use notations $\\widetilde{V}^{(i)}_l: E_i^{\\ot l}\\otimes \\mathcal {H}_{V}\\to \\mathcal{ H}_{V}$ defined by\n\t\\[ \\widetilde{V}^{(i)}_l (\\xi_1 \\ot \\cdots \\ot \\xi_l \\ot h) := V^{(i)} (\\xi_1) \\cdots V^{(i)}(\\xi_l) h\\]\n\twhere $\\xi_1, \\ldots, \\xi_l \\in E_{i}, h \\in \\mathcal{ H}_{V}$.\n\t\n\t \n\t \n\t \n\t \n\t \n\t The following definition of doubly\n\t commuting isometric representation is a generalization of  doubly $\\Lambda$-commuting row isometries \\cite{GP20} and non-commuting isometries \\cite{MP}.\n\\begin{definition}  \\label{dcom}\n\tA  CB-representation $(\\sigma, V^{(1)}, \\ldots,$ $ V^{(k)})$ of  $\\be$ on  $\\mathcal{H}_{V}$ is said to be {\\rm doubly\n\t\tcommuting isometric representation} (DCI-representation)  (cf. \\cite{S008}) if for each distinct $i,j \\in \\{1,\\ldots,k\\}$ we have\n\t\\begin{equation*}\\label{doubly}\\wV^{(j)^*} \\wV^{(i)} =\n\t(I_{E_j} \\ot \\wV^{(i)})  (t_{i,j} \\ot I_{\\mathcal{ H}_{V}})  (I_{E_i} \\ot \\wV^{(j)^*}),\n\t\\end{equation*} and each $\\widetilde{V}^{(i)}$ is an isometry. \n\\end{definition}\n The {\\rm Fock module} of  $\\mathbb{E},$ is given by\n\t$$\\mathcal{F}(\\mathbb{E}):=\\bigoplus_{\\mathbf{n} \\in \\mathbb{N}^k_0}\\mathbb{E}(\\mathbf{n}),$$ where the natural left action $\\phi_{\\infty}$  is given as $\\phi_{\\infty}(a)(\\oplus_{\\mathbf{n} \\in \\mathbb{N}_0^k} \\xi_{\\mathbf{n}})=\\oplus_{\\mathbf{n} \\in \\mathbb{N}_0^k} a \\cdot \\xi_{\\mathbf{n}},\\: \\xi_{\\mathbf{n}} \\in \\mathbb{E}(\\mathbf{n}), a \\in \\mathcal{B}.$\n\t Define for $\\mathbf{m}=(m_1, \\cdots, m_k) \\in \\mathbb{N}_0^k $, the map $V_{\\mathbf{m}}: \\mathbb{E}(\\mathbf{m}) \\to B(\\mathcal{H}_{V})$  by $V_{\\mathbf{m}}(\\xi_{\\mathbf{m}})h:=\\widetilde{V}_{\\mathbf{m}}(\\xi_{\\mathbf{m}} \\otimes h), ~ \\xi_{\\mathbf{m}} \\in \\mathbb{E}(\\mathbf{m}), h \\in \\mathcal{H}_{V},$ where $\\widetilde{V}_{\\mathbf{m}}:\\mathbb{E}(\\mathbf{m})\\otimes_{\\sigma} \\mathcal{H}_{V} \\to \\mathcal{H}_{V}$ is defined by\n$$\\widetilde{V}_{\\mathbf{m}}=\\widetilde{V}^{(1)}_{m_1}\\left(I_{E_1^{\\otimes m_1}} \\otimes\\widetilde{V}^{(2)}_{m_2}\\right) \\cdots \\left(I_{E_1^{\\otimes m_1} \\otimes \\cdots \\otimes E_{k-1}^{\\otimes {m_{k-1}}}} \\otimes\\widetilde{V}^{(k)}_{m_k}\\right).$$ \n\nNow we recall the definition of invariant subspace (cf. \\cite{SZ08}) for the covariant representation $(\\sigma, V^{(1)}, \\dots, V^{(k)})$.\n\\begin{definition}\t\n\t\\begin{itemize}\n\t\t\\item[$(1)$] Consider a CB-representation $(\\sigma, V^{(1)}, \\dots, V^{(k)})$ of $\\mathbb{E}$ on  $\\mathcal{H}_{V}$ and suppose  $\\mathcal K$ is a closed subspace of $ \\mathcal{ H}_{V}.$  Then we say $\\mathcal K$ is  $(\\sigma, V^{(1)}, \\dots, V^{(k)})$-{\\em invariant}   if it  is $\\sigma(\\mathcal M)$-invariant\n\t\n\t\n\t\tand, is invariant by each operator $V^{(i)}(\\xi_{i}),\\: \\xi_{i} \\in E_{i}, i \\in I_{k}.$  In addition, if  $ \\mathcal{K}^{\\bot}$ is   invariant by $V^{(i)}(\\xi_{i})$ for $\\xi_{i} \\in E_{i}, i \\in I_{k}$ then we say $\\mathcal{K}$ is $(\\sigma, V^{(1)}, \\dots, V^{(k)})$-{\\em reducing}.  Restricting naturally  this representation we get  another representation of $E$ on $\\mathcal{K}$ which will be denoted as $(\\sigma, V^{(1)}, \\dots, V^{(k)})|_{\\mathcal{K}}.$ \n\t\t\n\t\t\\item[$(2)$]  A closed  subspace $\\mathcal{W}$ of $\\mathcal{H}_{V}$ is called {\\em wandering} subspace  for $(\\sigma, V^{(1)}, \\dots, V^{(k)})$, if it is $\\sigma(\\mathcal{B})$-invariant and $\\mathcal{W}\\perp\\wt{V}_{\\mathbf n}(\\mathbb{E}({\\mathbf n} )\\ot \\mathcal{W})$ for every $n \\in \\mathbb{N}_{0}^{k}.$ The representation $(\\sigma, V^{(1)}, \\dots, V^{(k)})$ has {\\em generating wandering subspace property} (GWS-property) if there is a wandering subspace $\\mathcal{W}$ of $\\mathcal{H}_{V}$ satisfying\n\t\t$$\\mathcal{H}_{V}=\\displaystyle\\bigvee_{\\mathbf n\\in \\mathbb{N}_0^{k}}\\wt{V}_{\\mathbf n}(\\mathbb{E}({\\mathbf  n}) \\ot \\mathcal{W})$$\n\t\tand the corresponding  wandering subspace $\\mathcal{W}$\n\t\tis called generating wandering subspace (GWS). \n\t\\end{itemize}\n\t\n\\end{definition}\n\t\nConsider  pure DCI-representations $(\\sigma, V^{(1)}, \\dots, V^{(k)})$ and $(\\mu, T^{(1)}, \\dots, T^{(k)})$ of $\\mathbb{E}$ on the Hilbert spaces $\\mathcal{H}_{V}$ and $\\mathcal{H}_{T},$ respectively. Let $I_{k}:=\\{1,2,\\dots, k\\}.$ Then using \\cite[Corollary 3.5]{HV21}, we get $$\\mathcal{H}_{V}=\\bigoplus_{\\mathbf{n} \\in \\mathbb{N}_0^k}\\wt{V}_{\\mathbf{n}}(\\mathbb{E}(\\mathbf{n}) \\otimes \\mathcal{W}_{\\mathcal{H}_{V}} ) \\:\\: \\mbox{and} \\:\\: \\mathcal{H}_{T}=\\bigoplus_{\\mathbf{n} \\in \\mathbb{N}_0^k}\\wt{T}_{\\mathbf{n}}(\\mathbb{E}(\\mathbf{n}) \\otimes \\mathcal{W}_{\\mathcal{H}_{T}} ),$$ where $\\mathcal{W}_{\\mathcal{H}_{V}}$ and $\\mathcal{W}_{\\mathcal{H}_{T}}$ are the generating wandering subspaces for $V$ and $T,$ respectively.\n\nA bounded operator $A: \\mathcal{H}_{V}\\to \\mathcal{H}_{T}$ is called {\\it multi-analytic} (cf. \\cite{HV21}) if it satisfies \n\t\\begin{align*}\n\tAV^{(i)}(\\xi_i)h=T^{(i)}(\\xi_i)Ah \\hspace{0.7cm}\\mbox{and} \\hspace{0.7cm}A \\sigma(a)h= \\mu(a)Ah,\n\t\\end{align*}\n\twhere $\\xi_i \\in E_i, \\: h \\in \\mathcal{H}_{V},$ $ a \\in \\mathcal{B}$ and $ i \\in I_{k}.$ Each such $A$ is uniquely determined by  $\\Theta: \\mathcal{W}_{\\mathcal{H}_{V}} \\to \\mathcal{H}_{T}$ satisfying  $\\Theta \\sigma(a)h=\\mu(a)\\Theta h, h \\in \\mathcal{W}_{\\mathcal{H}_{V}},$ where $ \\Theta= A|_{\\mathcal{W}_{\\mathcal{H}_{V}}}.$ This follows  because for every $\\xi_{\\mathbf{n}} \\in \\mathbb{E}(\\mathbf{n}), h \\in \\mathcal{W}_{\\mathcal{H}_{V}}$ we have $AV_{\\mathbf{n}}(\\xi_{\\mathbf{n}})h=T_{\\mathbf{n}}(\\xi_{\\mathbf{n}})\\Theta h$ and $\\mathcal{H}_{V}=\\bigoplus_{\\mathbf{n} \\in \\mathbb{N}_0^k}\\wt{V}_{\\mathbf{n}}(\\mathbb{E}(\\mathbf{n}) \\otimes \\mathcal{W}_{\\mathcal{H}_{V}} ).$ Conversely, if we start with $\\Theta: \\mathcal{W}_{\\mathcal{H}_{V}} \\to \\mathcal{H}_{T}\\\\\\left( = \\bigoplus_{\\mathbf{n} \\in \\mathbb{N}_0^k}\\wt{T}_{\\mathbf{n}}(\\mathbb{E}(\\mathbf{n}) \\otimes \\mathcal{W}_{\\mathcal{H}_{T}} ) \\right)$ which  satisfies $\\Theta \\sigma(a)h=\\mu(a)\\Theta h, h \\in \\mathcal{W}_{\\mathcal{H}_{V}},$ then we get $M_{\\Theta}: \\mathcal{H}_{V} \\to \\mathcal{H}_{T}$ defined by the equation $$M_{\\Theta}V_{\\mathbf{n}}(\\xi_{\\mathbf{n}})h=T_{\\mathbf{n}}(\\xi_{\\mathbf{n}})\\Theta h=T_{\\mathbf{n}}(\\xi_{\\mathbf{n}})M_{\\Theta} h, \\:\\:\\: \\xi_{\\mathbf{n}} \\in \\mathbb{E}(\\mathbf{n}), h \\in \\mathcal{W}_{\\mathcal{H}_{V}},$$ which is multi-analytic.\n\tWe shall always consider $\\Theta$ such that $M_{\\Theta}$ is a contraction. It is easy to check that\n\t$$M_{\\Theta}\\left(\\bigoplus_{\\mathbf{n} \\in \\mathbb{N}^k_0}h_{\\mathbf{n}} \\right)=\\sum_{\\mathbf{n} \\in \\mathbb{N}^k_0} \\wt{T}_{\\mathbf{n}}(I_{\\mathbb{E}(\\mathbf{n})} \\otimes \\Theta)\\wt{V}_{\\mathbf{n}}^*h_n \\hspace{1cm} \\mbox{for}\\: \\bigoplus_{\\mathbf{n} \\in \\mathbb{N}^k_0}h_{\\mathbf{n}} \\in \\mathcal{H}_{V}.$$\n\n\n\tThe map $\\Theta,$ as defined above, will be called (cf. \\cite[Proposition 4.4]{HV21}) \n\t\\begin{enumerate}\n\t\t\\item[$(1)$]{\\it outer} if $\\overline{M_{\\Theta}\\mathcal{H}_{V}}=\\mathcal{H}_{T}$ (equivalently, $\\Theta \\mathcal{W}_{\\mathcal{H}_{V}} $ is cyclic for  $T,$ i.e., $$\\bigvee_{\\mathbf{n} \\in \\mathbb{N}^k_0}\\wt{T}_{\\mathbf{n}}(\\mathbb{E}(\\mathbf{n}) \\otimes  \\Theta \\mathcal{W}_{\\mathcal{H}_{V}} )=\\mathcal{H}_{T}).$$ \n\t\t\\item[$(2)$] {\\it inner} if $M_{\\Theta}$ is an isometry, (equivalently, $\\Theta \\mathcal{W}_{\\mathcal{H}_{V}} $ is a wandering subspace for\\\\ $T$  and $\\Theta$ is an isometry. )\n\\end{enumerate}\nIndeed, it follows that $\\Theta$ is inner and  outer if and only if $\\Theta$ is a unitary operator from $ \\mathcal{W}_{\\mathcal{H}_{V}}$ to $\\mathcal{W}_{\\mathcal{H}_{T}}.$\n\n\\begin{definition}\n\tLet $(\\sigma, V^{(1)}, \\dots, V^{(k)})$  be a pure isometric  representation of  $\\mathbb{E}$ on $\\mathcal{H}_{V}$  and $\\mathcal{K}$ be a $\\sigma$-invariant subspace (IS) of $\\mathcal{H}_{V}$, i.e., $\\sigma (b)\\mathcal{ K}\\subseteq\\mathcal{ K}$ for each $b \\in \\mathcal{B}.$ Then  $\\mathcal{K}$ is said to be  {\\rm quotient subspace} (QS) of $\\mathcal{H}_{V}$ if for each $ i\\in I_{k}$, $ \n\\widetilde{V}^{(i)*}\\mathcal{K}\\subseteq E_{i}\\ot \\mathcal{K}.\n$\\end{definition}\n\\begin{remark}\n\t Note that $\\mathcal{K}$ is $(\\sigma, V^{(1)}, \\dots, V^{(k)})$-IS  of $\\mathcal{H}_{V}$  if and only if  $\\mathcal{K}^{\\perp}$  is a QS of $\\mathcal{H}_{V}. $\n\\end{remark}\nIn particular, for $k=1,$ recall that  $(\\sigma , V)$ is said to be {\\it completely contractive}  if $I_{\\mathcal{H}_{V}}-\\widetilde{V}\\widetilde{V}^{*}\\geq 0 $ and {\\it pure (analytic)} if $\\{\\widetilde{V}_{m}^{*}\\}\\to 0$ as $m \\to \\infty,$ with respect to strong operator topology. Now according to Beurling's theorem \\cite{NFS}, $\\mathcal{L}$ is the image of an inner function if $\\mathcal{L}$ is a closed invariant subspace for the shift $M_z$ of the Hardy space $H^2(\\mathbb{D})$. The following  Beurling-Lax-Halmos type theorem \\cite[Theorem 4.5]{HV21} provides a characterization of  QS $\\mathcal{ K}$ of $\\mathcal{H} _V$  which is a generalization  of Popescu's version of Beurling-Lax theorem \\cite[Theorem 2.2]{G89}. \n\\begin{theorem}\\label{hs} Let $(\\sigma, V)$ be a pure isometric representation of  ${E}$ on $\\mathcal{H}_{V}$ and let $\\mathcal{K}$ be a  closed subspace of $\\mathcal{ H}_{V}$. Then $\\mathcal{K}$ is a QS of $\\mathcal{ H}_{V}$ if and only if there exist a Hilbert space $\\mathcal{W},$ a representation $\\sigma_1$ of $\\mathcal{B}$ on $\\mathcal{W}$ and an isometric multi-analytic operator $M_\\Theta: {\\mathcal{F}(E) \\otimes_{\\sigma_1} \\mathcal{W}} \\to \\mathcal{H}_{V}$ such that $$\\mathcal{K}=\\mathcal{H}_{V}\\ominus M_{\\Theta}(\\mathcal{F}(E) \\otimes_{\\sigma_1} \\mathcal{W}),$$ where  $\\mathcal{W}_{\\mathcal{F}(E) \\otimes \\mathcal{W}}$ is the generating wandering subspace for the induced representation $(\\rho,S)$ of $E$ on $\\mathcal{F}(E) \\otimes \\mathcal{W}_{\\mathcal{ K}}$  induced by $\\sigma|_{\\mathcal{W}_{\\mathcal{ K}}}$, where $\\mathcal{W}_{\\mathcal{ K}}$ is wandering subspace for $(\\sigma, V)|_{\\mathcal{K}}.$ \n\\end{theorem}\n For example if $\\mathcal{ K}$ is a QS of $\\mathcal{H}_{V}$ then the covariant representation  $(\\sigma|_{\\mathcal{ K}},V|_{\\mathcal{ K}}),$  is a pure completely contractive covariant representation of $E$ on $\\mathcal{K}.$ \n\n\n\n\nGenerally a QS  $\\mathcal{K}$ of $\\mathcal{H}_{V}$, where $(\\sigma, V^{(1)}, \\dots, V^{(k)})$ is a  pure DCI-representation, is not necessarily  have the Beurling-type representation \\cite{D89,WR,YR19}. In this article we give a description of QS  $\\mathcal{K}$ of $\\mathcal{H}_{V}$ based on \\cite{BDDS}. For a pure DCI-representation $(\\sigma, V^{(1)}, \\dots, V^{(k)})$ of  $\\mathbb{E}$ on a Hilbert space $\\mathcal{H}_{V},$ a  $\\sigma$-IS $\\mathcal{K}$  is said to be {\\it Beurling quotient subspace} (BQS) of $\\mathcal{H}_{V}$ if there exist a Hilbert space $\\mathcal{H}_W,$ a pure DCI-representation  $(\\pi, W^{(1)}, \\dots, W^{(k)})$ of $\\mathbb{E}$ on  $\\mathcal{H}_W$ and an isometric multi-analytic operator $M_\\Theta:{\\mathcal{H}_W} \\to \\mathcal{H}_{V}$ such that\n\\begin{equation*}\n\\mathcal{K}=\\mathcal{H}_{V}\\ominus M_{\\Theta}\\mathcal{H}_W,\n\\end{equation*} where $\\mathcal{W}_{\\mathcal{H}_W}$ is the generating wandering subspace for $(\\pi, W^{(1)}, \\dots, W^{(k)}).$ The following natural question arises: \n  \n\\begin{question}\\label{ques}\nWhich QS of $\\mathcal{H}_{V}$ admits Beurling representation ?\n\\end{question} \n To answer this question, consider a  pure DCI-representation  $(\\sigma, V^{(1)}, \\dots, V^{(k)})$ of  $\\mathbb{E}$ on  $\\mathcal{H}_{V}$ and let  $\\mathcal{K}$ be a QS of $\\mathcal{H}_{V}.$ Define  $$\\mu(a):=P_{\\mathcal{ K}}\\sigma(a){P_{\\mathcal{K}}}  \\:\\mbox{and} \\:\\:T^{(i)}(\\xi_{i}):=P_{\\mathcal{K}}V^{(i)}(\\xi_{i}){P_{\\mathcal{K}}},$$\nfor $\\xi_{i} \\in E_{i}$ and $a \\in \\mathcal{B},$ i.e.,\n$\\wT^{(i)}:=P_{\\mathcal{K}}\\widetilde{V}^{(i)}(I_{E_{i}}\\ot P_{\\mathcal{K}}),$ where $P_{\\mathcal{K}}$ is orthogonal projection of $\\mathcal{H}_{V}$ on $\\mathcal{ K}$  and  $i\\in I_{k}$. Then $(\\mu,T^{(1)},T^{(2)}, \\dots, T^{(k)}) $ is a contractive covariant representation of $\\mathbb{E}$ on $\\mathcal{K}.$ The following theorem  answers  Question \\ref{ques} which is the main result of Section \\ref{2}.\n\\begin{theorem}\\label{Beurling}\n\tLet $(\\sigma, V^{(1)}, \\dots, V^{(k)})$  be a pure DCI-representation of  $\\mathbb{E}$ on  $\\mathcal{H}_{V}.$ Let $\\mathcal{K}$ be a QS of $\\mathcal{H}_{V}$. Then $\\mathcal{K}$ is a BQS of $\\mathcal{H}_{V}$ if and only if \\begin{equation*}\n\t(I_{E_{j}}\\ot ( (I_{E_{i}}\\ot P_{\\mathcal{K}})- \\wT^{(i) *}\\wT^{(i)}))(t_{i,j} \\ot I_{\\mathcal{K}})(I_{E_{i}}\\ot ( (I_{E_{j}}\\ot P_{\\mathcal{K}})- \\wT^{(j) *}\\wT^{(j)}))=0.\n\t\\end{equation*}\n\\end{theorem}\n\n\nFor $k=2,$  $(\\sigma, V^{(1)}, V^{(2)})$   satisfies {\\it Brehmer-Solel condition} if \\begin{equation*}\n\\Delta_{*}(V):=I_{\\mathcal{H}}-\\widetilde{V}^{(1)}\\widetilde{V}^{(1)*}-\\widetilde{V}^{(2)}\\widetilde{V}^{(2)*}+\\widetilde{V}^{(1)}(I_{E_1} \\ot \\widetilde{V}^{(2)}\\widetilde{V}^{(2)*})\\widetilde{V}^{(1)*}\\geq 0.\n\\end{equation*}\nIn Section \\ref{3}, first we  give a geometrical proof of the regular dilation theorem for pure  completely contractive covariant representation $(\\sigma, V^{(1)}, V^{(2)},\\dots, V^{(k)})$ of $\\mathbb{E}$ on $\\mathcal{H}_{V}$ satisfying Brehmer-Solel condition.  Using the dilation theorem \\cite{MV93,CV95,S008,S009}  there exists a QS $\\mathcal{ K}$ of $\\mathcal{F}(\\mathbb{E}) \\ot \\mathcal{ H}$ such that   \\begin{equation}\\label{imp}\n(\\sigma, V^{(1)}, V^{(2)})\\equiv(\\rho|_{\\mathcal{K}}, S^{(1)}|_{\\mathcal{K}}, S^{(2)}|_{\\mathcal{K}}),\n\\end{equation}  where $(\\rho, S^{(1)}, S^{(2)})$ is the induced  representation of $\\mathbb{E}$ on $\\mathcal{F}(\\mathbb{E}) \\ot \\mathcal{ H}.$ But it is not necessary that $\\mathcal{K}$ obtained in this way is a BQS of $\\mathcal{F}(\\be) \\ot \\mathcal{H}.$ This motivates  Section \\ref{3}.\n\n\n\n\n\nConsider a  pure DCI-representation $(\\sigma, V^{(1)}, \\dots, V^{(k)})$ and the induced representation \\\\ $(\\rho, S^{(1)}, S^{(2)}, \\dots, S^{(k)})$  of  $\\mathbb{E}$ on the Hilbert spaces $\\mathcal{H}_{V}$ and $\\mathcal{F}(\\mathbb{E})\\otimes \\mathcal{H}$ respectively, where $\\mathcal{H}$ is a Hilbert space. Consider $M_{\\Theta}: \\mathcal{ H}_{V} \\to\\mathcal{F}(\\mathbb{E})\\ot \\mathcal{ H}   $ be  an isometric multi-analytic operator. Let us denote $\\mathcal{ K}_{\\Theta}= (M_{\\Theta}(\\mathcal{ H}_{V}))^{\\perp}$ and  $\\mathcal{S}_{\\Theta}= M_{\\Theta}(\\mathcal{ H}_{V})$  be BQS and BS, respectively. For each $i \\in I_k$, define  $$\\pi_{\\Theta}(a)=P_{\\mathcal{K}_{\\Theta}}\\rho(a)|_{{\\mathcal{K}_{\\Theta}}}  \\:\\mbox{and} \\:\\:W_{\\Theta}^{(i)}(\\xi_{i})=P_{\\mathcal{K}_{\\Theta}}S^{(i)}(\\xi_{i})|_{{\\mathcal{K}_{\\Theta}}},$$\nfor $\\xi_{i} \\in E_{i}$ and $a \\in \\mathcal{B}.$ Then  $W_{\\Theta }=(\\pi_{\\Theta}, W_{\\Theta}^{(1)}, W_{\\Theta}^{(2)}, \\dots, W_{\\Theta}^{(k)} )$ is a completely contractive covariant representation of $\\mathbb{E}$ on $\\mathcal{K}_{\\Theta}.$\nThe following theorem gives a necessary and sufficient condition when QS become BQS : \\begin{theorem}\n\tLet $(\\sigma, V^{(1)}, \\dots, V^{(k)})$  be a  completely contractive covariant representation of  $\\mathbb{E}$ on a Hilbert space $\\mathcal{H}_{V}.$ Then the following are equivalent:\\begin{enumerate}\n\t\t\\item $(\\sigma, V^{(1)}, \\dots, V^{(k)})\\equiv (\\pi_{\\Theta}, W_{\\Theta}^{(1)}, W_{\\Theta}^{(2)}, \\dots, W_{\\Theta}^{(k)} )$ for some BQS $\\mathcal{K}_{\\Theta}.$\n\t\t\\item  $(\\sigma, V^{(1)}, \\dots, V^{(k)})$ is pure and satisfies the Brehmer-Solel condition and \\begin{equation*}\n\t\t(I_{E_{j}}\\ot ( I_{E_{i}\\ot P_{\\mathcal{K}_{\\Theta}}}- \\wV^{(i) *}\\wV^{(i)}))(t_{i,j} \\ot I_{\\mathcal{K}_{\\Theta}})(I_{E_{i}}\\ot  (I_{E_{j}\\ot P_{\\mathcal{K}_{\\Theta}}}- \\wV^{(j) *}\\wV^{(j)}))=0,\n\t\t\\end{equation*} for distinct $i\\neq j.$ \n\t\\end{enumerate}\n\\end{theorem}\nSz.Nagy-Foias \\cite[Chapter VII, Theorem 2.3]{NFS} provides a necessary and sufficient for a nontrivial invariant subspace of the completely non-unitary contraction $V$ on a (separable) Hilbert space, if the characteristic function of $V$ admit a nontrivial factorization. Bercovici \\cite[Chapter 5, Proposition 1.21]{B88} extends this result and gave a description of the invariant subspaces of a functional model. Section \\ref{4} presents a characterization of a invariant subspaces of compression of a pure isometric covariant representation which is a generalization of \\cite[Chapter VII, Theorem 2.3]{NFS}  and \\cite[Chapter 5, Proposition 1.21]{B88}. \\begin{theorem}\n\tLet $(\\sigma, V)$ and $(\\mu, T)$ be pure isometric  representations of $E$ on Hilbert spaces $\\mathcal{H}_{V}$ and $\\mathcal{H}_{T},$ respectively.  Suppose ${\\mathcal{W}_{\\mathcal{H}_{V}}}$  and ${\\mathcal{W}_{\\mathcal{H}_{T}}}$ are the generating wandering subspace for $(\\sigma, V)$ and $(\\mu,T),$ respectively. Let ${M}_\\Theta:\\mathcal{H}_{V}\\to \\mathcal{H}_{T}$ be an isometric multi-analytic operator. Then  the covariant representation $(\\pi'_{\\Theta}, W'_{\\Theta})$ has an IS   if and only if there exist a Hilbert space $\\mathcal{H}_{R},$ a pure isometric  representation $(\\nu, R)$ of $E$ on $\\mathcal{H}_{R}$, and isometric multi-analytic operators ${M}_\\Phi: {\\mathcal{H}_{R}} \\to {\\mathcal{H}_{T}}$, ${M}_\\Psi: {\\mathcal{H}_{V}} \\to {\\mathcal{H}_{R}}$ such that  \\begin{equation*}\n\t{\\Theta}={\\Phi}{\\Psi}.\n\t\\end{equation*}\n\\end{theorem} More generally if $(\\sigma, V^{(1)}, \\dots, V^{(k)})$ and $(\\mu, T^{(1)}, \\dots, T^{(k)})$  be a DCI-representations of  $\\mathbb{E}$ on the Hilbert space $\\mathcal{H}_{V}$ and $\\mathcal{H}_{T},$ respectively and let ${M}_\\Theta:\\mathcal{H}_{V}\\to \\mathcal{H}_{T}$ be an isometric multi-analytic operator, then we can factorize $\\Theta$  in terms of inner operators.\n\n Moreover if we take $k=2,$ i.e., $(\\pi'_{\\Theta}, W_{\\Theta}^{'(1)},W_{\\Theta}^{'(2)})$-joint IS alone does not guarantee the factorization of an inner operator and this motivates Section \\ref{4}. Note that any reducing subspace for the covariant representation $(\\sigma, V^{(1)}, \\dots, V^{(k)})$  is doubly commuting subspace. However the converse need not be true in general. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Beurling Quotient Subspace For The Covariant Representations}\\label{2}\n\n\nIn this section, we prove a necessary and sufficient condition regarding when a QS becomes  BQS for DCI-representations of a product system. For this purpose we need some fundamental lemmas regarding BQS in terms of cross commutators. First we recall a Beurling-type characterization for  the doubly commuting invariant subspaces for DCI-representations.\n\\begin{thm}\\cite[Theorem 4.12]{HV21}\\label{MT5}\n\tLet $(\\sigma, V^{(1)}, \\dots, V^{(k)})$  be a pure DCI-representation of  $\\mathbb{E}$ on the Hilbert space  $\\mathcal{H}_{V}.$  Let $\\mathcal{K}$ be a  $(\\sigma, V^{(1)}, \\dots, V^{(k)})$-IS of $\\mathcal{H}_{V}$. Then  $\\mathcal{K}$ is a $(\\sigma, V^{(1)}, \\dots, V^{(k)})$-{\\rm doubly commuting subspace (DCS)}, that is,\n\t$${\\wt{V}|_{\\mathcal{K}}^{(i)}}^{*}{\\wt{V}|_{\\mathcal{K}}^{(j)}}=(I_{E_i} \\otimes {\\wt{V}|_{\\mathcal{K}}^{(j)}})(t_{j,i} \\otimes I_{\\mathcal{K}})(I_{E_j} \\otimes{\\wt{V}|_{\\mathcal{K}}^{(i)}}^{*})$$ where $i,j \\in I_k$ with $ i \\neq j,$   if and only if there exist a Hilbert space $\\mathcal{H}_{T}$, a pure DCI-representation $(\\mu, T^{(1)}, \\dots, T^{(k)})$ of $\\mathbb{E}$ on  $\\mathcal{H}_T$ and an isometric multi-analytic operator $M_\\Theta:{\\mathcal{H}_T} \\to \\mathcal{H}_{V}$ such that\n\t$$\\mathcal{K}=M_{\\Theta}\\mathcal{H}_T.$$ $\\mathcal{ K}$ is also called as {\\it Beurling subspace}. Indeed if $\\mathcal{K}=\\mathcal{H}_{V},$ then $\\Theta$ is outer.\n\\end{thm}\n\n \n\\begin{remark}\\label{1} Note that if $\\mathcal{K}$ is invariant subspace of $\\mathcal{ H}_{V},$ then for each $ i \\in I_{k},$ we have ${\\wt{V}|_{\\mathcal{K}}^{(i)}}^{*}=P_{E_{i}\\ot \\mathcal{ K}}{\\wt{V}^{(i)*}}.$ \tFurthermore if $\\mathcal{K}$ is a $(\\sigma, V^{(1)}, \\dots, V^{(k)})$-IS of $\\mathcal{ H}_{V},$ then ${\\wt{V}|_{\\mathcal{K}}^{(i)}}=P_{\\mathcal{ K}}{\\wt{V}|_{\\mathcal{K}}^{(i)}}$ and if $\\mathcal{K}$ is a $(\\sigma, V^{(1)}, \\dots, V^{(k)})$-co-invariant subspace of $\\mathcal{ H}_{V},$ then ${\\wt{V}|_{\\mathcal{K}^\\perp}^{(i)}}^{*}=P_{{E_{j}\\ot \\mathcal{K}^{\\perp}}}\\widetilde{V}^{(i)*}|_{\\mathcal{ K}^{\\perp}}.$\n\\end{remark}\n Consider a completely bounded representation $(\\sigma, V^{(1)}, \\dots, V^{(k)})$    of  $\\mathbb{E}$ on  $\\mathcal{H}_{V}$ and let $\\mathcal{K}$ be a subspace of $\\mathcal{ H}_{V}.$  \nFor each $i,j \\in I_{k},$  we denote the cross commutator of ${\\wt{V}|_{\\mathcal{K}}^{(j)}}^{*}$ and ${\\wt{V}|_{\\mathcal{K}}^{(i)}}$ by $[{\\wt{V}|_{\\mathcal{K}}^{(j)}}^{*}, {\\wt{V}|_{\\mathcal{K}}^{(i)}}]$ and  define it by \\begin{align*}\n&[{\\wt{V}|_{\\mathcal{K}}^{(j)}}^{*}, {\\wt{V}|_{\\mathcal{K}}^{(i)}}]:=\t{\\wt{V}|_{\\mathcal{K}}^{(j)}}^{*}{\\wt{V}|_{\\mathcal{K}}^{(i)}} - (I_{E_{j}} \\ot {\\wt{V}|_{\\mathcal{K}}^{(i)}})(t_{i,j} \\ot I_{\\mathcal{K}})(I_{E_{i}}\\ot {\\wt{V}|_{\\mathcal{K}}^{(j)}}^{*}),\n\\end{align*} where ${\\wt{V}|_{\\mathcal{K}}^{(i)}}:{E_{i}\\ot \\mathcal{ K}} \\to \\mathcal{ H}_{V}$ for each $ i \\in I_{k}.$\n\\begin{remark}\\label{11}\n\tIt follows from Theorem \\ref{MT5}  that  $\\mathcal{K}^{\\perp}$ is a  $(\\sigma, V^{(1)}, \\dots, V^{(k)})$-DCS of $\\mathcal{H}_{V}$ if and only if \\begin{align*}\n\t[\\widetilde{R}^{({j})*}, \\widetilde{R}^{(i)}]=0,\\:\\: \\mbox{for all}\\:\\: i\\neq j,\\:\\: i,j \\in I_{k}\n\t\\end{align*} where $\\widetilde{R}^{(i)}:={\\wt{V}|_{\\mathcal{K}^{\\perp}}^{(i)}} .$ In this case, we say that $\\mathcal{ K}$ is a BQS of $\\mathcal{H}_{V}.$ This gives the following lemma.\n\\end{remark}\n\n\\begin{lemma}\\label{Lemma 2.1}\tLet $(\\sigma, V^{(1)}, \\dots, V^{(k)})$  be a pure DCI-representation of  $\\mathbb{E}$ on  $\\mathcal{H}_{V}$ and let $\\mathcal{K}$ be a QS of $\\mathcal{H}_{V}.$ For each $i, j \\in I_{k}$, define \n \\begin{equation*}\nX_{i,j}:=(I_{E_{j}} \\ot P_{ \\mathcal{K}^{\\perp}} \\widetilde{V}^{(i)})(t_{i,j} \\ot P_{\\mathcal{K}})(I_{E_{i}}\\ot \\widetilde{V}^{(j)*} P_{\\mathcal{K}^\\perp}).\n\\end{equation*}\nThen $\\mathcal{K}$ is a BQS of $\\mathcal{H}_{V}$ if and only if $X_{i,j}=0$ for  $i \\neq j$.\n\\end{lemma}\t\n \\begin{proof}\n \tFor distinct $i, j\\in I_{k},$ using the DCS condition and Remark \\ref{1}\n  we obtain  \\begin{align*}[\\widetilde{R}^{({j})*}, \\widetilde{R}^{(i)}]&=[{\\wt{V}|_{\\mathcal{K}^{\\perp}}^{(j)}}^{*}, {\\wt{V}|_{\\mathcal{K}^{\\perp}}^{(i)}}]=\t{\\wt{V}|_{\\mathcal{K}^{\\perp}}^{(j)}}^{*}{\\wt{V}|_{\\mathcal{K}^{\\perp}}^{(i)}} - (I_{E_{j}} \\ot {\\wt{V}|_{\\mathcal{K}^{\\perp}}^{(i)}})(t_{i,j} \\ot I_{\\mathcal{K}^{\\perp}})(I_{E_{i}}\\ot {\\wt{V}|_{\\mathcal{K}^{\\perp}}^{(j)}}^{*})\\\\&=P_{{E_{j}\\ot \\mathcal{K}^{\\perp}}}\\widetilde{V}^{(j)*}{\\wt{V}|_{\\mathcal{K}^{\\perp}}^{(i)}}-P_{{E_{j}\\ot \\mathcal{K}^{\\perp}}}(I_{E_{j}} \\ot {\\wt{V}^{(i)}})(t_{i,j} \\ot P_{\\mathcal{K}^{\\perp}})(I_{E_{i}}\\ot {\\wt{V}|_{\\mathcal{K}^{\\perp}}^{(j)}}^{*})\\\\&=P_{{E_{j}\\ot \\mathcal{K}^{\\perp}}}(I_{E_{j}} \\ot {\\wt{V}^{(i)}})(t_{i,j} \\ot P_{\\mathcal{K}})(I_{E_{i}}\\ot {\\wt{V}|_{\\mathcal{K}^{\\perp}}^{(j)}}^{*}),\n \t\\end{align*}  where $P_{\\mathcal{K}^{\\perp}}$ is an orthogonal projection of $\\mathcal{H}_{V}$ on $\\mathcal{K}^{\\perp}$ and $P_{E_{j} \\ot \\mathcal{K}^{\\perp}}$ is an orthogonal projection of $E_{j} \\ot \\mathcal{H}_{V}$ on $E_{j} \\ot \\mathcal{K}^{\\perp}.$  Hence Remark \\ref{11} gives that $\\mathcal{ K}^{\\perp}$ is a DCS of $\\mathcal{ H}_{V}$  if and only if $${\\big((I_{E_{j}} \\ot P_{ \\mathcal{K}^{\\perp}})(I_{E_{j}}\\ot \\widetilde{V}^{(i)})(t_{i,j} \\ot P_{\\mathcal{K}})(I_{E_{i}}\\ot \\widetilde{V}^{(j)*})\\big)}|_{E_{i} \\ot \\mathcal{K}^{\\perp}}=0,$$ which means  $X_{i,j}=0.$\\end{proof}\n\n\n\n  Consider a DCI-representation $(\\sigma, V^{(1)}, \\dots, V^{(k)})$ of  $\\mathbb{E}$ on a Hilbert space $\\mathcal{H}_{V}$ and let $\\mathcal{K}$ be a QS of $\\mathcal{H}_{V}.$ Define  $$\\mu(a):=\\sigma(a)P_{\\mathcal{K}}  \\:\\mbox{and} \\:\\:T^{(i)}(\\xi_{i}):=P_{\\mathcal{K}}V^{(i)}(\\xi_{i})P_{\\mathcal{K}}, \\:\\:\\: \\text {for each}\\:\\: \\xi_{i} \\in E_{i},a \\in \\mathcal{B}, \\text{and}\\:\\:  i\\in I_{k}$$\n i.e.,\n $\\wT^{(i)}=P_{\\mathcal{K}}\\widetilde{V}^{(i)}(I_{E_{i}}\\ot P_{\\mathcal{K}}).$ Then $T=(\\mu,T^{(1)},T^{(2)}, \\dots, T^{(k)}) $ is a completely contractive covariant representation of $\\mathbb{E}$ on $\\mathcal{H}_{V}$ with having the commutant relation (\\ref{rep}). \n \n\n \n Throughout in this section we fix $i,j \\in I_{k}$ and assume that $i\\neq j$ and also $ \n\\mathbf{\\hat{m}}_{i}$ denote multi-indices in $\\mathbb{N}^{k}_{0}$ whose  $i^{th}$ entry is zero. The following lemma will take an important role in to prove main result of this section.\n\\begin{lemma}\\label{2.2}\n\tLet $(\\sigma, V^{(1)}, \\dots, V^{(k)})$  be a DCI-representation of  $\\mathbb{E}$ on  $\\mathcal{H}_{V}.$ Let $\\mathcal{K}$ be a QS of $\\mathcal{H}_{V}$. Then \\begin{equation*}\n\t[\\wT^{(i)}, \\wT_{\\mathbf{\\hat{m}}_{i}}^{*}]=(I_{\\mathbb{E}{(\\mathbf{\\hat{m}}_{i}})} \\ot P_{\\mathcal{K}})\\widetilde{V}_{\\mathbf{\\hat{m}}_{i}}^{*}P_{\\mathcal{K}^{\\perp}}\\widetilde{V}^{(i)}(I_{E_{i}}\\ot P_{\\mathcal{K}}).\n\t\\end{equation*}\n\\end{lemma}\n\\begin{proof}\n\tObserve that $\\widetilde{T}^{*}_{\\mathbf{m}}=\\widetilde{V}_{\\mathbf{m}}^{*}P_{\\mathcal{K}}$ and therefore $\\widetilde{T}_{\\mathbf{m}}=P_{\\mathcal{K}}\\widetilde{V}_{\\mathbf{m}}$   for all $\\mathbf{m}=(m_1,m_2,\\dots,m_k)\\in \\mathbb{N}^k_{0}.$  Substituting the values of  $\\wT_{\\mathbf{\\hat{m}}_{i}}^{*}$ and $\\wT^{(i)},$  we get\\begin{align*}\n\t&\t[\\wT^{(i)}, \\wT_{\\mathbf{\\hat{m}}_i}^{*}]\\\\&=(I_{\\mathbb{E}{(\\mathbf{\\hat{m}}_{i}})} \\ot P_{\\mathcal{K}}\\widetilde{V}^{(i)}(I_{E_{i}}\\ot P_{\\mathcal{K}}))(t_{i,\\mathbf{\\hat{m}}_{i}}\\ot I_{\\mathcal{H}_{V}})(I_{E_{i}}\\ot\\widetilde{V}_{\\mathbf{\\hat{m}}_{i}}^{*}P_{\\mathcal{K}})-\\widetilde{V}_{\\mathbf{\\hat{m}}_{i}}^{*}P_{\\mathcal{K}}\\widetilde{V}^{(i)}(I_{E_{i}}\\ot P_{\\mathcal{K}})\\\\&=(I_{\\mathbb{E}{(\\mathbf{\\hat{m}}_{i}})} \\ot P_{\\mathcal{K}} \\widetilde{V}^{(i)})(t_{i,\\mathbf{\\hat{m}}_{i}}\\ot I_{\\mathcal{H}_{V}})( I_{E_{i}}\\ot ((I_{\\mathbb{E}{(\\mathbf{\\hat{m}}_{i})}}\\ot P_{\\mathcal{K}})\\widetilde{V}_{\\mathbf{\\hat{m}}_{i}}^{*} P_{\\mathcal{K}}))-\\widetilde{V}_{\\mathbf{\\hat{m}}_{i}}^{*}P_{\\mathcal{K}}\\widetilde{V}^{(i)}(I_{E_{i}}\\ot P_{\\mathcal{K}})\\\\&=(I_{\\mathbb{E}{(\\mathbf{\\hat{m}}_{i}})} \\ot P_{\\mathcal{K}} \\widetilde{V}^{(i)})(t_{i,\\bf{\\hat{m}}_{i}}\\ot I_{\\mathcal{H}_{V}})( I_{E_{i}}\\ot \\widetilde{V}_{\\mathbf{\\hat{m}}_{i}}^{*} P_{\\mathcal{K}})-\\widetilde{V}_{\\mathbf{\\hat{m}}_{i}}^{*}P_{\\mathcal{K}}\\widetilde{V}^{(i)}(I_{E_{i}}\\ot P_{\\mathcal{K}})\\\\&=(I_{\\mathbb{E}{(\\mathbf{\\hat{m}}_{i}})} \\ot P_{\\mathcal{K}})\\widetilde{V}_{\\mathbf{\\hat{m}}_{i}}^{*} \\widetilde{V}^{(i)}( I_{E_{i}}\\ot P_{\\mathcal{K}})-(I_{\\mathbb{E}{(\\mathbf{\\hat{m}}_{i}})} \\ot P_{\\mathcal{K}})\\widetilde{V}_{\\mathbf{\\hat{m}}_{i}}^{*}P_{\\mathcal{K}}\\widetilde{V}^{(i)}(I_{E_{i}}\\ot P_{\\mathcal{K}})\\\\&=(I_{\\mathbb{E}{(\\mathbf{\\hat{m}}_{i}})} \\ot P_{\\mathcal{K}})\\widetilde{V}_{\\mathbf{\\hat{m}}_{i}}^{*}[I_{\\mathcal{H}_{V}}-P_{\\mathcal{K}}]\\widetilde{V}^{(i)}(I_{E_{i}}\\ot P_{\\mathcal{K}})\\\\&=(I_{\\mathbb{E}{(\\mathbf{\\hat{m}}_{i}})} \\ot P_{\\mathcal{K}})\\widetilde{V}_{\\mathbf{\\hat{m}}_{i}}^{*}P_{\\mathcal{K}^{\\perp}}\\widetilde{V}^{(i)}(I_{E_{i}}\\ot P_{\\mathcal{K}}),\n\t\\end{align*} which proves the lemma.\\end{proof}\n For each $i\\in I_{k},$ we have  \\begin{align}\\label{positive} 0 &\\leq (I_{E_{i}}\\ot P_{\\mathcal{K}})\\widetilde{V}^{(i) *}P_{\\mathcal{K}^{\\perp}}\\widetilde{V}^{(i)}(I_{E_{i}}\\ot P_{\\mathcal{K}}) \\nonumber\\\\&= (I_{E_{i}}\\ot P_{\\mathcal{K}})\\widetilde{V}^{(i) *}(I_{\\mathcal{H}_{V}}-P_{\\mathcal{K}})\\widetilde{V}^{(i)}(I_{E_{i}}\\ot P_{\\mathcal{K}})\\nonumber\\\\&=  (I_{E_{i}}\\ot P_{\\mathcal{K}})-\n \\wT^{(i) *}\\wT^{(i)} .\n \\end{align} Hence  $(I_{E_{i}}\\ot P_{\\mathcal{K}}-\\wT^{(i) *}\\wT^{(i)})^{\\frac{1}{2}}$ exists and let it be denoted by  $\\Delta{({T|_{\\mathcal{K}}^{(i)}})}.$  \n \n \nThe following lemma  relate $\\Delta{({T|_{\\mathcal{K}}^{(i)}})}$ to $[\\wT^{(i)}, \\wT_{\\mathbf{\\hat{m}}_{i}}^{*}].$  \n\\begin{lemma}\\label{lemma2.5}\n\tLet $\\mathbf{\\hat{m}}_{i} \\in \\mathbb{N}^k_{0}\\setminus\\{\\mathbf{0}\\}.$ Then there exist contractions $ X_{\\mathbf{\\hat{m}}_{i}}:E_{i} \\ot \\mathcal{K} \\to \\mathbb{E}(\\mathbf{\\hat{m}}_{i})\\ot \\mathcal{K}$ and $Y_{\\mathbf{\\hat{m}}_{i}}:  \\mathbb{E}(\\mathbf{\\hat{m}}_{i})\\ot \\mathcal{K} \\to E_{i} \\ot \\mathcal{K} $  such that $[\\wT^{(i)}, \\wT_{\\mathbf{\\hat{m}}_{i}}^{*}]= X_{\\mathbf{\\hat{m}}_{i}}\\Delta{({T|_{\\mathcal{K}}^{(i)}})}$ and $[\\wT_{\\mathbf{\\hat{m}}_{i}},\\wT^{(i)*}]=\\Delta{({T|_{\\mathcal{K}}^{(i)}})} Y_{\\mathbf{\\hat{m}}_{i}}.$\n\\end{lemma}\n\\begin{proof}\n\tUsing  Lemma \\ref{2.2} we get \\begin{align*}&\n\\Delta{({T|_{\\mathcal{K}}^{(i)}})}^{2}-[\\wT^{(i)}, \\wT_{\\mathbf{\\hat{m}}_{i}}^{*}]^{*}[\\wT^{(i)}, \\wT_{\\mathbf{\\hat{m}}_{i}}^{*}]\\\\&=(I_{E_{i}}\\ot P_{\\mathcal{K}})\\widetilde{V}^{(i) *}P_{\\mathcal{K}^{\\perp}}\\widetilde{V}^{(i)}(I_{E_{i}}\\ot P_{\\mathcal{K}})-(I_{E_{i}}\\ot P_{\\mathcal{K}})\\widetilde{V}^{(i)*}P_{\\mathcal{K}^{\\perp}}\\widetilde{V}_{\\mathbf{\\hat{m}}_{i}}(I_{\\mathbb{E}{(\\mathbf{\\hat{m}}_{i}})} \\ot P_{\\mathcal{K}})\\widetilde{V}_{\\mathbf{\\hat{m}}_{i}}^{*}P_{\\mathcal{K}^{\\perp}}\\widetilde{V}^{(i)}(I_{E_{i}}\\ot P_{\\mathcal{K}})\\\\&=(I_{E_{i}}\\ot P_{\\mathcal{K}})\\widetilde{V}^{(i) *}P_{\\mathcal{K}^{\\perp}}[I_{\\mathcal{H}_{V}}-\\widetilde{V}_{\\mathbf{\\hat{m}}_{i}}(I_{\\mathbb{E}{(\\mathbf{\\hat{m}}_{i})}} \\ot P_{\\mathcal{K}})\\widetilde{V}_{\\mathbf{\\hat{m}}_{i}}^{*}] P_{\\mathcal{K}^{\\perp}}\\widetilde{V}^{(i)}(I_{E_{i}}\\ot P_{\\mathcal{K}})\\\\& =((I_{E_{i}}\\ot P_{\\mathcal{K}})\\widetilde{V}^{(i) *}P_{\\mathcal{K}^{\\perp}})[I_{\\mathcal{H}_{V}}-\\widetilde{V}_{\\mathbf{\\hat{m}}_{i}}(I_{\\mathbb{E}{(\\mathbf{\\hat{m}}_{i})}} \\ot P_{\\mathcal{K}})\\widetilde{V}_{\\mathbf{\\hat{m}}_{i}}^{*}] ((I_{E_{i}}\\ot P_{\\mathcal{K}})\\widetilde{V}^{(i) *}P_{\\mathcal{K}^{\\perp}})^{*}.\n\t\\end{align*} As $\\widetilde{V}_{\\mathbf{\\hat{m}}_{i}}(I_{\\mathbb{E}{(\\mathbf{\\hat{m}}_{i}})} \\ot P_{\\mathcal{K}})$ is a contraction, $\n\tI_{\\mathcal{H}_{V}}-\\widetilde{V}_{\\mathbf{\\hat{m}}_{i}}(I_{\\mathbb{E}{(\\mathbf{\\hat{m}}_{i})}} \\ot P_{\\mathcal{K}})\\widetilde{V}_{\\mathbf{\\hat{m}}_{i}}^{*} \\geq 0,\n\t$ and hence \\begin{equation*}\n\t\\Delta{({T|_{\\mathcal{K}}^{(i)}})}^{2}-[\\wT^{(i)}, \\wT_{\\mathbf{\\hat{m}}_{i}}^{*}]^{*}[\\wT^{(i)}, \\wT_{\\mathbf{\\hat{m}}_{i}}^{*}] \\geq 0.\n\t\\end{equation*}That is $[\\wT^{(i)}, \\wT_{\\mathbf{\\hat{m}}_{i}}^{*}]^{*}[\\wT^{(i)}, \\wT_{\\mathbf{\\hat{m}}_{i}}^{*}]\\leq \t\\Delta{({T|_{\\mathcal{K}}^{(i)}})}^{2}.$ Therefore by Douglas's range inclusion theorem \\cite{D96}, there exists a contraction $ X_{\\mathbf{\\hat{m}}_{i}}:E_{i} \\ot \\mathcal{K} \\to \\mathbb{E}(\\mathbf{\\hat{m}}_{i})\\ot \\mathcal{K}$ such that $ [\\wT^{(i)}, \\wT_{\\mathbf{\\hat{m}}_{i}}^{*}]=X_{\\mathbf{\\hat{m}}_{i}}\\Delta({T|_{\\mathcal{K}}^{(i)}}).$  Since  $ [\\wT^{(i)},\\wT_{\\mathbf{\\hat{m}}_{i}}^{*}]^{*}=[\\wT_{\\mathbf{\\hat{m}}_{i}},\\wT^{(i)*}],$ which proves the second equality.\n\\end{proof}\n\\begin{lemma}\\label{lemma 2.4}\n\t\tConsider a  DCI-representation  $(\\sigma, V^{(1)}, \\dots, V^{(k)})$   of  $\\mathbb{E}$ on  $\\mathcal{H}_{V}$ and let $\\mathcal{K}$ be a QS of $\\mathcal{H}_{V}$. If \\begin{equation}\\label{hyp}\n\t\t(I_{E_{j}} \\ot \\Delta{({T|_{\\mathcal{K}}^{(i)}})}(t_{i,j} \\ot I_{\\mathcal{H}_{V}})\t(I_{E_{i}} \\ot \\Delta{({T|_{\\mathcal{K}}^{(j)}})}=0,\n\t\t\\end{equation} then, for each $\\mathbf{\\hat{m}}_{i},\\mathbf{\\hat{n}}_{j} \\in \\mathbb{N}^k_{0}\\setminus\\{\\mathbf{0}\\} ,$ \\begin{enumerate}\n\t\t\t\\item $(I_{E_j}\\ot (I_{\\mathbb{E}{(\\mathbf{\\hat{m}_i})}} \\ot P_{\\mathcal{K}}) \\widetilde{V}_{\\mathbf{\\hat{m}}_{i}}^{*})X_{i,j}(I_{E_i} \\ot \\widetilde{V}_{\\mathbf{\\hat{n}}_{j}} (I_{\\mathbb{E}{(\\mathbf{\\hat{n}}_j})} \\ot P_{\\mathcal{K}}))=0,\n\t\t\t$\n\t\t\t\\item $ (I_{E_j}\\ot (I_{E_{i}} \\ot P_{\\mathcal{K}})\\widetilde{V}^{(i)*})X_{i,j}(I_{E_i} \\ot \\widetilde{V}_{\\mathbf{\\hat{n}}_{j}} (I_{\\mathbb{E}{(\\mathbf{\\hat{n}}_j})} \\ot P_{\\mathcal{K}}))=0,$ and\n\t\t\t\\item $(I_{E_j}\\ot (I_{\\mathbb{E}{(\\mathbf{\\hat{m}_i})}} \\ot P_{\\mathcal{K}}) \\widetilde{V}_{\\mathbf{\\hat{m}}_{i}}^{*})X_{i,j} (I_{E_i} \\ot \\widetilde{V}^{(j)}( I_{E_{j}} \\ot P_{\\mathcal{K}}))=0.$\n\t\t\\end{enumerate}\n\\end{lemma}\n\\begin{proof} (1)  By  Lemma \\ref{lemma2.5}, there exist contractions $X_{\\mathbf{\\hat{m}}_{i}}$ and $Y_{\\mathbf{\\hat{n}}_{j}}$  such that $[\\wT^{(i)}, \\wT_{\\mathbf{\\hat{m}}_{i}}^{*}]= X_{\\mathbf{\\hat{m}}_{i}}\\Delta{({T|_{\\mathcal{K}}^{(i)}})}$ and $[\\wT_{\\mathbf{\\hat{n}}_{j}},\\wT^{(j)*}]= \\Delta{({T|_{\\mathcal{K}}^{(i)}})} Y_{\\mathbf{\\hat{n}}_{j}}$  and therefore  \\begin{align*}& (I_{E_j} \\ot[\\wT^{(i)}, \\wT_{\\mathbf{\\hat{m}}_{i}}^{*}] )(t_{i,j}\\ot I_{\\mathcal{H}_{V}})(I_{E_i} \\ot[\\wT^{(j)}, \\wT_{\\mathbf{\\hat{n}}_{j}}^{*}]^{*} )\\\\&=\n(I_{E_j} \\ot X_{\\mathbf{\\hat{m}}_{i}}\\Delta{({T|_{\\mathcal{K}}^{(i)}})} )(t_{i,j}\\ot I_{\\mathcal{H}_{V}})(I_{E_i} \\ot \\Delta{({T|_{\\mathcal{K}}^{(i)}})} Y_{\\mathbf{\\hat{n}}_{j}})\\\\&=0,\n\\end{align*} here the last equality follows by hypothesis (\\ref{hyp}).\n\nOn the other hand,  by using Lemma \\ref{2.2}, we obtain\n\\begin{align*}&\n(I_{E_j} \\ot[\\wT^{(i)}, \\wT_{\\mathbf{\\hat{m}}_{i}}^{*}] )(t_{i,j}\\ot I_{\\mathcal{H}_{V}})(I_{E_i} \\ot[\\wT^{(j)}, \\wT_{\\mathbf{\\hat{n}}_{j}}^{*}]^{*} )\\\\&=(I_{E_j} \\ot (I_{\\mathbb{E}{(\\mathbf{\\hat{m}}_{i}})} \\ot P_{\\mathcal{K}})\\widetilde{V}_{\\mathbf{\\hat{m}}_{i}}^{*}P_{\\mathcal{K}^{\\perp}}\\widetilde{V}^{(i)}(I_{E_{i}}\\ot P_{\\mathcal{K}}))(t_{i,j}\\ot I_{\\mathcal{H}_{V}})(I_{E_i} \\ot (I_{E_{j}}\\ot P_{\\mathcal{K}})\\widetilde{V}^{(j)*}P_{\\mathcal{K}^{\\perp}}\\widetilde{V}_{\\mathbf{\\hat{n}}_{j}}(I_{\\mathbb{E}{(\\mathbf{\\hat{n}}_{j})}} \\ot P_{\\mathcal{K}}))\\\\&=(I_{E_j}\\ot (I_{\\mathbb{E}{(\\mathbf{\\hat{m}_i})}} \\ot P_{\\mathcal{K}}) \\widetilde{V}_{\\mathbf{\\hat{m}}_{i}}^{*})X_{i,j}(I_{E_i} \\ot \\widetilde{V}_{\\mathbf{\\hat{n}}_{j}} (I_{\\mathbb{E}{(\\mathbf{\\hat{n}}_{j})}} \\ot P_{\\mathcal{K}})).\n\\end{align*} Thus we get (1).\n\n(2) Using hypothesis ({\\ref{hyp}}), we have \\begin{align*}&\n0=(I_{E_{j}} \\ot ((I_{E_{i}}\\ot P_{\\mathcal{K}})- \\wT^{(i) *}\\wT^{(i)}))(t_{i,j}\\ot I_{\\mathcal{H}_{V}})(I_{E_i} \\ot[\\wT^{(j)}, \\wT_{\\mathbf{\\hat{n}}_{j}}^{*}]^{*} )\\\\&=(I_{E_{j}} \\ot (I_{E_{i}}\\ot P_{\\mathcal{K}})\\widetilde{V}^{(i) *}P_{\\mathcal{K}^{\\perp}}\\widetilde{V}^{(i)}(I_{E_{i}}\\ot P_{\\mathcal{K}}))(t_{i,j}\\ot I_{\\mathcal{H}_{V}})(I_{E_i} \\ot (I_{E_{j}}\\ot P_{\\mathcal{K}})\\widetilde{V}^{(j)*}P_{\\mathcal{K}^{\\perp}}\\widetilde{V}_{\\mathbf{\\hat{n}}_{j}}(I_{\\mathbb{E}{(\\mathbf{\\hat{n}}_{j})}} \\ot P_{\\mathcal{K}}))\\\\&= (I_{E_j}\\ot( I_{E_{i}} \\ot P_{\\mathcal{K}}) \\widetilde{V}^{(i)*})X_{i,j}(I_{E_i} \\ot \\widetilde{V}_{\\mathbf{\\hat{n}}_{j}}(I_{\\mathbb{E}{(\\mathbf{\\hat{n}}_j})} \\ot P_{\\mathcal{K}})),\n\\end{align*} where the previous equality follows from Lemma \\ref{2.2}. Hence we proved (2).\n\n\n(3) In the similar way we can prove (3) as  we proved (2). \n\\end{proof}\n\nFor the proof of the main theorem first we show that $ \\mathcal{K}$ reduces $\\widetilde{V}^{(i)*}P_{\\mathcal{K}^{\\perp}}\\widetilde{V}^{(i)},$ for each $ i \\in I_{k}.$ For this it is enough to show that \\begin{equation}\\label{2.5}\n(I_{E_{i}}\\ot P_{\\mathcal{K}})(\\widetilde{V}^{(i)*}P_{\\mathcal{K}^{\\perp}}\\widetilde{V}^{(i)})=(\\widetilde{V}^{(i)*}P_{\\mathcal{K}^{\\perp}}\\widetilde{V}^{(i)})(I_{E_{i}}\\ot P_{\\mathcal{K}}), \\:\\:\\:  i\\in I_{k}.\n\\end{equation} \nConsider \\begin{align*}\n(I_{E_{i}}\\ot P_{\\mathcal{K}})\\widetilde{V}^{(i) *}P_{\\mathcal{K}^{\\perp}}\\widetilde{V}^{(i)}(I_{E_{i}}\\ot P_{\\mathcal{K}})&=\\widetilde{V}^{(i) *}P_{\\mathcal{K}^{\\perp}}\\widetilde{V}^{(i)}(I_{E_{i}}\\ot P_{\\mathcal{K}})-(I_{E_{i}}\\ot  P_{\\mathcal{K}^{\\perp}})\\widetilde{V}^{(i) *}P_{\\mathcal{K}^{\\perp}}\\widetilde{V}^{(i)}(I_{E_{i}}\\ot P_{\\mathcal{K}})\\\\&=\\widetilde{V}^{(i) *}P_{\\mathcal{K}^{\\perp}}\\widetilde{V}^{(i)}(I_{E_{i}}\\ot P_{\\mathcal{K}})-(I_{E_{i}}\\ot  P_{\\mathcal{K}^{\\perp}})\\widetilde{V}^{(i) *}\\widetilde{V}^{(i)}(I_{E_{i}}\\ot P_{\\mathcal{K}})\\\\&=\\widetilde{V}^{(i) *}P_{\\mathcal{K}^{\\perp}}\\widetilde{V}^{(i)}(I_{E_{i}}\\ot P_{\\mathcal{K}}),\n\\end{align*} as $(I_{E_{i}}\\ot  P_{\\mathcal{K}^{\\perp}})\\widetilde{V}^{(i) *}P_{\\mathcal{K}^{\\perp}}= (I_{E_{i}}\\ot  P_{\\mathcal{K}^{\\perp}})\\widetilde{V}^{(i) *}.$  After taking adjoint both sides, we obtain \\begin{equation*}\\label{2.4}\n(I_{E_{i}}\\ot P_{\\mathcal{K}})\\widetilde{V}^{(i) *}P_{\\mathcal{K}^{\\perp}}\\widetilde{V}^{(i)}(I_{E_{i}}\\ot P_{\\mathcal{K}})=(I_{E_{i}}\\ot P_{\\mathcal{K}})\\widetilde{V}^{(i) *}P_{\\mathcal{K}^{\\perp}}\\widetilde{V}^{(i)},\n\\end{equation*}thus $ \\mathcal{K}$ reduces $\\widetilde{V}^{(i)*}P_{\\mathcal{K}^{\\perp}}\\widetilde{V}^{(i)},$ for each $ i \\in I_{k}.$\n\\begin{theorem}\\label{Beurling}\nConsider a pure DCI-representation $(\\sigma, V^{(1)}, \\dots, V^{(k)})$   of  $\\mathbb{E}$ on $\\mathcal{H}_{V}$ and let $\\mathcal{K}$ be a   QS of $\\mathcal{H}_{V}$. Then $\\mathcal{K}$ is a  BQS of $\\mathcal{H}_{V}$ if and only if \\begin{equation*}\n\t(I_{E_{j}}\\ot ( (I_{E_{i}}\\ot P_{\\mathcal{K}})- \\wT^{(i) *}\\wT^{(i)}))(t_{i,j} \\ot I_{\\mathcal{H}_{V}})(I_{E_{i}}\\ot ( (I_{E_{j}}\\ot P_{\\mathcal{K}})- \\wT^{(j) *}\\wT^{(j)}))=0.\n\t\\end{equation*}\n\\end{theorem}\n\\begin{proof}\n\tFirst assume  $\\mathcal{K}$ to be a BQS of $\\mathcal{H}_{V}.$  Therefore by using Theorem \\ref{MT5} there exist a Hilbert space $\\mathcal{H}_W,$ a pure DCI-representation  $(\\pi, W^{(1)}, \\dots, W^{(k)})$ of $\\mathbb{E}$ on  $\\mathcal{H}_W$ and an isometric multi-analytic operator $M_\\Theta:{\\mathcal{H}_W} \\to \\mathcal{H}_{V}$ such that\n\t$$\\mathcal{K}=\\mathcal{H}_{V}\\ominus M_{\\Theta}\\mathcal{H}_W,$$\n\twhere $\\mathcal{W}_{\\mathcal{H}_W}$ is the generating wandering subspace for $(\\pi, W^{(1)}, \\dots, W^{(k)}).$  \n\tThen $P_{\\mathcal{K}^{\\perp}}=M_{\\Theta}M_{\\Theta}^{*}.$ Using Equations (\\ref{positive}) and (\\ref{2.5}) we get the following equality \\begin{equation}\\label{2.6}\n\t(I_{E_{i}}\\ot P_{\\mathcal{K}})- \\wT^{(i) *}\\wT^{(i)}=(I_{E_{i}}\\ot P_{\\mathcal{K}})\\widetilde{V}^{(i) *}P_{\\mathcal{K}^{\\perp}}\\widetilde{V}^{(i)}(I_{E_{i}}\\ot P_{\\mathcal{K}})= (\\widetilde{V}^{(i)*}P_{\\mathcal{K}^{\\perp}}\\widetilde{V}^{(i)})(I_{E_{i}}\\ot P_{\\mathcal{K}}),\n\t\\end{equation} where $ i \\in I_{k}.$ Note that  \\begin{align*}&\n(I_{E_{j}}\\ot ( (I_{E_{i}}\\ot P_{\\mathcal{K}})- \\wT^{(i) *}\\wT^{(i)}))(t_{i,j} \\ot I_{\\mathcal{H}_{V}})(I_{E_{i}}\\ot ( (I_{E_{j}}\\ot P_{\\mathcal{K}})- \\wT^{(j) *}\\wT^{(j)}))\\\\&=(I_{E_{j}}\\ot  \\widetilde{V}^{(i)*}P_{\\mathcal{K}^{\\perp}}\\widetilde{V}^{(i)}(I_{E_{i}}\\ot P_{\\mathcal{K}}))(t_{i,j} \\ot I_{\\mathcal{H}_{V}})(I_{E_{i}}\\ot \\widetilde{V}^{(j)*}P_{\\mathcal{K}^{\\perp}}\\widetilde{V}^{(j)}(I_{E_{j}}\\ot P_{\\mathcal{K}}))\\\\&=(I_{E_{j}}\\ot\\widetilde{V}^{(i)*} P_{\\mathcal{K}^{\\perp}}\\widetilde{V}^{(i)})(I_{E_{j}} \\ot I_{E_{i}}\\ot P_{\\mathcal{K}})(t_{i,j} \\ot I_{\\mathcal{H}_{V}})(I_{E_{i}}\\ot \\widetilde{V}^{(j)*}P_{\\mathcal{K}^{\\perp}}\\widetilde{V}^{(j)}(I_{E_{j}}\\ot P_{\\mathcal{K}}))\\\\& =(I_{E_{j}}\\ot\\widetilde{V}^{(i)*} P_{\\mathcal{K}^{\\perp}}\\widetilde{V}^{(i)})(t_{j,i} \\ot I_{\\mathcal{H}_{V}})(I_{E_{i}}\\ot (I_{E_{j}}\\ot P_{\\mathcal{K}})  \\widetilde{V}^{(j)*}P_{\\mathcal{K}^{\\perp}}\\widetilde{V}^{(j)}(I_{E_{j}}\\ot P_{\\mathcal{K}}))\\\\&=(I_{E_{j}}\\ot\\widetilde{V}^{(i)*} P_{\\mathcal{K}^{\\perp}}\\widetilde{V}^{(i)})(t_{j,i} \\ot I_{\\mathcal{H}_{V}})(I_{E_{i}}\\ot \\widetilde{V}^{(j)*}P_{\\mathcal{K}^{\\perp}}\\widetilde{V}^{(j)}(I_{E_{j}}\\ot P_{\\mathcal{K}})),\n\\end{align*} here the last inequality follows by (\\ref{2.5}).\nSince ${\\Theta}$ is an inner operator, $M_{\\Theta}$ is an isometric multi-analytic, i.e., $M_{\\Theta}^{*}M_{\\Theta}=I_{\\mathcal{H}_{W}} $ and \\begin{align}\\label{multi}\nM_{\\Theta}W^{(i)}(\\xi_i)h=V^{(i)}(\\xi_i)M_{\\Theta}h \\hspace{0.7cm}\\mbox{and} \\hspace{0.7cm}M_{\\Theta} \\pi(a)h= \\sigma(a)M_{\\Theta}h,\n\\end{align}\nwhere $\\xi_i \\in E_i, \\: h \\in \\mathcal{H}_{W},$ $ a \\in \\mathcal{B}$ and $i \\in I_{k}.$  Equation (\\ref{multi}) provides that \\begin{equation}\\label{2.7}\n(I_{E_{j}} \\ot M_{\\Theta}^{*})\\widetilde{V}^{(j)*}\\widetilde{V}^{(i)}(I_{E_{i}}\\ot M_{\\Theta})=\\widetilde{W}^{(j)*}\\widetilde{W}^{(i)}.\n\\end{equation}\nNow using  $P_{\\mathcal{K}^{\\perp}}=M_{\\Theta}M_{\\Theta}^{*}$, Equations (\\ref{2.6}) and (\\ref{2.7}), we obtain  \\begin{align*}&\n(I_{E_{j}}\\ot ( (I_{E_{i}}\\ot P_{\\mathcal{K}})- \\wT^{(i) *}\\wT^{(i)}))(t_{i,j} \\ot I_{\\mathcal{H}_{V}})(I_{E_{i}}\\ot ( (I_{E_{j}}\\ot P_{\\mathcal{K}})- \\wT^{(j) *}\\wT^{(j)}))\\\\&=(I_{E_{j}}\\ot\\widetilde{V}^{(i)*} P_{\\mathcal{K}^{\\perp}}\\widetilde{V}^{(i)})(t_{j,i} \\ot I_{\\mathcal{H}_{V}})(I_{E_{i}}\\ot \\widetilde{V}^{(j)*}P_{\\mathcal{K}^{\\perp}}\\widetilde{V}^{(j)}(I_{E_{j}}\\ot P_{\\mathcal{K}}))\\\\&=(I_{E_{j}}\\ot\\widetilde{V}^{(i)*} M_{\\Theta}M_{\\Theta}^{*})\\widetilde{V}^{(j)*}\\widetilde{V}^{(i)}(I_{E_{i}}\\ot M_{\\Theta}M_{\\Theta}^{*}\\widetilde{V}^{(j)}( I_{E_{j}}\\ot P_{\\mathcal{K}}))\\\\&=(I_{E_{j}}\\ot\\widetilde{V}^{(i)*} M_{\\Theta})\\widetilde{W}^{(j)*}\\widetilde{W}^{(i)}(I_{E_{i}}\\ot M_{\\Theta}^{*}\\widetilde{V}^{(j)} (I_{E_{j}}\\ot P_{\\mathcal{K})})\\\\&=(I_{E_{j}}\\ot\\widetilde{V}^{(i)*} M_{\\Theta})(I_{E_{j}} \\ot \\widetilde{W}^{(i)})(t_{j,i}\\ot I_{\\mathcal{H}_{W}})(I_{E_{i}}\\ot\\widetilde{W}^{(j)*})(I_{E_{i}}\\ot M_{\\Theta}^{*}\\widetilde{V}^{(j)} (I_{E_{j}}\\ot P_{\\mathcal{K}}))\\\\&=(I_{E_{j}}\\ot\\widetilde{V}^{(i)*})(I_{E_{j}}\\ot M_{\\Theta} \\widetilde{W}^{(i)})(t_{j,i}\\ot I_{\\mathcal{H}_{W}})(I_{E_{i}}\\ot\\widetilde{W}^{(j)*} M_{\\Theta}^{*})(I_{E_{i}}\\ot\\widetilde{V}^{(j)}(I_{E_{j}}\\ot P_{\\mathcal{K}}))\\\\&=(I_{E_{j}}\\ot I_{E_{i}}\\ot M_{\\Theta})(t_{j,i}\\ot I_{\\mathcal{H}_{W}})(I_{E_{i}}\\ot(I_{E_{j}}\\ot M_{\\Theta}^{*})(I_{E_{j}}\\ot P_{\\mathcal{K}}))\\\\&=(t_{i,j}\\ot I_{\\mathcal{H}_{W}})(I_{E_{i}}\\ot I_{E_{j}}\\ot M_{\\Theta} M_{\\Theta}^{*})(I_{E_{i}}\\ot I_{E_{j}}\\ot P_{\\mathcal{K}})=0.\n\\end{align*}\n\n\n\n\n\nFor the converse part, assume \\begin{equation*}\n(I_{E_{j}}\\ot ( (I_{E_{i}}\\ot P_{\\mathcal{K}})- \\wT^{(i) *}\\wT^{(i)}))(t_{i,j} \\ot I_{\\mathcal{ K}})(I_{E_{i}}\\ot ( (I_{E_{j}}\\ot P_{\\mathcal{K}})- \\wT^{(j) *}\\wT^{(j)}))=0.\n\\end{equation*} We have to show that  $\\mathcal{K}$ is a  BQS of $\\mathcal{H}_{V}$. For this we need to show $X_{i,j}=0$ (using Lemma \\ref{Lemma 2.1}). Define $\\mathcal{L}:=\\displaystyle\\bigvee_{\\mathbf{n} \\in \\mathbb{N}^k_0}\\widetilde{V}_{\\mathbf{n}}(\\mathbb{E}(\\mathbf{n})\\ot \\mathcal{K})$. First we observe that $\\mathcal{L}$ is a reducing subspace of $\\mathcal{H}_{V}$. Indeed, one can easily see that $\\mathcal{L}$ is a IS for $(\\sigma, V^{(1)}, \\dots, V^{(k)})$. Now for reducing, only we have to show that $\\widetilde{V}^{(i)*}\\widetilde{V}_{\\mathbf{n}}(\\mathbb{E}(\\mathbf{n})\\ot \\mathcal{K}) \\subseteq  E_{i} \\ot \\mathcal{L}.$ Consider \\begin{align*}\n\\widetilde{V}^{(i)*}\\widetilde{V}_{\\mathbf{n}}(\\mathbb{E}(\\mathbf{n})\\ot \\mathcal{K})&=\\widetilde{V}^{(i)*}\\widetilde{V}^{(i)}_{n_{i}}(I_{E_{i}^{\\ot n_i}}\\ot \\widetilde{V}_{\\mathbf{\\hat n}_{i}})(\\mathbb{E}(\\mathbf{n})\\ot \\mathcal{K})\\\\&=(I_{E_{i}} \\ot \\widetilde{V}^{(i)}_{n_{i}})(t_{i,n_{i}}\\ot I_{\\mathcal{H}_{V}})(I_{E_{i}^{\\ot n_i}}\\ot \\widetilde{V}^{(i)*})(I_{E_{i}^{\\ot n_i}}\\ot \\widetilde{V}_{\\mathbf{\\hat{n}}_{i}})(\\mathbb{E}(\\mathbf{n})\\ot \\mathcal{K}) \\\\&\\subseteq (I_{E_{i}} \\ot \\widetilde{V}^{(i)}_{n_{i}})(E_{i} \\ot E_{i}^{\\ot n_{i}}\\ot \\mathcal{K})\\subseteq  E_{i} \\ot \\mathcal{L},\n \\end{align*}  where $\\mathbf{e}_{i}$ is multi-indices with $ 1$ in the $i^{th}$ place and zero elsewhere. Hence $\\mathcal{L}$ reduces $(\\sigma, V^{(1)}, \\dots, V^{(k)}).$  Similarly $\\mathcal{L}^{\\perp}$ is also reducing subspace for $(\\sigma, V^{(1)}, \\dots, V^{(k)}).$ \n \n \n Therefore $(\\sigma, V^{(1)}, \\dots, V^{(k)})|_{\\mathcal{L}}$ and $(\\sigma, V^{(1)}, \\dots, V^{(k)})|_{\\mathcal{L}^{\\perp}}$ are  pure DCI-representations  for $i\\in I_{k}.$ Then using \\cite[Corollary 3.5]{HV21} there exist wandering subspaces $\\mathcal{W}$ of $\\mathcal{L}$ and  $\\mathcal{W}_{0}$ of $\\mathcal{L}^{\\perp}$ such that \\begin{equation*}\n\\mathcal{L}=\\bigoplus_{\\mathbf{n} \\in \\mathbb{N}^k_0}\\widetilde{V}_{\\mathbf{n}}(\\mathbb{E}(\\mathbf{n})\\ot \\mathcal{W})\\:\\:\\text{and}\\:\\: \\mathcal{L}^{\\perp}=\\bigoplus_{\\mathbf{n} \\in \\mathbb{N}^k_0}\\widetilde{V}_{\\mathbf{n}}(\\mathbb{E}(\\mathbf{n})\\ot \\mathcal{W}_{0}).\n\\end{equation*} Note that \\begin{align*}\n\\bigoplus_{\\mathbf{n} \\in \\mathbb{N}^k_0}\\widetilde{V}_{\\mathbf{n}}(\\mathbb{E}(\\mathbf{n})\\ot \\mathcal{W}_{\\mathcal{H}})=\\mathcal{H}_{V}=\\mathcal{L}\\oplus\\mathcal{L}^{\\perp}=\\bigoplus_{\\mathbf{n} \\in \\mathbb{N}^k_0}\\widetilde{V}_{\\mathbf{n}}(\\mathbb{E}(\\mathbf{n})\\ot (\\mathcal{W} \\oplus \\mathcal{W}_{0})).\n\\end{align*} Thus by uniqueness of the generating wandering subspace, we get $\\mathcal{W} \\oplus \\mathcal{W}_{0}=\\mathcal{W}_{\\mathcal{ H}}.$ Since $\\mathcal{K}\\subseteq\\mathcal{L},$ $\\mathcal{L}^{\\perp}=\\bigoplus_{\\mathbf{n} \\in \\mathbb{N}^k_{0}}\\widetilde{V}_{\\mathbf{n}}(\\mathbb{E}(\\mathbf{n})\\ot \\mathcal{W}_{0})\\subseteq \\mathcal{K}^{\\perp}.$ Hence, for each $ h \\in \\mathcal{L}^{\\perp}$ \\begin{equation*}\n(I_{E_{i}} \\ot P_{\\mathcal{K}})\\widetilde{V}^{(i)*}P_{\\mathcal{K}^{\\perp}}h=(I_{E_{i}} \\ot P_{\\mathcal{K}})\\widetilde{V}^{(i)*}h=0,\n\\end{equation*} as $ \\mathcal{L}^{\\perp}$ reduces $(\\sigma, V^{(1)}, \\dots, V^{(k)}).$  This proves that \\begin{equation*}\nX_{i,j}|_{E_{i} \\ot {\\mathcal{L}^{\\perp}}}=0.\n\\end{equation*} \n\nNow we only need to show $X_{i,j}|_{E_{i} \\ot {\\mathcal{L}}}=0,$ which is equivalent to show $X_{i,j}(I_{E_{i}} \\ot \\widetilde{V}_{\\mathbf{n}}(I_{\\mathbb{E}({\\mathbf{n}})}\\ot P_{\\mathcal{K}}))=0$\n for $ \\mathbf{n}\\in \\mathbb{N}^k_{0}.$ Suppose $\\mathbf{n}= \\mathbf0$, then $X_{i,j}(I_{E_{i}} \\ot \\widetilde{V}_{\\mathbf{n}}(I_{\\mathbb{E}({\\mathbf{n}})}\\ot P_{\\mathcal{K}}))=X_{i,j}(I_{E_{i}} \\ot P_{\\mathcal{K}})=0$ as $X_{i,j}=(I_{E_{j}} \\ot P_{ \\mathcal{K}^{\\perp}})(I_{E_{j}}\\ot \\widetilde{V}^{(i)})(t_{i,j} \\ot P_{\\mathcal{K}})(I_{E_{i}}\\ot \\widetilde{V}^{(j)*})(I_{E_{i}}\\ot P_{\\mathcal{K}^\\perp}),$ therefore we have to prove $X_{i,j}(I_{E_{i}} \\ot \\widetilde{V}_{\\mathbf{n}}(I_{\\mathbb{E}({\\mathbf{n}})}\\ot P_{\\mathcal{K}}))=0 $ only for  $ \\mathbf{n}\\in \\mathbb{N}^k_{0}\\setminus\\{\\mathbf{0}\\}.$ Furthermore for each $ h\\in \\mathcal{K}$, $ h_{0} \\in \\mathcal{L}^{\\perp},$ $\\xi_{i}\\in E_{i} $ and $\\eta_{\\mathbf{n}}\\in \\mathbb{E}(\\mathbf{n}), $ we have \\begin{align*}&\n \\langle X_{i,j}(I_{E_{i}} \\ot \\widetilde{V}_{\\mathbf{n}})(\\xi_{i}\\ot \\eta_{\\mathbf{n}}\\ot h), \\xi_{j} \\ot h_{0} \\rangle\\\\&=\\langle (I_{E_{j}} \\ot P_{ \\mathcal{K}^{\\perp}} \\widetilde{V}^{(i)})(t_{i,j} \\ot P_{\\mathcal{K}})(I_{E_{i}}\\ot \\widetilde{V}^{(j)*} P_{\\mathcal{K}^\\perp} \\widetilde{V}_{\\mathbf{n}})(\\xi_{i}\\ot \\eta_{\\mathbf{n}}\\ot h),\\xi_{j} \\ot h_{0} \\rangle\\\\&=\\langle (t_{i,j} \\ot P_{\\mathcal{K}})(I_{E_{i}}\\ot \\widetilde{V}^{(j)*})(I_{E_{i}}\\ot P_{\\mathcal{K}^\\perp})(I_{E_{i}} \\ot \\widetilde{V}_{\\mathbf{n}})(\\xi_{i}\\ot \\eta_{\\mathbf{n}}\\ot h),(I_{E_{j}}\\ot \\widetilde{V}^{(i)*}) \\xi_{j} \\ot h_{0}\\rangle=0.\n \\end{align*}  Thus $X_{i,j}(I_{E_{i}} \\ot \\widetilde{V}_{\\mathbf{n}})(E_{i} \\ot \\mathbb{E}(\\mathbf{n}) \\ot \\mathcal{K}) \\perp E_{j} \\ot \\mathcal{L}^{\\perp}.$ Therefore it suffices to show that \\begin{equation*}\nX_{i,j}(I_{E_{i}} \\ot \\widetilde{V}_{\\mathbf{n}})(E_{i} \\ot \\mathbb{E}(\\mathbf{n}) \\ot \\mathcal{K}) \\perp E_{j} \\ot \\mathcal{L},   \n \\end{equation*} which is equivalent to prove that  \\begin{equation*}\n X_{i,j}(I_{E_{i}} \\ot \\widetilde{V}_{\\mathbf{n}})(E_{i} \\ot \\mathbb{E}(\\mathbf{n}) \\ot \\mathcal{K}) \\perp E_{j} \\ot \\widetilde{V}_{\\mathbf{m}}(\\mathbb{E}(\\mathbf{m})\\ot \\mathcal{K}),\\:\\:\\mathbf{m},\\mathbf{n}\\in \\mathbb{N}^k_{0}\\setminus\\{\\mathbf{0}\\},\n \\end{equation*}  as $\\text{ran}  X_{i,j}\\subseteq E_{j} \\ot \\mathcal{ K}^{\\perp},$  the above equation is obvious  for $ \\mathbf{m}= \\mathbf0.$ For each $ h\\in \\mathcal{K}$, $\\xi_{i}\\in E_{i}$  and $\\eta_{\\mathbf{n}}\\in \\mathbb{E}(\\mathbf{n}), $ \\begin{align*}&\n \\langle X_{i,j}(I_{E_{i}} \\ot \\widetilde{V}_{\\mathbf{n}}(I_{\\mathbb{E}({\\mathbf{n}})}\\ot P_{\\mathcal{K}}))(\\xi_{i}\\ot \\eta_{\\mathbf{n}}\\ot h),  I_{E_{j}} \\ot \\widetilde{V}_{\\mathbf{m}}(I_{\\mathbb{E}({\\mathbf{m}})}\\ot P_{\\mathcal{K}})(\\xi_{j}\\ot \\eta_{\\mathbf{m}}\\ot h)\\rangle\\\\&=\\langle (I_{E_{j}} \\ot (I_{\\mathbb{E}({\\mathbf{m}})} \\ot P_{\\mathcal{K}}) \\widetilde{V}_{\\mathbf{m}}^{*}) X_{i,j}(I_{E_{i}} \\ot \\widetilde{V}_{\\mathbf{n}}(I_{\\mathbb{E}({\\mathbf{n}})}\\ot P_{\\mathcal{K}}))(\\xi_{i}\\ot \\eta_{\\mathbf{n}}\\ot h),  (\\xi_{j}\\ot \\eta_{\\mathbf{m}}\\ot h)\\rangle\n \\end{align*} and hence we only need to show that \\begin{equation}\\label{main}\n (I_{E_{j}} \\ot (I_{\\mathbb{E}({\\mathbf{m}})} \\ot P_{\\mathcal{K}}) \\widetilde{V}_{\\mathbf{m}}^{*}) X_{i,j}(I_{E_{i}} \\ot \\widetilde{V}_{\\mathbf{n}}(I_{\\mathbb{E}({\\mathbf{n}})}\\ot P_{\\mathcal{K}}))=0.\n \\end{equation} We start by showing the following equation \\begin{equation}\\label{main2}\n (I_{E_{j}}\\ot\\widetilde{V}^{(i)*})X_{i,j}(I_{E_{i}}\\ot \\widetilde{V}^{(j)})=0.\n \\end{equation}By using hypothesis and Equation (\\ref{positive}) we have \n \\begin{align*}&\n0= (I_{E_{j}}\\ot ( (I_{E_{i}}\\ot P_{\\mathcal{K}})- \\wT^{(i) *}\\wT^{(i)}))(t_{i,j} \\ot I_{\\mathcal{H}_{V}})(I_{E_{i}}\\ot ( (I_{E_{j}}\\ot P_{\\mathcal{K}})- \\wT^{(j) *}\\wT^{(j)}))\\\\&=(I_{E_{j}}\\ot  (I_{E_{i}}\\ot P_{\\mathcal{K}})\\widetilde{V}^{(i)*}P_{\\mathcal{K}^{\\perp}}\\widetilde{V}^{(i)}(I_{E_{i}}\\ot P_{\\mathcal{K}}))(t_{i,j} \\ot I_{\\mathcal{H}_{V}})(I_{E_{i}}\\ot (I_{E_{j}}\\ot P_{\\mathcal{K}})\\widetilde{V}^{(j)*}P_{\\mathcal{K}^{\\perp}}\\widetilde{V}^{(j)}(I_{E_{j}}\\ot P_{\\mathcal{K}}))\\\\&=(I_{E_{j}}\\ot  (I_{E_{i}}\\ot P_{\\mathcal{K}})\\widetilde{V}^{(i)*}P_{\\mathcal{K}^{\\perp}}\\widetilde{V}^{(i)})(t_{i,j} \\ot I_{\\mathcal{H}_{V}})(I_{E_{i}}\\ot (I_{E_{j}}\\ot P_{\\mathcal{K}})\\widetilde{V}^{(j)*}P_{\\mathcal{K}^{\\perp}}\\widetilde{V}^{(j)}(I_{E_{j}}\\ot P_{\\mathcal{K}}))\\\\&=(I_{E_{j}}\\ot  \\widetilde{V}^{(i)*}P_{\\mathcal{K}^{\\perp}}\\widetilde{V}^{(i)}(I_{E_{i}}\\ot P_{\\mathcal{K}}))(t_{i,j} \\ot I_{\\mathcal{H}_{V}})(I_{E_{i}}\\ot (I_{E_{j}}\\ot P_{\\mathcal{K}})\\widetilde{V}^{(j)*}P_{\\mathcal{K}^{\\perp}}\\widetilde{V}^{(j)})\\\\&=(I_{E_{j}}\\ot  \\widetilde{V}^{(i)*}P_{\\mathcal{K}^{\\perp}}\\widetilde{V}^{(i)})(t_{i,j} \\ot P_{\\mathcal{K}})(I_{E_{i}}\\ot \\widetilde{V}^{(j)*}P_{\\mathcal{K}^{\\perp}}\\widetilde{V}^{(j)})\\\\&= (I_{E_{j}}\\ot\\widetilde{V}^{(i)*})X_{i,j}(I_{E_{i}}\\ot \\widetilde{V}^{(j)}).\n \\end{align*} Now for  $ \\mathbf{m}, \\mathbf{n}\\in \\mathbb{N}^k_{0}\\setminus\\{\\mathbf{0}\\}$ with $m_{i},n_{j}\\neq0$ and using $\\widetilde{V}_{\\mathbf{m}}=\\widetilde{V}^{(i)}(I_{E_{i}} \\ot \\widetilde{V}_{{m_{i}\\mathbf{e}_{i}-\\mathbf{e}_{i}}})(I_{\\mathbb{E}({m_{i}\\mathbf{e}_i})}\\ot \\widetilde{V}_{\\mathbf{\\hat{m}}_{i}})$ and  $\\widetilde{V}_{\\mathbf{n}}=\\widetilde{V}^{(j)}(I_{E_{j}} \\ot \\widetilde{V}_{{n_{j}\\mathbf{e}_{j}-\\mathbf{e}_{j}}})(I_{\\mathbb{E}({n_{j}\\mathbf{e}_{j}})}\\ot \\widetilde{V}_{\\mathbf{\\hat{n}}_{j}})$ in the left hand side of the Equation (\\ref{main}), we obtain \\begin{align*}&\n(I_{E_{j}} \\ot ( I_{\\mathbb{E}({\\mathbf{m}})} \\ot P_{\\mathcal{K}}) \\widetilde{V}_{\\mathbf{m}}^{*}) X_{i,j}(I_{E_{i}} \\ot \\widetilde{V}_{\\mathbf{n}}(I_{\\mathbb{E}({\\mathbf{n}})}\\ot P_{\\mathcal{K}}))\\\\&=(I_{E_{j}} \\ot (I_{\\mathbb{E}({\\mathbf{m}})} \\ot P_{\\mathcal{K}})(I_{E({m_{i}\\mathbf{e}_i})}\\ot \\widetilde{V}_{\\mathbf{\\hat{m}}_{i}} ^{*})(I_{E_{i}} \\ot \\widetilde{V}_{{m_{i}\\mathbf{e}_{i}-\\mathbf{e}_{i}}}^{*})\\widetilde{V}^{(i)*})X_{i,j}\\\\&\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:(I_{E_{i}} \\ot\\widetilde{V}^{(j)}(I_{E_{j}} \\ot \\widetilde{V}_{{n_{j}\\mathbf{e}_{j}-\\mathbf{e}_{j}}})(I_{\\mathbb{E}({n_{j}\\mathbf{e}_{j}})}\\ot \\widetilde{V}_{\\mathbf{\\hat{n}}_{j}})(I_{\\mathbb{E}({\\mathbf{n}})}\\ot P_{\\mathcal{K}}))=0,\n \\end{align*} here the last equality follows by the Equation (\\ref{main2}). Now we will discuss the remaining cases : $ \\mathbf{m}, \\mathbf{n}\\in \\mathbb{N}^k_{0}\\setminus\\{\\mathbf{0}\\}$ where \n \\begin{enumerate}\n \t\\item $m_{i}=n_{j}=0$, \n \t\\item $m_{i}\\neq 0$ and $n_{j}=0,$ \n \t\\item $m_{i}= 0$, $n_{i}\\neq 0$. \n \\end{enumerate} \n   In first case when we substitute these values in left hand side of the Equation (\\ref{main}), we get \\begin{align*}&(I_{E_{j}} \\ot I_{\\mathbb{E}({\\mathbf{m}})} \\ot P_{\\mathcal{K}})(I_{E_{j}} \\ot \\widetilde{V}_{\\mathbf{m}}^{*}) X_{i,j}(I_{E_{i}} \\ot \\widetilde{V}_{\\mathbf{n}}(I_{\\mathbb{E}({\\mathbf{n}})}\\ot P_{\\mathcal{K}}))\\\\&=\n   (I_{E_{j}} \\ot I_{\\mathbb{E}({\\mathbf{\\hat{m}}_{i}})} \\ot P_{\\mathcal{K}})(I_{E_{j}} \\ot \\widetilde{V}_{\\mathbf{\\hat{m}}_{i}}^{*}) X_{i,j}(I_{E_{i}} \\ot \\widetilde{V}_{\\mathbf{\\hat{n}}_{j}}(I_{\\mathbb{E}({\\mathbf{\\hat{n}}_{j}})}\\ot P_{\\mathcal{K}}))=0,\n   \\end{align*} by using first part of Lemma \\ref{lemma 2.4}.\n     \n    For the (2), substitute $\\widetilde{V}_{\\mathbf{m}}=\\widetilde{V}^{(i)}(I_{E_{i}} \\ot \\widetilde{V}_{\\mathbf{m}-\\mathbf{e}_{i}})$,  $m_{i}\\neq 0$ and $n_{j}=0$ in left hand side of Equation (\\ref{main}), we get \\begin{align*}&\n   (I_{E_{j}} \\ot I_{\\mathbb{E}({\\mathbf{m}})} \\ot P_{\\mathcal{K}} (I_{E_{i}} \\ot \\widetilde{V}_{\\mathbf{m}-\\mathbf{e}_{i}}^{*})\\widetilde{V}^{(i)*}) X_{i,j}(I_{E_{i}} \\ot \\widetilde{V}_{\\mathbf{\\hat{n}}_{i}}(I_{\\mathbb{E}({\\mathbf{\\hat{n}}_{i}})}\\ot P_{\\mathcal{K}}))\\\\&=(I_{E_{j}} \\ot I_{\\mathbb{E}({\\mathbf{m}})} \\ot P_{\\mathcal{K}}(I_{E_{i}} \\ot \\widetilde{V}_{\\mathbf{m}-\\mathbf{e}_{i}}^{*}) (I_{E_{i}}\\ot P_{\\mathcal{K}})\\widetilde{V}^{(i)*}) X_{i,j}(I_{E_{i}} \\ot \\widetilde{V}_{\\mathbf{\\hat{n}}_{i}}(I_{\\mathbb{E}({\\mathbf{\\hat{n}}_{i}})}\\ot P_{\\mathcal{K}}))=0,\n    \\end{align*} by using second part of Lemma \\ref{lemma 2.4}. Using the same technique, which is used in (2) and third part of Lemma \\ref{lemma 2.4}, we can easily prove the Equation (\\ref{main})for $m_{i}= 0$ and $n_{i}\\neq 0.$ This completes the proof. Hence $\\mathcal{K}$ is a  BQS of $\\mathcal{H}_{V}.$\n\\end{proof}\n\n\\section{Geometrical approach to isometric dilation}\\label{3}\n\n\t \n\\subsection{Dilation theory for single pure contractive representation} Muhly and Solel in \\cite{MS98} gave the isometric dilation of  completely contractive covariant representation. In this subsection, we construct the isometric dilation of pure completely contractive covariant representation using the geometrical approach based on \\cite{AC14}. First we recall some basic definitions that will be needed.\n\n\nThe following definition of induced representation is a generalization of the multiplication operator $M_z\\otimes I_{\\mathcal{H}}$ on the vector valued Hardy spaces $H^2(\\mathbb D)\\otimes \\mathcal{H}$:\n\\begin{definition}\n\t\\begin{enumerate}\n\t\t\\item For $\\xi \\in E$, we define the {\\it creation operator} $T_{\\xi}$ on $\\mathcal{F}(E)$ by $$T_{\\xi}(\\eta):=\\xi \\otimes \\eta, \\:\\: \\eta \\in E^{\\otimes n}, n \\in \\mathbb N_0.$$\n\t\t\\item  Let $\\pi $ denote a representation of a $C^*$-algebra $\\mathcal{B}$ on a Hilbert space $\\mathcal{H}$. Let $E$ be a $C^*$-correspondence over $\\mathcal M.$ The isometric covariant representation $(\\rho, S)$ of $E$ on the Hilbert space $\\mathcal{F}(E)\\otimes_{\\pi}\\mathcal{H}$ defined as\n\t\t\\begin{align*}\n\t\t\\rho(a):&=\\phi_{\\infty}(a) \\otimes I_{\\mathcal{H}} \\:\\:, a \\in \\mathcal{B}&\\\\\n\t\tS(\\xi):&=T_{\\xi}\\otimes I_{\\mathcal{H}} ,\\:\\: \\xi \\in E.\n\t\t\\end{align*}\n\t\tis called an {\\it induced representation} (cf. \\cite{R74}) (induced  by $\\pi$).\n\t\\end{enumerate}\n\\end{definition}\n\n\n\\begin{definition}\n\tLet $(\\sigma, V)$ and  be the  completely contractive covariant representation of $E$ on  the Hilbert space $\\mathcal{H}_{V}$ and $(\\mu, T)$ be isometric representation of $E$ on $\\mathcal{H}_{T},.$ We say that  $(\\mu,T)$ is an isometric dilation (cf. \\cite {MS98}) of $(\\sigma, V)$ if there exists an isometry $\\Pi: \\mathcal{H}_{V} \\rightarrow \\mathcal{H}_{T}$ such that   $\\Pi \\sigma(a)=\\mu(a)\\Pi, $ and $(I_E \\otimes \\Pi)\\wt{V}^*=\\wt{T}^*\\Pi, $ for $a \\in \\mathcal{B}.$\n\\end{definition}\n\n Consider the following natural question: Which type of covariant representation $(\\sigma, V)$ of $E$ on $\\mathcal{H}_{V}$ is  unitarily equivalent to the restriction of the induced representation  $(\\rho, S)$ of $E$ on  $\\mathcal{F}(E)\\otimes_{\\pi}\\mathcal{H}$? Since $(\\rho, S)$ is completely contractive, $(\\sigma, V)$ is completely contractive representation. Therefore we need to find only an isometry $\\Pi_V:\\mathcal{H}_{V}\\to \\mathcal{F}(E)\\otimes_{\\pi}\\mathcal{H}$ such that \\begin{align} \\label{sid}\n \\Pi_V \\sigma(a)=\\rho(a)\\Pi_V\\:\\: \\mbox{and} \\:\\: (I_{E} \\ot \\Pi_V)\\widetilde{V}^{*}=\\widetilde{S}^{*}\\Pi_V, \\:\\:\\:a \\in \\mathcal{B} . \n \\end{align}  If such an isometry exists, then the required QS of  $\\mathcal{F}(E)\\ot \\mathcal{H}$ will be $\\Pi_{V}(\\mathcal{ H}_{V}).$\n \n Suppose such an isometry $\\Pi_V$ exists. Observe that  $\\Pi_{V}=\\bigoplus_{n=0}^{\\infty}\\Pi_V^{n},$ where $\\Pi_V^n:\\mathcal{H}_{V}\\to E^{\\ot n}\\ot \\mathcal{H}$ defined by $\\Pi_V^nh=P_{{E^{\\ot n}}\\ot \\mathcal{H}}\\Pi_Vh, h \\in \\mathcal{H}_{V}, n \\in \\mathbb{N}_0.$ Now using (\\ref{sid}), we obtain $\\Pi_V^{n+1}=(I_{E } \\ot \\Pi_V^n)\\widetilde{V}^{*},$ for every $n \\in \\mathbb{N}_{0}.$ Indeed \\begin{align*}\n (I_{E} \\ot \\Pi_V^n)\\widetilde{V}^{*}&=(I_{E}\\ot P_{{E^{\\ot n}}\\ot \\mathcal{H}})(I_{E}\\ot \\Pi_V)\\widetilde{V}^{*} =(I_{E}\\ot P_{{E^{\\ot n}}\\ot \\mathcal{H}})\\widetilde{S}^{*}\\Pi_V\\\\&= P_{{E^{\\ot n+1}}\\ot \\mathcal{H}}\\Pi_V=\\Pi_V^{n+1}.\n \\end{align*} Therefore, we can easily see by induction $ \\Pi_V^{n}=(I_{E^{\\ot n}}\\ot \\Pi_V^{0})\\widetilde{V}_{n}^{*}.$ Thus, it's sufficient  to find $ \\Pi_V^{0}$ only,  since the remaining $\\Pi_V^{n}$ will be determined by $ (I_{E^{\\ot n}}\\ot \\Pi_V^{0})\\widetilde{V}_{n}^{*}.$ Since $ \\Pi_V$ is an isometry, we have \\begin{align}\\label{3.1}\n \\|h\\|^{2}=\\| \\Pi_Vh\\|^{2}&=\\sum_{n =0}^{\\infty}\\| \\Pi_V^{n}h\\|^{2}=\\sum_{n =0}^{\\infty}\\|(I_{E^{\\ot n}}\\ot \\Pi_V^{0})\\widetilde{V}_{n}^{*}h\\|^{2}\n \\end{align} for all $ h \\in \\mathcal{H}_{V}.$ \n Note that \\begin{align*}\n (I_{E} \\ot \\Pi_{V})\\widetilde{V}^{*}= (I_{E} \\ot\\bigoplus_{n=0}^{\\infty}(I_{E^{\\ot n}}\\ot \\Pi_V^{0})\\widetilde{V}_{n}^{*})\\widetilde{V}^*=\\bigoplus_{n=1}^{\\infty}(I_{E^{\\ot n}}\\ot \\Pi_V^{0})\\widetilde{V}_{n}^{*},\n \\end{align*}\n and since $(I_{E} \\ot \\Pi_{V})$ is an isometry,  we get \\begin{align}\\label{3.2}\n \\|\\widetilde{V}^{*}h\\|^{2}=\\|(I_{E} \\ot \\Pi_{V})\\widetilde{V}^{*}h\\|^{2}=\\sum_{n=1}^{\\infty}\\|(I_{E^{\\ot n}}\\ot \\Pi_V^{0})\\widetilde{V}_{n}^{*}h\\|^{2}.\n \\end{align}\n \n \n Finally using Equations (\\ref{3.1}) and (\\ref{3.2}), we have $ \\|\\Pi_V^{0}h\\|^{2}=\\|h\\|^{2}-\\|\\widetilde{V}^{*}h\\|^{2}, \\:\\:h \\in \\mathcal{H}_{V}.$  Hence the simplest choice for   $\\Pi_V^{0}=\\Delta_{*}({V})$ and $\\mathcal{H}=\\mathcal{D}_{ *,V}$ where  $\\Delta_{*}(V) =(I_{\\mathcal{H}_{V}}-\\widetilde{V}\\widetilde{V}^{*})^{\\frac{1}{2}}$ and $\\mathcal{D}_{ *,V}=\\overline{\\Delta_{*}(V) \\mathcal{H}_{V}}.$  Then for $ h \\in \\mathcal{H}_{V}$,\\begin{align*}\n \\| \\Pi_Vh\\|^{2}=\\sum_{n=0}^{\\infty}\\|(I_{E^{\\ot n}}\\ot \\Pi_V^{0})\\widetilde{V}_{n}^{*}h\\|^{2}=\\sum_{n=0}^{\\infty}(\\|\\widetilde{V}_{n}h\\|^{2}-\\|\\widetilde{V}_{n+1}h\\|^{2})=\\|h\\|^{2}-\\lim_{n \\rightarrow \\infty}\\|\\widetilde{V}_{n}h\\|^{2}.\n \\end{align*} Hence $\\Pi_{V}$ is  an isometry if and only if $(\\sigma,V)$ is pure. Since  $\\mathcal{K}:=\\Pi_V \\mathcal{H}_{V}$ is IS under $\\widetilde{S}^{*}$, $\\mathcal{K}$ is a QS of $\\mathcal{F}(E)\\ot \\mathcal{H}.$  Therefore $(\\sigma, V)$ is isomorphic to  $(\\rho, S)|_\\mathcal{K},$ equivalently \\begin{equation}\\label{imp1}\n (\\sigma, V)\\equiv(\\rho|_{\\mathcal{K}}, S|_{\\mathcal{K}}).\n \\end{equation} In other words, we can say a pure, completely contractive covariant representation $(\\sigma, V)$ of $E$ on $\\mathcal{H}_{V}$ dilates to an induced representation $(\\rho, S).$ \n  The above discussion, which is based on \\cite[Theorem 3]{AC14} and \\cite[Theorem 2.22.]{V11}, gives a   geometric proof of the following Sz.Nagy-Foias type dilation.\n  \\begin{theorem}\\label{dilation}\n\tLet $(\\sigma, V)$  be a pure completely contractive covariant representation  of $E$ on $\\mathcal{H}_{V}.$ Then $(\\sigma, V)$ is an unitarily equivalent to  the  restriction of the induced representation $(\\rho, S)$ to $\\Pi_V \\mathcal{H}_{V},$ a QS of $\\mathcal{F}(E)\\otimes_{\\pi}\\mathcal{H}$, where $\\Pi_V:\\mathcal{H}_{V}\\to \\mathcal{F}(E)\\otimes_{\\pi}\\mathcal{H}$, is an isometry.\n\\end{theorem} \nNote that from Theorem \\ref{MT5} it follows that $\\mathcal{K}=\\Pi_V \\mathcal{H}_{V}$ is a BQS of $\\mathcal{F}(E)\\ot \\mathcal{H}$, therefore there exist a Hilbert space $\\mathcal{H}_{T}$, a pure DCI-representations  $(\\mu, T)$ of ${E}$ on $\\mathcal{H}_T$ and an isometric multi-analytic operator $M_\\Theta:{{\\mathcal{H}_{T}}} \\to \\mathcal{F}(E)\\ot \\mathcal{H}$ such that\n$$\\mathcal{K}=\\mathcal{F}(E)\\ot \\mathcal{H}\\ominus M_{\\Theta}\\mathcal{H}_{T}.$$ \n\n\n\\subsection{Regular Dilation}\n  The regular  isometric dilation of completely contractive representation $(\\sigma, V^{(1)}, V^{(2)},\\dots, V^{(k)})$ of the product systems $\\be$ over $\\mathbb{N}^k_0$  on the Hilbert space $\\mathcal{H}_{V}$ is discussed in \\cite{S008, S009}. In this subsection, we discuss geometrical construction of dilation in the pure case based on \\cite{AC14}. \n\n\n\nWe start with the  definition  of induced representation  which is a generalization of the multiplication operators $M_{z_i} \\otimes I_{\\mathcal{H}}$ on the vector valued Hardy space $H^2_{\\mathcal{H}}(\\mathbb{D}^k).$  \n\\begin{definition}\n\tLet $\\pi$ be a representation of $\\mathcal{B}$ on  $\\mathcal{H}.$  Define an isometric  covariant representation $(\\rho, S^{(1)},S^{(2)}, \\dots , S^{(k)})$ of $\\mathbb{E}$ on $\\mathcal{F}(\\mathbb{E})\\otimes_{\\pi} \\mathcal{H}$ (cf. \\cite{SZ08}) by  $$\\rho(a)=\\phi_{\\infty}(a) \\otimes I_{\\mathcal{H}}, \\:\\:\\mbox{and}\\:\\: S^{(i)}(\\xi_i)=V_{{\\xi}_i} \\otimes I_{\\mathcal{H}},\\: i\\in I_{k}, \\xi_i \\in E_i, a \\in \\mathcal{B},$$ where $V_{\\xi_i}$  denotes the creation operator on $\\mathcal{F}(\\mathbb{E})$ determined by $\\xi_i,$ that is, $V_{\\xi_i}(\\eta)=\\xi_i \\otimes \\eta,$ where $\\eta \\in \\mathcal{F}(\\mathbb{E})$ and  $\\phi_{\\infty}$  denotes the canonical left action of $\\mathcal{B}$ on $\\mathcal{F}(\\mathbb{E}).$ It is easy to see that the above isometric covariant representation $(\\rho,S^{(1)}, \\dots ,S^{(k)})$ is doubly commuting  and it is called {\\rm induced representation} of $\\mathbb{E}$ induced by $\\pi.$ Any covariant representation of $\\mathbb{E}$ which is isomorphic to $(\\rho,S^{(1)}, \\dots ,S^{(k)})$ is also called an {\\rm induced representation}.\n\\end{definition}\n\\begin{definition}\n\tLet $(\\sigma, V^{(1)}, V^{(2)},\\dots, V^{(k)})$  be the  completely contractive covariant representation of $\\mathbb{E}$ on  the Hilbert space $\\mathcal{H}_{V}$ and  $(\\mu, T^{(1)}, T^{(2)},\\dots T^{(k)})$  isometric representation of $\\mathbb{E}$ on  the Hilbert space $\\mathcal{H}_{T}$ . We say that    $(\\mu, T^{(1)}, T^{(2)},\\dots T^{(k)})$ is an isometric dilation (cf. \\cite{S009}) of $(\\sigma, V^{(1)}, V^{(2)},\\dots, V^{(k)})$  if there exists an isometry $\\Pi: \\mathcal{H}_{V} \\rightarrow \\mathcal{H}_{T}$ such that $\\Pi \\sigma(a)=\\mu(a)\\Pi, $ and $(I_E \\otimes \\Pi)\\wt{V}^{(i)*}=\\wt{T}^{(i)*}\\Pi, $ for $a \\in \\mathcal{B}$ and $i \\in I_{k}.$ \n\\end{definition}\nWe want to find out a covariant representation $(\\sigma, V^{(1)}, V^{(2)},\\dots, V^{(k)})$ of $\\mathbb{E}$ on the Hilbert space $\\mathcal{H}_{V} $ such that it is unitarily equivalent to the restriction of the induced representation $(\\rho,S^{(1)}, \\dots ,S^{(k)})$ to QS of $\\mathcal{F}(\\mathbb{E})\\otimes\\mathcal{H}.$  Since   $(\\rho,S^{(1)}, \\dots ,S^{(k)})$ is completely contractive, the representation $(\\sigma, V^{(1)}, V^{(2)},\\dots, V^{(k)})$ of $\\mathbb{E}$ on the Hilbert space $\\mathcal{H}_{V}$ is  completely contractive.   \nTherefore, consider a completely contractive covariant representation $(\\sigma, V^{(1)}, V^{(2)},\\dots, V^{(k)})$ of $\\mathbb{E}$ on  $\\mathcal{H}_{V}.$ We need to prove that there exists a isometry $\\Pi_V:\\mathcal{H}_{V}\\to \\mathcal{F}(\\mathbb{E})\\otimes_{\\pi}\\mathcal{H}$ such that \n\\begin{align}\\label{dilate}\n&\\Pi_V\\sigma(a)=\\rho(a)\\Pi_V \\:\\:\\:\\:\\mbox{and}\\:\\:\\:\\: (I_{E_{i}} \\ot \\Pi_V)\\widetilde{V}^{(i)*}=\\widetilde{S}^{(i)*}\\Pi_V,  \n\\end{align} where  $a \\in \\mathcal{B}$ and $ i \\in I_{k}.$ If such an isometry exists, then the needed QS of  $\\mathcal{F}(\\mathbb{E})\\ot \\mathcal{H}$ will equal its range, $\\Pi_{V}(\\mathcal{ H}_{V}).$\nSuppose that such an isometry $\\Pi_{V}$ exists. Note that $\\Pi_{V}=\\bigoplus_{\\mathbf{n}\\in\\mathbb{N}_{0}^{k}}\\Pi_V^{\\mathbf{n}},$ where $\\Pi_V^\\mathbf{n}:\\mathcal{H}_{V}\\to \\mathbb{E}({\\mathbf{n}})\\ot \\mathcal{H}$ defined by $\\Pi_V^\\mathbf{n}h=P_{{\\mathbb{E}(\\mathbf{n})}\\ot \\mathcal{H}}\\Pi_Vh, h \\in \\mathcal{H}_{V}.$ Now using Equation(\\ref{dilate}), we obtain \\begin{equation}\\label{regular}\n\\Pi_V^{\\mathbf{n}+\\mathbf{e}_{i}}=(I_{E_{i} } \\ot \\Pi_V^{\\mathbf{n}})\\widetilde{V}^{(i)* }\n\\end{equation} for every $\\mathbf{n} \\in \\mathbb{N}_{0}^{k}$ and $ i \\in I_{k}.$  Indeed \\begin{align*}\n(I_{E_{i} } \\ot \\Pi_V^{\\mathbf{n}})\\widetilde{V}^{(i)* }&=(I_{E _{i}} \\ot P_{{\\mathbb{E}(\\mathbf{n})}\\ot \\mathcal{H}})(I_{E _{i}} \\ot \\Pi_V)\\widetilde{V}^{(i)*} =(I_{E _{i}} \\ot P_{{\\mathbb{E}(\\mathbf{n})}\\ot \\mathcal{H}})\\widetilde{S}^{(i)*}\\Pi_V\\\\&=(I_{E_{i}}\\ot  P_{{\\mathbb{E}(\\mathbf{n})}\\ot \\mathcal{H}})\\Pi_V= P_{{\\mathbb{E}(\\mathbf{n}+\\mathbf{e}_{i})}\\ot \\mathcal{H}}\\Pi_V=\\Pi_V^{\\mathbf{n}+\\mathbf{e}_{i}},\n\\end{align*} where $\\mathbf{e}_{i}:=(0,0,\\dots,1,0,\\dots 0),$ i.e., $1$ at $i^{th}$ place and $0$ otherwise. More generally, for $ {m_{i}}\\in \\mathbb{N}_{0} $ we have \\begin{align*} \n\\Pi_V^{\\mathbf{n}+{m_{i}}\\mathbf{e}_{i}}=(I_{E_{i}^{\\ot m_i} } \\ot \\Pi_V^{\\mathbf{n}})\\widetilde{V}^{(i)* }_{m_i}.\n\\end{align*}\nIt follows  from the  above equality, we obtain \n\\begin{align*}\n\\Pi_V^{\\mathbf{n}}=(I_{\\mathbb{E}(\\mathbf{n})}\\ot \\Pi_V^{\\mathbf{0}})\\widetilde{V}_{\\mathbf{n}}^{*}.\n\\end{align*}\nIndeed, for $\\mathbf{n}=({n_1,n_2,\\dots,n_{k}}) \\in \\mathbb{N}_{0}^{k}$ we have\n\\begin{align*}\n\\Pi_V^{\\mathbf{n}}&=(I_{E_{1}^{\\ot {{n_{1}}}} } \\ot \\Pi_V^{\\mathbf{n}-{n_{1}}\\mathbf{e}_1})\\widetilde{V}^{(1)* }_{{n_{1}}}\\\\&=(I_{E_{1}^{\\ot{{n_1}}} } \\ot (I_{E_{2}^{\\ot {{n_{2}}}}}\\ot\\Pi_V^{\\mathbf{n}-({n_1}\\mathbf{e}_1+{n_{2}}\\mathbf{e}_2)} )\\widetilde{V}^{(2)* }_{{n_{2}}})\\widetilde{V}^{(1)* }_{{n_{1}}}\\\\&=(I_{E_{1}^{\\ot {{n_{1}}}} } \\ot (I_{E_{2}^{\\ot {{n_{2}}}}}\\ot(I_{E_{3}^{\\ot {{n_{3}}}}}\\ot \\Pi_V^{\\mathbf{n}-({n_{1}}\\mathbf{e}_1+{n_{2}}\\mathbf{e}_2+{n_{3}}\\mathbf{e}_3)}) \\widetilde{V}^{(3)* }_{{n_{3}}} )\\widetilde{V}^{(2)* }_{{n_{2}}})\\widetilde{V}^{(1)* }_{{n_{1}}}=\\cdots\\\\&=(I_{\\mathbb{E}(\\mathbf{n})}\\ot \\Pi_V^{\\mathbf{0}})\\widetilde{V}_{\\mathbf{n}}^{*}.\n\\end{align*}Thus we get $ \\Pi_V^{\\mathbf{n}}=(I_{\\mathbb{E}(\\mathbf{n})}\\ot \\Pi_V^{\\mathbf{0}})\\widetilde{V}_{\\mathbf{n}}^{*}.$ Therefore,  it is  sufficient to find  $ \\Pi_V^{\\mathbf{0}}$ only, since the remaining $\\Pi_V^{\\mathbf{n}}$ will be determined by $ (I_{\\mathbb{E}(\\mathbf{n})}\\ot \\Pi_V^{\\mathbf{0}})\\widetilde{V}_{\\mathbf{n}}^{*}.$ Since $\\Pi_V$ is an isometry, we have \\begin{align}\n\\|h\\|^{2}=\\| \\Pi_Vh\\|^{2}&=\\sum_{\\mathbf{n} \\in \\mathbb{N}_{0}^{k}}\\| \\Pi_V^{\\mathbf{n}}h\\|^{2}=\\sum_{\\mathbf{n} \\in \\mathbb{N}_{0}^{k}}\\|(I_{\\mathbb{E}(\\mathbf{n})}\\ot \\Pi_V^{\\mathbf{0}})\\widetilde{V}_{\\mathbf{n}}^{*}h\\|^{2},\n\\end{align} for all $ h \\in \\mathcal{H}_{V}.$ \n\nSuppose $u=\\{u_1,u_2,\\dots, u_r\\}\\subseteq I_{k}$ such that $u_1< u_2< \\dots < u_r.$ Define $\\mathbf{e}(u):=\\mathbf{e}_{u_1}+\\mathbf{e}_{u_2}+\\dots+\\mathbf{e}_{u_r}.$ \nNote that  \\begin{align*}\n(I_{\\mathbb{E}(\\mathbf{e}(u))} \\ot \\Pi_{V})\\widetilde{V}^{*}_{\\mathbf{e}(u)}= (I_{\\mathbb{E}(\\mathbf{e}(u))}  \\ot\\bigoplus_{\\mathbf{n} \\in \\mathbb{N}_{0}^{k}}(I_{\\mathbb{E}(\\mathbf{n})}\\ot \\Pi_V^{\\mathbf{0}})\\widetilde{V}_{\\mathbf{n}}^{*})\\widetilde{V}^{*}_{\\mathbf{e}(u)}.\n\\end{align*}\n Since $(I_{\\mathbb{E}(\\mathbf{e}(u))} \\ot \\Pi_{V})$ is an isometry, we get \\begin{align}\\label{F}\n\\|\\widetilde{V}_{\\mathbf{e}(u)}^{*}h\\|^{2}=\\|(I_{\\mathbb{E}(\\mathbf{e}(u))} \\ot \\Pi_{V})\\widetilde{V}_{\\mathbf{e}(u)}^{*}h\\|^{2}=\\sum_{\\mathbf{n} \\in \\mathbb{N}_{0}^{k}}\\|(I_{\\mathbb{E}(\\mathbf{e}(u))} \\ot(I_{\\mathbb{E}(\\mathbf{n})}\\ot \\Pi_V^{\\mathbf{0}})\\widetilde{V}_{\\mathbf{n}}^{*})\\widetilde{V}^{*}_{\\mathbf{e}(u)}h\\|^{2},\\:\\:\\:h \\in \\mathcal{H}_{V}.\n\\end{align} Denote by $|u|$ the cardinality of $u.$ It follows from Equation ({\\ref{F}}) that \\begin{align*}\n\\sum_{u\\subseteq I_{k}}(-1)^{|u|}\\|\\widetilde{V}_{\\mathbf{e}(u)}^{*}h\\|^{2}&=\\sum_{u\\subseteq I_{k}}(-1)^{|u|}\\left(\\sum_{\\mathbf{n} \\in \\mathbb{N}_{0}^{k}}\\| (I_{\\mathbb{E}(\\mathbf{e}(u))}   \\ot(I_{\\mathbb{E}(\\mathbf{n})}\\ot \\Pi_V^{\\mathbf{0}})\\widetilde{V}_{\\mathbf{n}}^{*})\\widetilde{V}^{*}_{\\mathbf{e}(u)}h\\|^{2}\\right)\\\\&=\\sum_{\\mathbf{m} \\in \\mathbb{N}^k_0}\\left({\\sum_{{\\mathbf{n}\\leq\\mathbf{m}},{\\max\\{m_j-n_j\\}\\leq 1}}}(-1)^{|\\mathbf{m}|-|\\mathbf{n}|} \\| (I_{\\mathbb{E}(\\mathbf{m})}\\ot \\Pi_V^{\\mathbf{0}})\\widetilde{V}_{\\mathbf{m}}^{*}h\\|^{2}\\right)\\\\&=\\|\\Pi_V^{\\mathbf{0}}h\\|^{2}.\n\\end{align*}\nHence \\begin{align*}\n\\sum_{u\\subseteq I_{k}}(-1)^{|u|}\\|\\widetilde{V}_{\\mathbf{e}(u)}^{*}h\\|^{2}\\geq 0,\\:\\:\\:\\:h\\in \\mathcal{H}_{V},\n\\end{align*}  equivalently \\begin{align*}\n\\sum_{u\\subseteq I_{k}}(-1)^{|u|}\\widetilde{V}_{\\mathbf{e}(u)}\\widetilde{V}_{\\mathbf{e}(u)}^{*}\\geq 0.\n\\end{align*}\nNow, define  $\\Delta_{*}(V)=(\\sum_{u\\subseteq I_{k}}(-1)^{|u|}\\widetilde{V}_{\\mathbf{e}(u)}\\widetilde{V}_{\\mathbf{e}(u)}^{*})^{\\frac{1}{2}}$ and $\\mathcal{D}_{*,V}=\\overline{Im\\Delta_*(V)}.$ Therefore we can choose   for $\\mathcal{H}$ is $\\mathcal{D}_{*,V}$ and  $\\Pi_V^{\\mathbf{0}}=\\Delta_{*}(V),$ thus  $ \\Pi_V^{\\mathbf{n}}=(I_{\\mathbb{E}(\\mathbf{n})}\\ot \\Delta_{*}(V))\\widetilde{V}_{\\mathbf{n}}^{*}.$ Then for each $ h \\in \\mathcal{H}_{V},$ we obtain \\begin{align*}\n\\| \\Pi_Vh\\|^{2}&=\\sum_{\\mathbf{n} \\in \\mathbb{N}_{0}^{k}}\\| \\Pi_V^{\\mathbf{n}}h\\|^{2}=\\lim_{p\\rightarrow \\infty}\\sum_{{\\mathbf{n} \\in \\mathbb{N}_{0}^{k}},{\\max n_j\\leq p-1}}\\|(I_{\\mathbb{E}(\\mathbf{n})}\\ot \\Pi_V^{\\mathbf{0}})\\widetilde{V}_{\\mathbf{n}}^{*}h\\|^{2}\\\\&=\\lim_{p \\rightarrow \\infty}\\sum_{{\\mathbf{n} \\in \\mathbb{N}_{0}^{k}},{\\max n_j\\leq p-1}}\\sum_{{u}\\subseteq{I_{k}}}(-1)^{|u|}\\|(I_{\\mathbb{E}(\\mathbf{n})}\\ot\\widetilde{V}^{*}_{\\mathbf{e}(u)} )\\widetilde{V}_{\\mathbf{n}}^{*}h\\|^{2}\\\\&=\\lim_{p \\rightarrow \\infty}\\sum_{{\\mathbf{m} \\in \\mathbb{N}_{0}^{k}},{\\max m_j\\leq p}}\\left( \\sum_{\\mathbf{n}\\leq \\mathbf{m},\\max \\{{m_j-n_j}\\}\\leq 1,{\\max n_j\\leq p-1}}(-1)^{|\\mathbf{m}|-|\\mathbf{n}|}\\|\\widetilde{V}_{\\mathbf{m}}^{*}h\\|^{2}\\right).\n\\end{align*} Observe that \\begin{align*}\n\\sum_{\\mathbf{n}\\leq \\mathbf{m},\\max \\{{m_j-n_j}\\}\\leq 1,{\\max n_j\\leq p-1}}(-1)^{|\\mathbf{m}|-|\\mathbf{n}|}=0,\n\\end{align*} whenever $0< m_i< p,$ for all $i \\in I_{k}.$ If $m_i=0$ or $m_i=p,$ then \\begin{align*}\n\\sum_{\\mathbf{n}\\leq \\mathbf{m},\\max \\{{m_j-n_j}\\}\\leq 1,{\\max n_j\\leq p-1}}(-1)^{|\\mathbf{m}|-|\\mathbf{n}|}=(-1)^{|{\\supp\\mathbf{m}}|}.\n\\end{align*} Therefore\n\\begin{align*}\n\\| \\Pi_Vh\\|^{2}=\\lim_{p \\rightarrow \\infty}\\sum_{u\\subseteq I_{k}}(-1)^{|u|}\\|\\widetilde{V}_{\\mathbf{e}(u)}^{*p}h\\|^{2}=\\|h\\|^{2}+\\lim_{p \\rightarrow \\infty}\\sum_{\\emptyset\\neq {u\\subseteq I_{k}}}(-1)^{|u|}\\|\\widetilde{V}_{\\mathbf{e}(u)}^{*p}h\\|^{2},\\:\\:\\:\\: h \\in \\mathcal{ H}_{V}.\n\\end{align*}\nHence $\\Pi_{V}$ will be an isometry if $SOT-\\lim_{p \\rightarrow \\infty}\\widetilde{V}^{({j}) *} _p=0$ for $j\\in I_{k}.$ \n\n\nNow we define  the following Brehmer-Solel condition.\n\\begin{definition}\n\tLet $(\\sigma, V^{(1)}, V^{(2)},\\dots, V^{(k)})$ be  a pure, completely contractive representation of $\\mathbb{E}$ on $\\mathcal{H}_{V}.$ Then  we say it is satisfy Brehmer-Solel condition (cf. \\cite{S008}) if it satisfies the following condition\\begin{align*}\n\t\\sum_{u \\subseteq I_k}(-1)^{|u|}\\widetilde{V}_{\\mathbf{e}(u)}\\widetilde{V}_{\\mathbf{e}(u)}^{*}\\geq 0.\n\t\\end{align*}\n\\end{definition} \nThe above observations yields the following theorem which is generalization of \\cite[Theorem 4]{AC14}.\n\\begin{theorem}\\label{6}\n\tLet $(\\sigma, V^{(1)}, V^{(2)},\\dots, V^{(k)})$  be a  completely contractive covariant representation of $\\mathbb{E}$ on $\\mathcal{H}_{V},$ satisfying Brehmer-Solel condition. Then there exists  $\\Pi_V:\\mathcal{H}_{V}\\to \\mathcal{F}(\\mathbb{E})\\otimes_{\\pi}\\mathcal{D}_{ *,V},$ satisfy \\begin{align*}\n\t&\\Pi_V\\sigma(a)=\\rho(a)\\Pi_V \\:\\:\\:\\:\\mbox{and}\\:\\:\\:\\: (I_{E_{i}} \\ot \\Pi_V)\\widetilde{V}^{(i)*}=\\widetilde{S}^{(i)*}\\Pi_V,  \n\t\\end{align*} where  $a \\in \\mathcal{B}$ and $ i \\in I_{k}$  such that \\begin{align*}\n\t\\| \\Pi_Vh\\|^{2}=\\lim_{p \\rightarrow \\infty}\\sum_{u\\subseteq I_{k}}(-1)^{|u|}\\|\\widetilde{V}_{\\mathbf{e}(u)}^{*p}h\\|^{2}=\\|h\\|^{2}+\\lim_{p \\rightarrow \\infty}\\sum_{\\emptyset\\neq {u\\subseteq I_{k}}}(-1)^{|u|}\\|\\widetilde{V}_{\\mathbf{e}(u)}^{*p}h\\|^{2},\\:\\:\\:\\: h \\in \\mathcal{ H}_{V}.\n\t\\end{align*}\n\\end{theorem}\nIn particular if we take a pure completely contractive covariant representation  then we obtain the following corollary which is  a generalization of \\cite[Theorem 6]{AC14}.\n\\begin{corollary}\\label{dilationN}\n\tLet $(\\sigma, V^{(1)}, V^{(2)},\\dots, V^{(k)})$  be a pure, completely contractive covariant representation of $\\mathbb{E}$ on the Hilbert space $\\mathcal{H}_{V},$ satisfying Brehmer-Solel condition. Then it is an unitarily equivalent to  the  restriction of the induced representation $(\\rho,S^{(1)}, \\dots ,S^{(k)})$ to QS $ \\Pi_V \\mathcal{H}_{V}$  of $\\mathcal{F}(\\mathbb{E})\\otimes\\mathcal{D}_{ *,V}$ where $\\Pi_V:\\mathcal{H}_{V}\\to \\mathcal{F}(\\mathbb{E})\\otimes_{\\pi}\\mathcal{D}_{ *,V}$ is an isometry. Therefore it dilates the induced representation $(\\rho, S^{(1)}, S^{(2)}, \\dots, S^{(k)}).$ \n\\end{corollary} \n\nIn particular there exists an isometry $\\Pi_V:\\mathcal{H}_{V}\\to \\mathcal{F}(\\mathbb{E})\\otimes_{\\pi}\\mathcal{H}$ satisfying Equation (\\ref{dilate}).\n Then $\\mathcal{ K}=\\Pi_{V}\\mathcal{ H}_{V}$ is a QS of $\\mathcal{F}(\\mathbb{E})\\ot \\mathcal{H},$ therefore  $(\\sigma, V^{(1)}, V^{(2)},\\dots, V^{(k)})$ is isomorphic to  $(P_{\\mathcal{ K}}\\rho, P_{\\mathcal{ K}}S^{(1)}, P_{\\mathcal{ K}}S^{(2)}, \\dots, P_{\\mathcal{ K}}S^{(k)})|_{\\mathcal{K}}.$\n \n \n \n \n \n   Note that if $k=1,$ then $\\mathcal{K}$ will be a BQS under $(\\rho, S)$ by Theorem \\ref{hs}, i.e. there exist a Hilbert space $\\mathcal{ H'}$ and a representation $\\Psi$ and an multi-analytic operator $M_{\\Theta}: \\mathcal{F}(E)\\ot \\mathcal{ H'} \\to \\mathcal{F}(E)\\ot \\mathcal{ H}  $ such that $\\mathcal{ K}_{\\Theta}= (M_{\\Theta}(\\mathcal{F}(E)\\ot_{\\Psi} \\mathcal{ H'}))^{\\perp}$\n \\begin{equation*}\n (\\sigma,V)\\equiv (P_{(M_{\\Theta}(\\mathcal{F}(E)\\ot_{\\Psi} \\mathcal{ H'}))^{\\perp}} \\rho, P_{(M_{\\Theta}(\\mathcal{F}(E)\\ot_{\\Psi} \\mathcal{ H'}))^{\\perp}}S)|_{(M_{\\Theta}(\\mathcal{F}(E)\\ot_{\\Psi} \\mathcal{ H'}))^{\\perp}}.\n \\end{equation*}  The space $\\mathcal{ K}_{\\Theta}$ is said to be {\\it model space} corresponding to $(\\sigma,V).$ Therefore each pure completely contractive covariant representation is unitarily equivalent to compression of the induced representation to the model space. \n\n\n\nLet $(\\sigma, V^{(1)}, \\dots, V^{(k)})$  be a  completely contractive covariant representation of  $\\mathbb{E}$ on  $\\mathcal{H}_{V}.$ \nNow in the $k >1$ case, consider $M_{\\Theta}:  \\mathcal{H}_{V} \\to\\mathcal{F}(\\mathbb{E})\\ot \\mathcal{ H}   $ be  an isometric multi-analytic operator. Let  $\\mathcal{ K}_{\\Theta}= (M_{\\Theta}(\\mathcal{H}_{V}))^{\\perp}$ and  $\\mathcal{S}_{\\Theta}= M_{\\Theta}(\\mathcal{H}_{V})$  which are BQS and BS, respectively. For each $i \\in I_k$, define  $$\\pi_{\\Theta}(a)=P_{\\mathcal{K}_{\\Theta}}\\rho(a)|_{{\\mathcal{K}_{\\Theta}}}  \\:\\mbox{and} \\:\\:W_{\\Theta}^{(i)}(\\xi_{i})=P_{\\mathcal{K}_{\\Theta}}S^{(i)}(\\xi_{i})|_{{\\mathcal{K}_{\\Theta}}},$$\nfor $\\xi_{i} \\in E_{i}$ and $a \\in \\mathcal{B}.$ Then  $W_{\\Theta }=(\\pi_{\\Theta}, W_{\\Theta}^{(1)}, W_{\\Theta}^{(2)}, \\dots, W_{\\Theta}^{(k)} )$ is a completely contractive covariant representation of $\\mathbb{E}$ on $\\mathcal{K}_{\\Theta}.$ Indeed, \\begin{align*}\n(\\phi^{(i)}(a)\\ot I_{\\mathcal{F}(\\mathbb{E})\\otimes \\mathcal{H}})\\widetilde{W}_{\\Theta}^{ (i)*}&=(\\phi^{(i)}(a)\\ot I_{\\mathcal{F}(\\mathbb{E})\\otimes \\mathcal{H}})(I_{E}\\ot P_{\\mathcal{K}_{\\Theta}})\\widetilde{S}^{ (i)*}P_{\\mathcal{K}_{\\Theta}}\\\\&=(I_{E}\\ot P_{\\mathcal{K}_{\\Theta}})(\\phi^{(i)}(a)\\ot I_{\\mathcal{F}(\\mathbb{E})\\otimes \\mathcal{H}})\\widetilde{S}^{ (i)*}P_{\\mathcal{K}_{\\Theta}}\\\\&=(I_{E}\\ot P_{\\mathcal{K}_{\\Theta}})\\widetilde{S}^{ (i)*}\\rho(a)P_{\\mathcal{K}_{\\Theta}}\\\\&=(I_{E}\\ot P_{\\mathcal{K}_{\\Theta}})\\widetilde{S}^{ (i)*}P_{\\mathcal{K}_{\\Theta}}\\rho(a)P_{\\mathcal{K}_{\\Theta}}=\\widetilde{W}_{ \\Theta}^{(i)*}\\pi_{\\Theta}(a),\n\\end{align*} where the last equality follows by the fact that $\\mathcal{K}_{\\Theta}$ is a QS of ${\\mathcal{F}(\\mathbb{E})\\otimes \\mathcal{H}}.$ Now using the definition of QS we have \\begin{align*}\n\\sum_{{u}\\subseteq{I_{k}}}(-1)^{|u|}\\widetilde{W}_{\\mathbf{e}(u)} \\widetilde{W}_{\\mathbf{e}(u)}^{*}=\\sum_{{u}\\subseteq{I_{k}}}(-1)^{|u|}P_{\\mathcal{K}_\\Theta}\\widetilde{S}_{\\mathbf{e}(u)}\\widetilde{S}_{\\mathbf{e}(u) }^{*}P_{\\mathcal{K}_\\Theta},\n\\end{align*} Therefore $(\\sigma, V^{(1)}, \\dots, V^{(k)})$ satisfies Brehmer-Solel condition if and only if $W_{\\Theta}$  satisfies the  Brehmer-Solel condition whenever they are equivalent. One can now ask which type of covariant representation are unitarily  equivalent to $W_{\\Theta }$ on BQS, the following theorem gives answer to  this question.\n\\begin{theorem}\nLet $(\\sigma, V^{(1)}, \\dots, V^{(k)})$  be a  completely contractive covariant representation of  $\\mathbb{E}$ on a Hilbert space $\\mathcal{H}_{V}.$ Then the following are equivalent:\\begin{enumerate}\n\t\\item $(\\sigma, V^{(1)}, \\dots, V^{(k)})\\equiv (\\pi_{\\Theta}, W_{\\Theta}^{(1)}, W_{\\Theta}^{(2)}, \\dots, W_{\\Theta}^{(k)} )$ for some BQS $\\mathcal{K}_{\\Theta}.$\n\t\\item  $(\\sigma, V^{(1)}, \\dots, V^{(k)})$ is pure and satisfies the Brehmer-Solel condition and \\begin{equation*}\n\t(I_{E_{j}}\\ot ( I_{E_{i}\\ot P_{\\mathcal{K}_{\\Theta}}}- \\wV^{(i) *}\\wV^{(i)}))(t_{i,j} \\ot I_{\\mathcal{K}_{\\Theta}})(I_{E_{i}}\\ot  (I_{E_{j}\\ot P_{\\mathcal{K}_{\\Theta}}}- \\wV^{(j) *}\\wV^{(j)}))=0,\n\t\\end{equation*} for distinct $i\\neq j.$ \n\t\\end{enumerate}\n\\end{theorem}\n\\begin{proof}\n\tThe proof  follows from Theorem \\ref{Beurling}  and Corollary \\ref{dilationN}.\n\\end{proof}\n\\section{Factorizations and invariant subspaces}\\label{4}\n \nThis section establishes a relationship between invariant subspaces of compression of a pure isometric covariant representation with factorization of the corresponding inner function.\n \n Let $(\\sigma, V)$ and $(\\mu, T)$ be pure isometric  representations of $E$ on Hilbert spaces $\\mathcal{H}_{V}$ and $\\mathcal{H}_{T},$ respectively. Let ${M}_\\Theta:\\mathcal{H}_{V}\\to \\mathcal{H}_{T}$ be an isometric multi-analytic operator.\n Let us denote  $\\mathcal{K}_{\\Theta}=\\mathcal{H}_{T}\\ominus M_{\\Theta} \\mathcal{H}_{V},$ and $ M_{\\Theta} \\mathcal{H}_{V}$  be BQS and Beurling subspace of $\\mathcal{H}_{T}$ corresponding to $\\Theta,$ respectively.\n Define  $$\\pi_{\\Theta}'(a):=P_{\\mathcal{K}_{\\Theta }}\\mu(a)|_{{\\mathcal{K}_{\\Theta}}}  \\:\\mbox{and} \\:\\:W'_{\\Theta}(\\xi)=P_{\\mathcal{K}_{\\Theta}}T(\\xi)|_{{\\mathcal{K}_{\\Theta}}},$$\n for $\\xi \\in E$ and $a \\in \\mathcal{B}.$ Then $(\\pi'_{\\Theta}, W^{\\prime}_{\\Theta})$ is a covariant representation of $E$ on $ \\mathcal{K}_{\\Theta }\\subseteq\\mathcal{H}_{T}.$\n \nThe following theorem  presents a characterization of a invariant subspaces of compression of a pure isometric covariant representation which is a generalization of  Sz.Nagy-Foias \\cite[Chapter VII, Theorem 2.3]{NFS}  and  Bercovici \\cite[Chapter 5, Proposition 1.21]{B88}.\n\\begin{theorem}\\label{Ber}\n\tLet $(\\sigma, V)$ and $(\\mu, T)$ be pure isometric  representations of $E$ on Hilbert spaces $\\mathcal{H}_{V}$ and $\\mathcal{H}_{T},$ respectively.  Suppose ${\\mathcal{W}_{\\mathcal{H}_{V}}}$  and ${\\mathcal{W}_{\\mathcal{H}_{T}}}$ are the generating wandering subspace for $(\\sigma, V)$ and $(\\mu,T),$ respectively. Let ${M}_\\Theta:\\mathcal{H}_{V}\\to \\mathcal{H}_{T}$ be an isometric multi-analytic operator. Then  the covariant representation $(\\pi'_{\\Theta}, W'_{\\Theta})$ has an IS   if and only if there exist a Hilbert space $\\mathcal{H}_{R},$ a pure isometric  representation $(\\nu, R)$ of $E$ on $\\mathcal{H}_{R}$, isometric multi-analytic operators ${M}_\\Phi: {\\mathcal{H}_{R}} \\to {\\mathcal{H}_{T}}$, ${M}_\\Psi: {\\mathcal{H}_{V}} \\to {\\mathcal{H}_{R}}$ such that  \\begin{equation*}\n\t{\\Theta}={\\Phi}{\\Psi}.\n\t\\end{equation*}\n\\end{theorem}\n\n\\begin{proof}\n\t\n\t\n(\t$\\implies$)\n \tLet $\\mathcal{S}$ be a nontrivial $(\\pi'_{\\Theta}, W'_{\\Theta})$-IS of $\\mathcal{K}_{\\Theta} \\subseteq \\mathcal{H}_{T},$ then $\\mathcal{S}\\oplus M_\\Theta \\mathcal{H}_{V}$ is a $(\\mu, T)$-IS of $\\mathcal{H}_{T}.$ Since $(\\mu, T)$ is pure,  using Theorem \\ref{MT5}, there exist a Hilbert space $\\mathcal{H}_{R}$, a pure isometric  representation $(\\nu, R) $ of $E$ on $\\mathcal{H}_{R}$ and an isometric multi-analytic operator $M_\\Phi: {\\mathcal{H}_{R}} \\to {\\mathcal{H}_{T}}$ such that \\begin{equation*}\\label{quotient}\n\\mathcal{S}\\oplus M_\\Theta \\mathcal{H}_{V}=M_{\\Phi}{\\mathcal{H}_{R}},\n\t\\end{equation*} where $ \\mathcal{W}_{\\mathcal{H}_{R}}$ is the  generating wandering subspace for $(\\nu,R).$ Note that $M_{\\Theta}\\mathcal{H}_{V}\\subseteq M_{\\Phi}\\mathcal{H}_{R}$ and $\\mathcal{S}=M_{\\Phi}\\mathcal{H}_{R}\\ominus M_{\\Theta}\\mathcal{H}_{V}.$   Then by Douglas's range  inclusion theorem \\cite{D96}, there exists a contraction $Z: \\mathcal{H}_{V} \\to \\mathcal{H}_{R}$ such that \\begin{equation*}\n\tM_{\\Theta}=M_{\\Phi}Z.\n\t\\end{equation*}  Using multi-analytic property of $M_{\\Phi}$ and $M_{\\Theta}$, we obtain \\begin{equation*}\nM_{\\Phi}Z\\widetilde{V}=M_{\\Theta}\\widetilde{V}=\\wT(I_{E}\\ot M_{\\Theta})=\\wT(I_{E}\\ot M_{\\Phi} Z)=M_{\\Phi}\\widetilde{R}(I_{E} \\ot Z).\n\t\\end{equation*} Thus $\\widetilde{R}(I_{E} \\ot Z)=Z\\widetilde{V}$ and  similarly $\\nu(a)Z=Z\\sigma(a),$ for $ a \\in \\mathcal{B},$ i.e.,   $Z$ is multi-analytic, therefore $Z=M_{\\Psi}$, for some inner operator $\\Psi: \\mathcal{W}_{\\mathcal{H}_{V}} \\to {\\mathcal{H}_{R}}.$ Indeed, ${\\Theta}={\\Phi}{\\Psi}$ and  $M_{\\Phi}$ and $M_{\\Theta}$ are isometries, we deduce\n\t\\begin{equation*}\n\t\\|h\\|=\\|M_{\\Theta}h\\|=\\|M_{\\Phi}M_{\\Psi}h\\|=\\|M_{\\Psi}h\\|,\\:\\:\\: h\\in \\mathcal{ H}_{V}.\n\t\\end{equation*} \n\t\n\t\n($\\impliedby$)\t\nOn the other hand, suppose there exist a Hilbert space $\\mathcal{H}_{R},$ a pure isometric representation $(\\nu, R)$ of $E$ on $\\mathcal{H}_{R}$ and  isometric multi-analytic operators  $M_\\Psi: {\\mathcal{H}_{V}} \\to {\\mathcal{H}_{R}}$ and $M\\Phi: {\\mathcal{H}_R} \\rightarrow  {\\mathcal{H}_T}$ such that   \\begin{equation*}\n\t{\\Theta}={\\Phi}{\\Psi}.\n\t\\end{equation*}  Since  ${\\Theta}={\\Phi}{\\Psi},$ we have $M_{\\Theta}(\\mathcal{H}_V) \\subseteq M_\\Phi(\\mathcal{H}_R).$  Define $\\mathcal{S}:=M_{\\Phi}(\\mathcal{H}_R)\\ominus M_{\\Theta}(\\mathcal{H}_V).$ Clearly $\\mathcal{S}$ is a closed subspace of $\\mathcal{K}_{\\Theta}.$ Also note that $\\mathcal{K}_{\\Theta}\\ominus \\mathcal{S}=\\mathcal{K}_{\\Phi}.$ \n Since $\\widetilde{W'^{*}_{\\Theta}}=(I_{E}\\ot P_{\\mathcal{K}_{\\Theta}})\\widetilde{T}^{*}P_{\\mathcal{K}_{\\Theta}},$ and $\\mathcal{K}_{\\Phi}\\subseteq\\mathcal{K}_{\\Theta}$, we have $\\mathcal{K}_{\\Phi}$ is $ \\widetilde{W'^{*}_{\\Theta}} $-IS. Indeed let $ h \\in \\mathcal{K}_{\\Phi}$ \\begin{align*}\n\t \\widetilde{W'^{*}_{\\Theta}}h=(I_{E}\\ot P_{\\mathcal{K}_{\\Theta}})\\widetilde{T}^{*}P_{\\mathcal{K}_{\\Theta}}h=\\widetilde{T}^{*}P_{\\mathcal{K}_{\\Theta}}h=\\widetilde{T}^{*}h=\\widetilde{T}^{*}P_{\\mathcal{K}_{\\Phi}}h= \\widetilde{W'^{*}_{\\Phi}}h\\in \\mathcal{K}_{\\Phi},\n\t \t\\end{align*} where  $\\widetilde{W'^{*}_{\\Phi}}=(I_{E}\\ot P_{\\mathcal{K}_{\\Phi}})\\widetilde{T}^{*}P_{\\mathcal{K}_{\\Phi}},$ Also in similar way  $\\mathcal{S}^{\\perp}$ is $ \\widetilde{W'^{*}_{\\Theta}} $-IS. That is  $\\mathcal{S}\n\t \t$ is an $(\\pi'_{\\Theta},W'_{\\Theta})$-IS.\n\\end{proof} We are now ready for the multi-variable analog of Theorem \\ref{Ber}. The following theorem is a generalization of \\cite[Theorem 4.4]{BDDS}.\t  \n\\begin{theorem}\\label{44}\nLet $(\\sigma, V^{(1)}, \\dots, V^{(k)})$ and $(\\mu, T^{(1)}, \\dots, T^{(k)})$  be pure DCI-representations of  $\\mathbb{E}$ on Hilbert spaces $\\mathcal{H}_{V}$ and $\\mathcal{H}_{T},$ respectively. Let ${M}_\\Theta:\\mathcal{H}_{V}\\to \\mathcal{H}_{T}$ be an isometric multi-analytic operator. Then the following are equivalent:\n\\begin{enumerate}\n\t\\item There exist a Hilbert space $\\mathcal{H}_{R},$ a pure DCI-representation $(\\nu, R^{(1)},\\dots,R^{(k)}) $ of $\\mathbb{E}$ on $\\mathcal{H}_{R}$ and isometric multi-analytic operators ${M}_\\Phi: {\\mathcal{H}_{R}} \\to {\\mathcal{H}_{T}}$, ${M}_\\Psi: {\\mathcal{H}_{V}} \\to {\\mathcal{H}_{R}}$ such that \\begin{equation*}\n\t{\\Theta}={\\Phi}{\\Psi}.\n\t\\end{equation*}\n\t\\item  There exists  $(\\pi'_{\\Theta}, W^{'(1)}_{\\Theta},\\dots, W^{'(k)}_{\\Theta})$-IS $\\mathcal{S}\\subseteq \\mathcal{K}_{\\Theta}$  such that $\\mathcal{S}\\oplus\\mathcal{S}_{\\Theta}$ is a Beurling subspace of $\\mathcal{H}_{T}.$\n\t\\item There exists  $(\\pi'_{\\Theta}, W^{'(1)}_{\\Theta},\\dots, W^{'(k)}_{\\Theta})$-IS $\\mathcal{S}\\subseteq \\mathcal{K}_{\\Theta}$ of $\\mathcal{H}_{T}$ such that \\begin{equation*}\n\t(I_{E_{j}}\\ot ( (I_{E_{i}}\\ot P_{\\mathcal{K}_{\\Theta} \\ominus \\mathcal{S}})- \\widetilde{U}^{(i) *}\\widetilde{U}^{(i)}))(t_{i,j} \\ot I_{\\mathcal{H}_{T}})(I_{E_{i}}\\ot ( (I_{E_{j}}\\ot P_{\\mathcal{K}_{\\Theta} \\ominus \\mathcal{S}})- \\widetilde{U}^{(j) *}\\widetilde{U}^{(j)}))=0,\n\t\\end{equation*} where  $\\widetilde{U}^{(i)}=P_{\\mathcal{K}_{\\Theta} \\ominus \\mathcal{S}}\\widetilde{W'_{\\Theta}}^{(i)}{_{|_{{E_{i}}\\ot {\\mathcal{K}_{\\Theta} \\ominus \\mathcal{S}}}}},$ for each $i\\in I_{k}.$\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{proof}\n Let us denote  $\\mathcal{K}_{\\Theta}=\\mathcal{H}_{T}\\ominus M_{\\Theta} \\mathcal{H}_{V},$ and $ M_{\\Theta} \\mathcal{H}_{V}$  be BQS and Beurling subspace of $\\mathcal{H}_{T}$ corresponding to $\\Theta,$ respectively.\n\tFor each $i \\in I_{k}$, define  $$\\pi'_{\\Theta}(a):=P_{\\mathcal{K}_{\\Theta}}\\mu(a)|_{{\\mathcal{K}_{\\Theta}}}  \\:\\mbox{and} \\:\\:W_{\\Theta}^{'(i)}(\\xi_{i}):=P_{\\mathcal{K}_{\\Theta}}T^{(i)}(\\xi_{i})|_{{\\mathcal{K}_{\\Theta}}},$$\n\tfor $\\xi_{i} \\in E_{i}$ and $a \\in \\mathcal{B}.$ Then $(\\pi'_{\\Theta}, W_{\\Theta}^{'(1)}, W_{\\Theta}^{'(2)}, \\dots, W_{\\Theta}^{'(k)} )$ is a covariant representation of $\\mathbb{E}$ on $\\mathcal{K}_{\\Theta} \\subseteq \\mathcal{H}_{T}.$\n\t\n\t\n\t\n\t\n\t\n\t\t$(1) \\implies (2)$ : Suppose there exist  a Hilbert space $\\mathcal{H}_{R},$ a pure DCI-representation $(\\nu, R^{(1)},\\dots,R^{(k)}) $ of $\\mathbb{E}$ on $\\mathcal{H}_{R}$ and  isometric multi-analytic operators ${M}_\\Phi: {\\mathcal{H}_{R}} \\to {\\mathcal{H}_{T}}$, ${M}_\\Psi: {\\mathcal{H}_{V}} \\to {\\mathcal{H}_{R}}$ such that $\t{\\Theta}={\\Phi}{\\Psi}.$ Therefore $M_{\\Theta}\\mathcal{H}_V \\subseteq M_\\Phi\\mathcal{H}_R,$ we can define $\\mathcal{S}:=M_{\\Phi}\\mathcal{H}_R\\ominus M_{\\Theta}\\mathcal{H}_V.$ Clearly $\\mathcal{S}$ is a closed subspace of $\\mathcal{K}_{\\Theta}.$ Also note that $\\mathcal{K}_{\\Theta}\\ominus \\mathcal{S}=\\mathcal{K}_{\\Phi}.$\tSince $\\widetilde{W'_{\\Theta}}^{(i)*}=(I_{E_{i}} \\ot P_{\\mathcal{ K}_{\\Theta}})\\widetilde{T}^{(i)*}P_{\\mathcal{K}_{\\Theta}},$ and $\\mathcal{K}_{\\Phi}\\subseteq\\mathcal{K}_{\\Theta}$, we have $\\mathcal{K}_{\\Phi}$ is $\\pi'_{\\Theta}$ and $ \\widetilde{W'_{\\Theta}}^{(i)*} $-IS for each $i\\in I_{k}.$ Indeed, let $ h \\in \\mathcal{K}_{\\Phi}$ \\begin{align*}\n\t\\widetilde{W'_{\\Theta}}^{(i)*}h=(I_{E}\\ot P_{\\mathcal{K}_{\\Theta}})\\widetilde{T}^{(i)*}P_{\\mathcal{K}_{\\Theta}}h=\\widetilde{T}^{(i)*}P_{\\mathcal{K}_{\\Theta}}h=\\widetilde{T}^{(i)*}h=\\widetilde{T}^{(i)*}P_{\\mathcal{K}_{\\Phi}}h= \\widetilde{W'_{\\Phi}}^{(i)*}h\\in \\mathcal{K}_{\\Phi}.\n\t\\end{align*} Also in similar way  $\\mathcal{S}^{\\perp}$ is $ \\widetilde{W'_{\\Theta}}^{(i)} $-IS. That is  $\\mathcal{S}\n\t$ is an $(\\pi_{\\Theta}, W^{'(1)}_{\\Theta},\\dots, W^{'(k)}_{\\Theta})$-IS.\n\t\n\t$(2) \\implies (1)$: Let $\\mathcal{S}$ be a $(\\pi'_{\\Theta}, W^{'(1)}_{\\Theta},\\dots, W^{'(k)}_{\\Theta})$-IS of $\\mathcal{H}_{T}$ such that $\\mathcal{S}\\oplus M_\\Theta \\mathcal{H}_{V}$  is a DCS of $\\mathcal{H}_{T}$. Therefore by using Theorem \\ref{MT5}, there exist a pure DCI-representation $(\\nu, R^{(1)},\\dots,R^{(k)}) $ of $\\mathbb{E}$ on $\\mathcal{H}_{R}$ and an isometric multi-analytic operator $M_\\Phi: {\\mathcal{H}_{R}} \\to {\\mathcal{H}_{T}}$ such that \\begin{equation}\\label{quotient 11}\n\t\\mathcal{S}\\oplus M_\\Theta \\mathcal{H}_{V}=M_{\\Phi}{\\mathcal{H}_{R}}.\n\t\\end{equation} Thus $M_{\\Theta}\\mathcal{H}_{V}\\subseteq M_{\\Phi}\\mathcal{H}_{R},$ and hence by Douglas's range  inclusion theorem, there exists a contraction $Z: \\mathcal{H}_{V} \\to \\mathcal{H}_{R}$ such that \\begin{equation*}\n\tM_{\\Theta}=M_{\\Phi}Z.\n\t\\end{equation*} Using multi-analytic property of $M_{\\Phi}$ and $M_{\\Theta}$, we get \\begin{equation*}\n\tM_{\\Phi}\\widetilde{R}^{(i)}(I_{E} \\ot Z)=\\wT^{(i)}(I_{E}\\ot M_{\\Phi} Z)=\\wT^{(i)}(I_{E}\\ot M_{\\Theta})=M_{\\Theta}\\widetilde{V}^{(i)}=M_{\\Phi}Z\\widetilde{V}^{(i)},\\:\\:\\: \\text{for each}\\:\\: i \\in I_{k}.\n\t\\end{equation*} Thus $\\widetilde{R}^{(i)}(I_{E} \\ot Z)=Z\\widetilde{V}^{(i)}$, similarly $\\nu(a)Z=Z\\sigma(a),$ for $ a \\in \\mathcal{B}.$ Then there exists an inner operator $\\Psi: \\mathcal{W}_{\\mathcal{H}_{V}} \\to {\\mathcal{H}_{R}}$ such that $Z= {M}_{\\Psi}.$ Indeed, \t${\\Theta}={\\Phi}{\\Psi}$ and using  $M_{\\Phi}$ and $M_{\\Theta}$ are isometries, we deduce\n\t\\begin{equation*}\n\t\\|h\\|=\\|M_{\\Theta}h\\|=\\|M_{\\Phi}M_{\\Psi}h\\|=\\|M_{\\Psi}h\\|,\\:\\:\\: h\\in \\mathcal{ H}_{V}.\n\t\\end{equation*}\n\t\n\t\n\t\n\t$(1) \\implies (3)$ : Suppose there exist a pure DCI-representation $(\\nu, R^{(1)},\\dots,R^{(k)}) $ of $\\mathbb{E}$ on $\\mathcal{H}_{R}$ and isometric multi-analytic operators $M_\\Phi: {\\mathcal{H}_{R}} \\to {\\mathcal{H}_{T}}$, $M_\\Psi: {\\mathcal{H}_{V}} \\to {\\mathcal{H}_{R}}$ such that $\t{\\Theta}={\\Phi}{\\Psi}.$ Define $\\mathcal{S} =M_{\\Phi}\\mathcal{H}_R\\ominus M_{\\Theta}\\mathcal{H}_V$ and then $\\mathcal{K}_{\\Theta}\\ominus \\mathcal{S}=\\mathcal{K}_{\\Phi}.$  Therefore  $\\mathcal{K}_{\\Phi}$ is a BQS of $\\mathcal{H}_{T}$  using Theorem \\ref{Beurling}, we get  \\begin{equation*}\n\t(I_{E_{j}}\\ot ( (I_{E_{i}}\\ot P_{\\mathcal{K}_{\\Phi}})- \\widetilde{U}^{(i) *}\\widetilde{U}^{(i)}))(t_{i,j} \\ot I_{\\mathcal{H}_{T}})(I_{E_{i}}\\ot ( (I_{E_{j}}\\ot P_{\\mathcal{K}_{\\Phi}})- \\widetilde{U}^{(j) *}\\widetilde{U}^{(j)}))=0,\n\t\\end{equation*} where  $\\widetilde{U}^{(i)}=P_{\\mathcal{K}_{\\Phi}}\\widetilde{W'_{\\Theta}}^{(i)}{_{|_{{E_{i}}\\ot {\\mathcal{K}_{\\Phi}}}}},$ for each $i\\in I_{k}.$\n\t\n\t\n \n\t\n\t\n\t \n\t\n\t\t$(3) \\implies (2)$: Since $\\widetilde{W'_{\\Theta}}^{(i)}=(I_{E}\\ot P_{\\mathcal{K}_{\\Theta}})\\widetilde{T}^{(i)*}|_{{\\mathcal{K}_{\\Theta}}},$ it follows that $\\mathcal{K}_{\\Theta}\\ominus \\mathcal{S}$  is a QS of $\\mathcal{H}_{T}$. Therefore by hypothesis and using Theorem \\ref{Beurling}, $\\mathcal{K}_{\\Theta}\\ominus \\mathcal{S}$  is  a BQS. Now observe that \\begin{align*}\n\t \\mathcal{K}_{\\Theta}\\ominus \\mathcal{S}= \\mathcal{H}_{T}\\cap (M_{\\Theta}(\\mathcal{H}_{V})\\cup \\mathcal{S})^{\\perp}= \\mathcal{H}_{T} \\ominus (M_{\\Theta}(\\mathcal{H}_{V}) \\oplus\\mathcal{S}),\n\t\t\\end{align*} which implies that $\\mathcal{S}\\oplus\\mathcal{S}_{\\Theta}$ is a Beurling submodule of $\\mathcal{H}_{T}.$\n\t\\end{proof}\nIn the above theorem we discussed only non-trivial $(\\pi'_{\\Theta}, W^{'(1)}_{\\Theta},\\dots, W^{'(k)}_{\\Theta})$-IS $\\mathcal{S}\\subseteq \\mathcal{K}_{\\Theta}$ of $\\mathcal{K}_{\\Theta}$. Let us discuss trivial case :\\begin{enumerate}\n \\item First take $\\mathcal{S} =\\{0\\}$. Recall that $\\mathcal{S}=M_{\\Phi}\\mathcal{H}_R\\ominus M_{\\Theta}\\mathcal{H}_V$  therefore $M_{\\Phi}\\mathcal{H}_R\\ominus M_{\\Theta}\\mathcal{H}_V=\\{0\\}$, it follows that $M_\\Phi\\mathcal{H}_{R}\\subseteq M_\\Theta\\mathcal{H}_{V}.$ Since \t${\\Theta}={\\Phi}{\\Psi}$, $M_\\Theta\\mathcal{H}_{V}\\subseteq M_\\Phi\\mathcal{H}_{R}.$ Hence \\begin{align*}\n M_\\Phi\\mathcal{H}_{R}=M_\\Theta\\mathcal{H}_{V}= M_{\\Phi}M_{\\Psi}\\mathcal{H}_{V}, \n \\end{align*} we obtain $M_{\\Psi}\\mathcal{H}_{V}=\\mathcal{H}_{R}.$ Thus ${\\Psi}$ is a unitary.\n \\item Now take $\\mathcal{S} =\\mathcal{K}_{\\Theta}$. Therefore \\begin{align*}\n M_{\\Phi}\\mathcal{H}_R\\ominus M_{\\Theta}\\mathcal{H}_V=\\mathcal{K}_{\\Theta}=\\mathcal{H}_{T} \\ominus M_{\\Theta}\\mathcal{H}_{V},\n \\end{align*} it implies that  we obtain $ M_{\\Phi}\\mathcal{H}_R=\\mathcal{H}_{T}.$ Thus ${\\Phi}$ is a unitary.\n\\end{enumerate} Therefore from the above discussion, if $\\mathcal{S}$ is  a non-trivial  $(\\pi'_{\\Theta}, W^{'(1)}_{\\Theta},\\dots, W^{'(k)}_{\\Theta})$-IS of $ \\mathcal{K}_{\\Theta}$ if and only if isometric multi-analytic operators ${M}_\\Phi: {\\mathcal{H}_{R}} \\to {\\mathcal{H}_{T}}$, ${M}_\\Psi: {\\mathcal{H}_{V}} \\to {\\mathcal{H}_{R}}$ are not unitary.\nThe following corollary is a generalization of \\cite[Corollary 4.5]{BDDS}.\n\\begin{corollary}\nLet $(\\sigma, V^{(1)}, \\dots, V^{(k)})$ and $(\\mu, T^{(1)}, \\dots, T^{(k)})$  be  pure DCI-representations of  $\\mathbb{E}$ on  Hilbert spaces $\\mathcal{H}_{V}$ and $\\mathcal{H}_{T},$ respectively.  Let ${M}_\\Theta: {\\mathcal{H}_{V}} \\to {\\mathcal{H}_{T}}$ be an isometric multi-analytic operator. Then  isometric multi-analytic operators ${M}_\\Phi: {\\mathcal{H}_{R}} \\to {\\mathcal{H}_{T}}$, ${M}_\\Psi: {\\mathcal{H}_{V}} \\to {\\mathcal{H}_{R}},$ coming from Theorem (\\ref{44}), are non-unitary if and only if the following holds: \n\\begin{enumerate}\n\t\\item $\\mathcal{S}$ is a non-trivial $(\\pi'_{\\Theta}, W^{'(1)}_{\\Theta},\\dots, W^{'(k)}_{\\Theta})$-IS of $\\mathcal{K}_{\\Theta}.$ \n\t\\item $\\mathcal{S}$ is not a Beurling subspace of $\\mathcal{H}_{T}.$\n\t\\item  $\\mathcal{K}_{\\Theta}\\ominus \\mathcal{S}$ does not reduces $(\\mu, T^{(1)}, \\dots, T^{(k)})$.\n\t\\end{enumerate}  \n\\end{corollary}\n\n\n\\begin{proof}\n\tFirst assume isometric multi-analytic operators ${M}_\\Phi: {\\mathcal{H}_{R}} \\to {\\mathcal{H}_{T}}$, ${M}_\\Psi: {\\mathcal{H}_{V}} \\to {\\mathcal{H}_{R}}$ are non-unitary.  As by above discussion, if $\\mathcal{S}$ is  a non-trivial  $(\\pi'_{\\Theta}, W^{'(1)}_{\\Theta},\\dots, W^{'(k)}_{\\Theta})$-IS of $ \\mathcal{K}_{\\Theta}$ if and only if  isometric multi-analytic operators ${M}_\\Phi$, ${M}_\\Psi $  are not unitary. This proves (1).\n\t\n\t\n\t\n\t Let if possible $\\mathcal{S}$ is a Beurling subspace of $\\mathcal{H}_{T}$. Therefore by using Theorem \\ref{MT5}, there exist a pure DCI-representation $(\\mu_1, R_1^{(1)},\\dots,R_1^{(k)}) $ of $\\mathbb{E}$ on $\\mathcal{H}_{R_1}$ and an isometric multi-analytic operator $M_{\\Phi_1}: {\\mathcal{H}_{R_1}} \\to {\\mathcal{H}_{T}}$ such that $\\mathcal{S}=M_{\\Phi_1}{\\mathcal{H}_{R_1}}.$ But \\begin{align}\\label{inv}\n\t\\mathcal{S}=M_{\\Phi}\\mathcal{H}_R\\ominus M_{\\Theta}\\mathcal{H}_V=M_{\\Phi_1}{\\mathcal{H}_{R_1}},\n\t\\end{align} we have $M_{\\Phi_1}{\\mathcal{H}_{R_1}} \\subseteq M_{\\Phi}\\mathcal{H}_R,$ and hence by Douglas's range  inclusion theorem, there exists a contraction $Z: \\mathcal{H}_{R_1} \\to \\mathcal{H}_{R}$ such that \\begin{equation*}\n\tM_{\\Phi_1}=M_{\\Phi}Z.\n\t\\end{equation*}\n\t\n\tUsing multi-analytic property of $M_{\\Phi}$ and $M_{\\Phi_1}$, we get \\begin{equation*}\nM_{\\Phi}\\widetilde{R}^{(i)}(I_{E} \\ot Z)=\\wT^{(i)}(I_{E}\\ot M_{\\Phi} Z)=\\wT^{(i)}(I_{E}\\ot M_{\\Phi_1})=M_{\\Phi_1}\\widetilde{R}_{1}^{(i)}=M_{\\Phi}Z\\widetilde{R}_{1}^{(i)},\\quad i \\in I_{k}.\n\\end{equation*} Thus $\\widetilde{R}^{(i)}(I_{E} \\ot Z)=Z\\widetilde{R}_{1}^{(i)}$, similarly $\\nu(a)Z=Z\\mu_1(a),$ for $ a \\in \\mathcal{B}.$ Then there exists inner operator $\\Phi_2: \\mathcal{W}_{\\mathcal{H}_{R_1}} \\to \\mathcal{W}_{\\mathcal{H}_{R}}$ such that $Z= {M}_{\\Phi_{2}}.$ Indeed, $\\Phi_1=\\Phi\\Phi_2$ and   $M_{\\Phi}$ and $M_{\\Phi_1}$ are isometries, we deduce\n\t\\begin{equation*}\n\t\\|h\\|=\\|M_{\\Phi_1}h\\|=\\|M_{\\Phi}M_{\\Phi_2}h\\|=\\|M_{\\Phi_2}h\\|,\\:\\:\\: h\\in \\mathcal{ H}_{R_{1}}.\n\t\\end{equation*} By using Equation (\\ref{inv}) \\begin{align*}  \\mathcal{S}=M_{\\Phi_1}{\\mathcal{H}_{R_1}}\n\tM_{\\Phi}(M_{\\Phi_2}{\\mathcal{H}_{R_1}})=M_{\\Phi}\\mathcal{H}_R\\ominus M_{\\Theta}\\mathcal{H}_V&=M_{\\Phi}\\mathcal{H}_R\\ominus M_{\\Phi}M_{\\Psi}\\mathcal{H}_V\\\\&=M_{\\Phi}(\\mathcal{H}_R\\ominus M_{\\Psi}\\mathcal{H}_V),\n\t\\end{align*} we obtain $M_{\\Phi_2}{\\mathcal{H}_{R_1}}=\\mathcal{H}_R\\ominus M_{\\Psi}\\mathcal{H}_V=\\mathcal{K}_{\\Psi}.$ It implies that $\\mathcal{K}_{\\Psi}$ is  $(\\nu, R^{(1)}, \\dots, R^{(k)})$-reducing subspace of $\\mathcal{H}_{R}$ It implies that $\\mathcal{K}_{\\Psi}$ is trivial subspace which means that $\\Psi$ is an unitary. But $\\Psi$ is not unitary operator, thus it is contradiction. Hence  $\\mathcal{S}$ is not a Beurling subspace of $\\mathcal{H}_{T}$. This proves (2). \n\n\n\n\nSuppose  $\\mathcal{K}_{\\Theta}\\ominus \\mathcal{S}$  reduces $(\\mu, T^{(1)}, \\dots, T^{(k)}).$ Then $(\\mathcal{K}_{\\Theta}\\ominus \\mathcal{S})^{\\perp}= \\mathcal{S}\\oplus \\mathcal{S}_{\\Theta}$ is also reduces $(\\mu, T^{(1)}, \\dots, T^{(k)}).$ On the other hand, By using Equation (\\ref{quotient 11})  $\\mathcal{S}\\oplus \\mathcal{S}_{\\Theta}=M_{\\Phi}\\mathcal{H}_{R},$ it follows that $\\Phi$ is an unitary, which is contradiction as $\\Phi$ is not unitary inner operator.\n\nFor the converse part, suppose $\\mathcal{S}$ is a non-trivial $(\\pi_{\\Theta}, W^{(1)}_{\\Theta},\\dots, W^{(k)}_{\\Theta})$-IS of $\\mathcal{K}_{\\Theta}.$ Since $\\Theta=\\Phi\\Psi$ and $\\Theta$ is non constant, both $\\Phi$ and $\\Psi$ can not be unitaries. It remains to show that $\\Phi$ and $\\Psi$ can not be  isometry operators. Let if possible  $\\Phi\\equiv X_1$ for some non-unitary isometry $X_1$ and $\\Psi$ is not unitary. Then \\begin{align*}\n\\mathcal{S}\\oplus \\mathcal{S}_{\\Theta}=M_{\\Phi}\\mathcal{H}_{R}=X_1\\mathcal{H}_{R}=\\mathcal{H}_{X_1R}\n\\end{align*} and hence $\\mathcal{S}\\oplus \\mathcal{S}_{\\Theta}$ reduces $(\\mu, T^{(1)}, \\dots, T^{(k)}),$ which contradicts to (3). On the other hand, if $\\Psi\\equiv X_2$ for some non-unitary isometry $X_2$ and $\\Phi$ is not unitary. Then\\begin{align*}\n\\mathcal{S}= M_{\\Phi}(\\mathcal{H}_R\\ominus M_{\\Psi}\\mathcal{H}_V)=M_{\\Phi}(\\mathcal{H}_R\\ominus(X_2\\mathcal{H}_V))=M_{\\Phi}\\mathcal{H}_{R-X_{2}V}\n\\end{align*} is a Beurling subspace of $\\mathcal{H}_{T},$ which contradicts to (2). This completes the proof. \n\\end{proof}\n\\subsection*{Acknowledgment}\nAzad Rohilla is supported by a UGC fellowship (File No: 16-6(DEC.2017)\n\/2018(NET\/CSIR)). Shankar Veerabathiran thanks ISI Bangalore for the visiting scientist position. Harsh Trivedi is supported by MATRICS-SERB  \nResearch Grant, File No: MTR\/2021\/000286, by the Science and Engineering Research Board\n(SERB), Department of Science \\& Technology (DST), Government of India. We thank Jaydeb Sarkar for making us aware of reference \\cite{AC14}. We acknowledge the Centre for Mathematical \\& Financial Computing and the DST-FIST grant for the financial support for the computing lab facility under the scheme FIST ( File No: SR\/FST\/MS-I\/2018\/24) at the LNMIIT, Jaipur. \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{INTRODUCTION}\nThe discovery of graphene in 2004 \\cite{Novoselov2014Science} was a significant breakthrough in materials science and since then there has been sustained research interest in graphene and a rush to discover other stable two-dimensional (2D) materials. 2D materials have the thickness of one or a few atomic layers and have markedly different material properties than their bulk counterparts due to the quantum confinement effect. Since graphene, there have been several advances in the field of 2D materials such as the discovery of semiconducting monolayer transition metal dichalcogenides (TMDs) \\cite{2_mak_lee_hone_shan_heinz_2010, 2_wang_kalantar-zadeh_kis_coleman_strano_2012}. The growing number and variety of 2D materials has fuelled interest in the use of 2D materials in the development of novel nanoscale devices.\n\nA recent development is the experimental isolation of a single layer of bulk black phosphorus, also known as phosphorene \\cite{Brent2014CC,Buscema2014NC,Castellanos-Gomez20142DM,Li2014NN,Liu2014AN,Qiao2014NC,Rodin2014PRL,Zhu2014PRL,Reich2014Nature,Churchill2014NN,Koenig2014APL,Fei2014NL,Fei2015PRB,Tran2014PRB,Tran2014PRB2,Guo2014JPCC,Peng2014PRB,Peng2014MRE,Wang2015PRB,Cai2014SR}. Like graphene, phosphorene was first obtained by mechanical exfoliation \\cite{Li2014NN,Liu2014AN}. Liquid exfoliation, which is a scalable process, has also been demonstrated as a possible alternative means of producing phosphorene \\cite{Brent2014CC}. Several experiments have demonstrated that phosphorene is a direct-gap semiconductor and also has a high hole mobility\\cite{Li2014NN,Xia2014NC,Liu2014AN}. These characteristics make phosphorene attractive for use in electronic and optoelectronic devices \\cite{Xia2014NC}. Field effect transistors (FET) based on few-layer phosphorene were shown to have high on\/off ratios \\cite{Li2014NN}. It also has the potential to be used as an anode material in lithium-ion batteries \\cite{Li2015NL}. Another potential application is in thin film excitonic solar cells as phosphorene has a predicted band gap in the visible region \\cite{Dai2014JPCL}. Excitonic solar cells (XSCs) based on some 2D materials, such as MoS$_2$, WSe$_2$, graphene, h-BN, SiC$_2$ and bilayer phosphorene, are potentially seen as the new generation of thin film solar cells\\cite{1_tsai_su_chang_tsai_chen_wu_li_chen_he_2014, 2_pospischil_furchi_mueller_2014,Bernardi2012ACS,Dai2014JPCL,Zhou2013NL,Britnell2013Science}, and they might have higher efficiencies than existing XSCs which typically have less than 10\\% efficiency \\cite{1_green_emery_hishikawa_warta_dunlop_2015}. Till now, despite the limitations of fabrication methodologies for such 2D solar cells, there has been some progress on the fabrication of few-layer heterostructures such as graphene-WS${_2}$\\cite{1_georgiou_jalil_belle_britnell_gorbachev_morozov_kim_gholinia_haigh_makarovsky_2012, 2_tan_avsar_balakrishnan_koon_taychatanapat_ofarrell_watanabe_taniguchi_eda_castro_neto_2014}, graphene-MoS${_2}$\\cite{Britnell2012Science} and phosphorene-MoS${_2}$\\cite{Deng2014ACS}.\n\nIn this paper we study the excited-state properties of monolayer phosphorene, and evaluate the viability of monolayer phosphorene as one building block of of an excitonic solar cell heterostructure. For the other building-block material in the heterostructure, the semiconducting monolayer TMDs, which have been extensively researched, are chosen. The semiconducting TMDs considered here include semiconducting MoS${_2}$, MoSe${_2}$, MoTe${_2}$, WS${_2}$, WSe${_2}$, WTe${_2}$, TiS${_2}$ and ZrS${_2}$. This paper is organized as follows: In \\textbf{Sec. II}, we first introduce the structures of geometrically optimized monolayer phosphorene and computational details. \\textbf{Sec. III} is the main part of results and discussions, which includes three subsections: In \\textbf{SubSec. A}, we show the excited-state properties of monolayer phosphorene, including in quasi-particle band structures and optical spectra. In \\textbf{SubSec. B}, we calculate the excited-state properties of 8 semiconducting TMDs. In \\textbf{SubSec. C}, the band alignment of phosphorene and TMDs is presented. Then the power conversion efficiencies (PCE) of the phosphorene-TMD heterostructures are discussed. In \\textbf{Sec. IV}, we conclude our studies.\n\n\n\n\\section{Structures and COMPUTATIONAL DETAILS}\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.7\\textwidth]{Fig1.eps}\\\\\n\\caption{(Color online) GW-calculated (solid lines) and HSE06-calculated (dash lines) band structure of monolayer phosphorene. The Fermi level is set on the top of the valance band. The ball and stick model of phosphorene (light color of the upper layer) with primitive cell and the reciprocal lattice with the high symmetry points are shown in inset.}\n\\label{fig1}\n\\end{figure}\n\nPhosphorene has a puckered honeycomb structure as shown in inset of \\textbf{Figure 1}. The underlying lattice is rectangular which leads to anisotropy in the band structure and optical properties. The calculated lattice constants are $a = 4.58$~\\AA~and $b = 3.30$~\\AA~which are consistent with results in literature \\cite{Liu2014AN}. The calculations were performed using the Vienna Ab-initio Simulation Package (VASP) \\cite{16_kresse_furth_1996, 15_kresse_furth_1996, 13_kresse_hafner_1993, 14_kresse_hafner_1994}, and a projector augmented wave (PAW) basis set was used \\cite{1_blochl_1994, 2_kresse_joubert_1999}. Geometry optimisation was done using density function theory (DFT) with the generalised gradient approximation (GGA) using the Perdew-Burke-Ernzerhof (PBE) exchange correlation functional. A $14\\times10\\times1$ $k$-point grid was used for phosphorene while an $11\\times11\\times1$ $k$-point grid was used for TMDs. The geometry was relaxed until the force acting on the atoms was less than 0.01 eV\/atom. To ensure that the interlayer interaction is negligible, the out of plane lattice parameter, which is perpendicular to the plane of the material, was set as at least 15 \\AA.\nBand gaps and band structures were calculated using the GW method. The band gaps\/structures were also calculated through the screened exchange hybrid density functional by Heyd, Scuseria, and Ernzerhof (HSE06) for reference. A high number of empty conductions bands is necessary for the convergence of the absolute band positions\\cite{Liang2013APL}. A total of 1024 bands were used for phosphorene and 1536 bands were used for TMDs. Single shot GW calculations, $G_{0}W_{0}$, were performed on phosphorene to obtain the ground state band energies. For TMDs, one eigenvalue update is performed to obtain the expected direct band gap of trigonal prismatic TMDs \\cite{Shi2013PRB,Cheiwchanchamnangij2012PRB, Ramasubramaniam2012PRB, Qiu2013PRL}. The band structure was then interpolated from Wannier functions rather than evaluated directly at discrete $k$-points. This was done using the WANNIER90 library \\cite{12_mostofi_yates_lee_souza_vanderbilt_marzari_2008} and the VASP2WANNIER interface. The optical gap was calculated by solving the Bethe-Salpeter equation (BSE). The GW method has more computational demands and the band gaps converge with a much smaller number of bands\\cite{Liang2013APL}. Thus to streamline optical calculations, the $G_{0}W_{0}$ calculation was run with 128 bands for phosphorene and 192 bands for TMDs before solving the BSE. This produced the frequency dependent dielectric tensor that was used to calculate the absorption spectrum. By solving the BSE, electron-hole interactions such as excitons are accounted for in the dielectric tensor. For phosphorene the $x$- and $y$-components of the dielectric tensor were treated separately because of the anisotropy but for TMDs, the average value of the $x$- and $y$-components was used due to crystal symmetry.\n\n\n\n\\section{RESULTS AND DISCUSSIONS}\n\\subsection{Excited-state properties of phosphorene}\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.75\\textwidth]{Fig2.eps}\\\\\n\\caption{(Color online) GW-calculated (solid lines) and HSE06-calculated (dashed lines) absorption spectrum of phosphorene for \\emph{armchair} polarised light. The energy values of first absorption peak, observed in the experiment\\cite{Li2014NN}, is presented by a black solid circle for reference.}\n\\label{fig2}\n\\end{figure}\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.6\\textwidth]{Fig3.eps}\\\\\n\\caption{(Color online) (a) The band structures of MoS${_2}$ with a ball and stick model of the trigonal prismatic geometry. (b) The band structures of TiS${_2}$ with a ball and stick model of the octahedral geometry. The reciprocal lattice with high symmetry points is shown in the inset. The Fermi level is set on the top of the valance band.}\n\\label{fig3}\n\\end{figure}\nThe calculated band structure of phosphorene is shown in \\textbf{Figure 1}. The GW calculation predicts a band gap value of 2.15 eV and the gap is approximately located at the $\\Gamma$ point, which is much higher than the DFT-calculated band gap of 0.91 eV \\cite{Qiao2014NC}. The conduction band minimum (CBM) and the valence band maximum (VBM) are not exactly aligned in the GW-calculated band structure, but they are sufficiently close to be considered as a direct band gap \\cite{Tran2014PRB2}. A similar computational approach by Tran \\textit{et al}. predicted a band gap of 2.0 eV and a comparable profile of the band structure \\cite{Tran2014PRB2}. The CBM position is around -4.25 eV with respect the the vacuum level. Besides the GW method, the hybrid density functional HSE06 was used to calculate the band structures and band gap for reference in \\textbf{Fig. 1}. As can be seen, the HSE06-calculated band gap is 0.6 eV lower than the GW-calculated gap, which is in agreement with previous calculations \\cite{Qiao2014NC}.\n\n\nThe optical gap of phosphorene is calculated using the GW-BSE approach and is determined to be 1.6 eV. This is the first optical peak of the absorption spectrum for light polarised along the armchair direction as seen in \\textbf{Fig. 2}. This optical gap is slightly larger than the experimental photoluminescence measurement value of 1.45 eV of phosphorene \\cite{Liu2014AN}, but smaller than the HSE06-calculated value of 1.8 eV. The optical gap of 1.6 eV is much lower than the electronic band gap of 2.15 eV which suggests that significant excitonic effects are present in phosphorene. The exciton binding energy of 0.55 eV in monolayer phosphorene is quite huge. Both the self-energy corrected large electronic gap and small optical gap of phosphorene indicate significant many-electron effect in phosphorene.\n\n\n\\subsection{Excited-state properties of TMDs}\n\n\nUsing a similar approach to the above, the geometry, band positions and band structures of 8 semiconducting TMDs are calculated. A summary of the lattice constants, absolute positions of the valence band maximum (VBM) and conduction band minimum for the different materials is shown in \\textbf{Table 1}. The TMDs are categorised into either trigonal prismatic or octahedral TMDs, both of which have a hexagonal lattice but different coordination of the atoms within the unit cell. The two different structures are shown in the inset of \\textbf{Fig. 3}. For trigonal prismatic TMDs, the lattice constant is largely determined by the size chalcogen atom which increases with atomic number (sulphur to selenium to tellurium). Furthermore, as can be seen, phosphorene has a large exciton binding energy compared to TMDs because of its unique quasi 1D band dispersions.\n\n\n\\begin{table*}\n\\centering\n\\caption{Summary of the lattice constants $a$ and $b$, the band gap $E_g$, valence band maximum $E_{VBM}$, conduction band minimum $E_{CBM}$ and optical gap $E_{opt}$}\\label{tab:res}\n\\begin{tabular}{c c c c c c c c c}\n\\toprule\nMaterial & Lattice & Structure & $a$ (\\AA) & $b$ (\\AA) & $E_g$ (eV) & $E_{VBM}$ (eV) & $E_{CBM}$ (eV) & $E_{opt}$ (eV) \\\\\nPhosphorene & Rectangular & - & 4.58 & 3.30 & 2.15 & -6.40 & -4.25 & 1.6\\\\\nMoS${_2}$  & Hexagonal & Trigonal Prismatic & 3.16 & - & 2.68 & -6.57 & -3.89 & 2.3\\\\\nMoSe${_2}$ & Hexagonal & Trigonal Prismatic & 3.29 & - & 2.39 & -5.97 & -3.58 & 2.2\\\\\nMoTe${_2}$ & Hexagonal & Trigonal Prismatic & 3.52 & - & 1.74 & -5.43 & -3.69 & 1.7\\\\\nWS${_2}$   & Hexagonal & Trigonal Prismatic & 3.16 & - & 2.94 & -6.47 & -3.53 & 2.5\\\\\nWSe${_2}$  & Hexagonal & Trigonal Prismatic & 3.26 & - & 2.70 & -5.84 & -3.14 & 2.6\\\\\nWTe${_2}$  & Hexagonal & Trigonal Prismatic & 3.52 & - & 1.98  & -5.35 & -3.37 & 1.9\\\\\nTiS${_2}$  & Hexagonal & Octahedral & 3.37 & - & 1.88 & -6.80 & -4.88 & -\\\\\nZrS${_2}$  & Hexagonal & Octahedral & 3.58 & - & 2.65 & -7.56 & -4.81 & -\\\\\n\\end{tabular}\n\\end{table*}\n\n\n\n\nThe trigonal prismatic TMDs have a direct band gap the the $K$ point [see \\textbf{Supplementary Materials}]. They have the similar band structures. The band structure of MoS${_2}$ is shown in \\textbf{Fig. 3(a)} as an example. For trigonal prismatic TMDs, the band gap decreases with increasing chalcogen atomic number. The CBM position also decreases with increasing chalcogen atomic number. These trends and the band gap values are consistent with other similar studies of these materials \\cite{Gong2013APL,Liang2013APL,Kang2013APL, Ding2011PBCM,Shi2013PRB}. Octahedral TMDs have an indirect band gap between the $\\Gamma$ and $M$ points [see \\textbf{Supplementary Materials}]. This is also in agreement with band profiles calculated in previous studies. \\cite{Liang2013APL,Gong2013APL,6_ivanovskaya_2005,Li2014RSC,Kang2013APL,Jiang2012JPCC}. The band structure of TiS${_2}$ is shown in \\textbf{Figure 3(b)}.\n\n\n\\subsection{Band offset and PCE of excitonic solar cells}\n\n\n\n\\begin{figure}[htp]\n\\centering\n\\includegraphics[width=0.8\\textwidth]{Fig4.eps}\\\\\n\\caption{(Color online) Band alignment of phosphorene with semiconducting monolayer TMDs.}\n\\label{fig6}\n\\end{figure}\n\nNote that many important properties and potential device applications of semiconductors are not determined entirely by the band gap only. The band alignment and corresponding band offsets (the relative band-edge energies) of two or more semiconductors are other fundamental\/critical parameters in the design of heterojunction devices \\cite{Wei1998APL,Zhang2000PRL,Jiang2012JPCC,Kang2013APL,Liang2013APL}, for example, the 2D heterostructure devices for photocatalytic water splitting \\cite{Jiang2012JPCC,Kang2013APL,Liang2013APL}, field effect transistors \\cite{Gong2013APL} and \\textit{p-n} diodes\\cite{Deng2014ACS}. Chemical trends of the band offesets provide a useful tool for predicting catalytic ability of TMDs-based heterojunctions. \\textbf{Figure 4} shows the band alignment (using the vacuum level as reference) of phosphorene with 8 semiconducting monolayer TMDs. It can be seen that the CBM of trigonal prismatic TMDs is higher than that of monolayer phosphorene, while the CBM of octahedral TMDs is lower than phosphorene. Thus, trigonal prismatic TMDs could function as the donor whereas octahedral TMDs would function as the acceptor in heterostructure with phosphorene. The optical gap of 8 TMDs is calculated, which is determined from the absorption spectrum. In most cases, the optical gap is slightly lower than the electronic band gap as shown in \\textbf{Table 1}. Notice that because of strong many-electron effects in some 2D materials, such as phosphorene and MoS$_2$, we might not get the accurate band offset parameters in some cases without considering the excited-state effect. For example, the HSE-calculated CBM band energy of monolayer MoS$_2$ is -4.21~eV \\cite{Guo2014JPCC} (-4.25~eV \\cite{Kang2013APL}) and phosphorene is -3.94~eV \\cite{Guo2014JPCC} (-3.92~eV \\cite{Cai2014SR}). Thus, phoshporene (MoS$_2$) is the donor (acceptor) in the phosphorene-MoS$_2$ 2D heterojunction \\cite{Guo2014JPCC} and the predicted PCE is 17.5\\% \\cite{Guo2014JPCC}. However, if we consider many-electron effects, the GW-calculated CBM band energy of monolayer MoS$_2$ is -3.89~eV (-3.74~eV \\cite{Liang2013APL}) and phosphorene is -4.25~eV. Interestingly, phoshporene (MoS$_2$) becomes the acceptor (donor) instead and the predicted PCE is reducing to 10\\%. Regarding to other 2D materials without strong many-electron effects, both HSE and GW can give similar band offset \\cite{Liang2013APL,Kang2013APL}.\n\nA model developed by Scharber et al. for organic solar cells \\cite{1_scharber_2006} and later adapted for exciton based 2D solar cells \\cite{Bernardi2012ACS,Dai2014JPCL} is used to predict the maximum PCE, $\\eta$, based on the fill-factor, $\\beta_{FF}$, open circuit voltage, $V_{oc}$, and short circuit current, $J_{sc}$.\n\\begin{align}\n\\eta = \\frac{\\beta_{FF}V_{oc}J_{sc}}{P_{\\mathrm{solar}}}\n\\end{align}\nwhere $P_{\\mathrm{solar}}$ is the total incident solar power per unit area based on the Air Mass (AM) 1.5 solar spectrum \\cite{1_astm_2012, 2_RReDC_2014}. The fill factor is the ratio of power output at the maximum power point to the product of the open circuit voltage and the short circuit current. The fill factor is estimated to be $0.65$ from literature. The $V_{oc}$, in units of V, and $J_{sc}$, in units of A\/$\\mathrm{m}^2$, are estimated in the limit of 100\\% external quantum efficiency as\n\\begin{align}\nV_{oc} &= \\frac{1}{e}\\left( E_{opt}^d - \\Delta E_{CBM} - 0.3 \\right) \\label{eq:voc}\\\\\nJ_{sc} &= e\\int_{E_{opt}^d}^\\infty \\frac{(\\hbar \\omega)}{\\hbar \\omega} d \\hbar \\omega\n\\end{align}\nwhere $e$ is the elementary charge, $E_{opt}^d$ is the donor optical gap, $\\Delta E_{CBM}$ is the conduction band offset and $P(\\hbar \\omega)$ is AM 1.5 solar spectrum. In equation \\ref{eq:voc}, the constant 0.3 eV is an empirical parameter that estimates losses due to energy conversion kinetics.\n\n\\begin{figure}[htp]\n\\centering\n\\includegraphics[width=0.7\\textwidth]{Fig5.eps}\\\\\n\\caption{(Color online) Power conversion efficiency of potential excitonic thin film solar cell heterojunctions. The labels indicate the material that complements phosphorene.}\n\\label{fig7}\n\\end{figure}\n\n\n\nUsing this model the TMDs are paired with phosphorene. The material with the lower CBM is the acceptor and the material with the higher CBM is the donor. Phosphorene is the donor when paired with both octahedral TMDs and the acceptor when paired with trigonal prismatic TMDs. The maximum PCE values for these eight heterostructures are marked on \\textbf{Fig. 5}. Of the eight heterostructures, phosphorene-ZrS${_2}$ and MoTe${_2}$-phosphorene have the highest PCE value of 12\\%. This efficiency is higher than that achieved by existing excitonic solar cells, and comparable to the proposed 2D g-SiC$_2$\/GaN (14.2\\%), PCBM\/CBN (10-20\\%)\\cite{Bernardi2012ACS}, and bilayer-phosphorene\/MoS$_2$ (16-18\\%) solar cells.\n\n\\begin{figure} [htp]\n\\centering\n\\includegraphics[width=0.8\\textwidth]{Fig6.eps}\\\\\n\\caption{(Color online) (a) The band structure of phosphorene under applied 2\\% armchair compressed strain. The Fermi level is set on the top of the valance band. The dashed black lines are the band structure of phosphorene without strain for reference. (b) The schematic diagram of bilayer strained phosphorene-free phophorene heterostructure. The strain of the 1$^{\\textrm{st}}$ layer can be induced by the substrate, whereas the 2$^{\\textrm{nd}}$ layer is unstrained because of the weak van der Waals interaction.}\n\\label{fig7}\n\\end{figure}\nActually, a further observation based on this model is that a solar cell with a phosphorene donor could have maximum PCE values of up to about 20\\% with an appropriate choice for the acceptor. The conduction band offset (CBO) between phosphorene and TiS${_2}$ is 0.63 eV while the CBO between phosphorene and ZrS${_2}$ is 0.56 eV for the two cases here where monolayer phosphorene is the donor. This translates to a significant drop in $V_{oc}$ in the model which results in a lower PCE. In addition to looking out for new materials that have a better band alignment with phosphorene, means of tuning the properties of both the donor and acceptor can be considered. For example, strained phosphorene may be used as the acceptor material because the strain effect is a well-known method to tune the band structure of materials \\cite{Fei2014NL,Fei2015PRB}. The band structures of different strained monolayer phosphorene are shown in \\textbf{Supplementary Materials}. Here, we choose the phosphorene with 2\\% compressed strain (along the armchair direction) as a donor. Its band structure is shown in \\textbf{Fig. 6}. For comparison, the band structure of phosphorene without strain is also presented in the same figure. As can be seen, the position of CMB can be effectively tuned by the strain effect. 2\\% compressed armchair strain can shift the CBM down to the Fermi level around 0.11 eV because the compressed strain enhances the interaction of hybridized $s-p$ orbitals of P atoms, which contribute to the CBM. The calculated PCE value of -2\\% strained-phosphorene\/phosphorene can be around 20\\% as shown by the white solid circle in \\textbf{Fig. 5}. Based on the theoretical studies of mechanical properties of monolayer phosphorene, the mechanical stability of phosphorene can be up to under 30\\% strain \\cite{Wei2014APL,Peng2014PRB}. The 2\\% compressed strain of a monolayer phorphorene can be realized by the substrate effect in the experiment [see Fig. 6(b)]. Meanwhile, there is no strain on the second deposited monolayer phosphorene because of the weak van der Waals interaction between the two phosphorene layers. This can realize the strain\/non-strain phosphorene heterostructure.\n\nAlternatively multilayer structures of phosphorene or TMDs may also be considered to increase the overall power conversion per unit area. Our calculated optical gap of phosphorene is 1.6 eV which is at the edge of the infrared region. Therefore, heterostructures with a phosphorene donor may be considered for multijunction solar cells absorbing photons across the entire visible spectrum. Such cells incorporate several junctions that aim to absorb different portions of the solar spectrum so as to maximise total absorption [See \\textbf{Supplementary Materials}]. Given the anisotropy of phosphorene, the stacking orientation in multilayer structures may be a significant factor \\cite{Dai2014JPCL}.\n\n\n\n\\section{CONCLUSIONS}\nIn conclusion, through GW calculations, the band structures and optical spectra of monolayer phosphorene have been calculated. The electronic gap of 2.15 eV and the optical gap of 1.6 eV are desirable for solar cell applications because of the strong exciton binding energy. When paired with ZrS${_2}$ or MoTe${_2}$ the power conversion efficiency of the excitonic solar cells can be as high as 12\\%. There is further potential to improve the PCE of phosphorene based solar cells substantially by tuning the materials, such as through the strain and multi-stacking effect, to achieve a better band alignment.\n\n\n\\section{ACKNOWLEDGEMENT}\n\n\nAuthors thank Minggang Zeng for his helpful discussion. This project was supported by the Computational Condensed Matter Physics Laboratory, NUS. Computational resources were provided by Centre for Advanced 2D Materials, NUS.\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\label{sec:intro}Introduction}\nModern proton driven accelerator applications, such as neutron spallation sources and high energy hadron colliders, demand increasingly higher beam currents.\nA common solution is to inject \\ensuremath{\\text{H}^-\\ } ions via the charge exchange process, which overcomes the limit imposed by Liouville's theorem in circular machines~\\cite{dimov1996}.\nSuch high beam currents present a challenge for conventional, invasive beam diagnostics, which may be damaged by the high beam powers.\nContinuous online monitoring of the beam parameters also requires diagnostics that has minimal influence on the beam.\nFor this purpose, a laserwire provides an inherently indestructible and essentially non-invasive probe, that replaces the mechanical counterpart, such as a wire or slit.\nThe laser neutralizes a small fraction of the \\ensuremath{\\text{H}^-\\ } beam via the photo-detachment process, generating free electrons and \\ensuremath{\\text{H}^0\\ } atoms.\nScanning the laser transversely across the particle beam and measuring either of the products of the interaction provides information on the beam properties.\\\\\nLaserwire diagnostics for measuring an \\ensuremath{\\text{H}^-\\ } beam were originally built at Los Alamos National Laboratory~\\cite{Cottingame1985, Connolly1992}, and subsequently developed at facilities including the Brookhaven National Laboratory LINAC~\\cite{bnl93}, and the Spallation Neutron Source at Oak Ridge National Laboratory~\\cite{Liu2010,Liu2012}, and more recently considered for future applications such as the Front End Test Stand (FETS) at Rutherford Appleton Laboratory, UK~\\cite{Gabor2005, Gibson2013}.\\\\\nExisting systems typically use lasers with peak powers above the megawatt level and free-space beam transport. Such laser systems need frequent maintenance, can be susceptible to stability and alignment issues, and also require a high level of safety measures. In contrast, the laserwire system presented in this paper has been designed to use a fibre laser with comparatively low pulse power in the kilowatt range. By transporting the laser light in an optical fibre to the interaction region the instrument is compact, has a stable output in terms of spatial and temporal properties and is unaffected by mechanical vibrations or misalignments. This has been demonstrated in \\cite{Liu2013}, with a picosecond laser used to measure the longitudinal beam profile. The instrument developed for the LINAC4 at CERN will be used to measure the transverse beam emittance.\\\\\nLINAC4 is currently being commissioned in the framework of the LHC (Large Hadron Collider) Injector Upgrade (LIU)~\\cite{Hanke_LIU}. It will replace the existing proton accelerator, LINAC2, as the injector to the Proton Synchrotron Booster (PSB). After its completion expected by the end of 2016, the 90\\,m long linac will produce an \\ensuremath{\\text{H}^-\\ } beam with mean current of 40~mA and energy of 160\\,MeV \\cite{TDR2006}. The transverse emittance of the LINAC4 beam must be monitored during operation to optimize injection into the PSB. Since conventional invasive diagnostics are unsuited to continuous online monitoring of the LINAC4 beam, instead, the development of a non-invasive laserwire instrument was proposed~\\cite{Hofmann2013}. To achieve this goal, a collaboration between the CERN Beam Instrumentation group and the UK FETS beam diagnostics team was established~\\cite{Gibson2013}. As the first step in this ongoing development, a laserwire emittance scanner has been prototyped and tested at the 3\\,MeV beam energy during the first commissioning phase of LINAC4.\\\\\nThis paper describes the configuration of the laserwire prototype and presents first results from transverse emittance measurements of the 3\\,MeV beam. The instrument design is first reviewed in Sec.~\\ref{sec:design}, which includes simulation studies that instruct the system requirements. In Sec.~\\ref{sec:exp_setup} the installed setup is detailed and the essential components are characterized in Sec.~\\ref{sec:calibration}. The emittance measurement results are presented and compared with a conventional diagnostic in Sec~\\ref{sec:results}. Finally, the paper provides an overview of future developments.\n\\section{\\label{sec:design}Instrument design}\n\\subsection{\\label{sec:concept}Principle of operation}\nThe operational principle of a laserwire instrument is to cross an accelerated beam of particles with a laser beam, such that the charged particles and photons can interact in a finely controlled overlap region, as shown schematically in Fig.~\\ref{fig:concept}. The fundamental process exploited for \\ensuremath{\\text{H}^-\\ } beam diagnostics is photo-detachment; the weakly bound outer electron in the \\ensuremath{\\text{H}^-\\ } ion is permanently ejected if a sufficiently energetic photon is absorbed. The binding energy for the outermost electron is very low (0.75\\,eV), and so is detached by photons with a wavelength of less than 1.65\\,$\\mu$m. Only a small transverse section of the particle bunch is sampled, as the laser is stepped through a series of y-positions. The distribution of the particles neutralized (\\ensuremath{\\text{H}^0\\ } atoms) at each laser position is measured by a downstream detector, after being separated from the main beam by a bending dipole magnet. Merging each measured \\ensuremath{\\text{H}^0\\ } profile with the vertical laser position provides the necessary information to reconstruct the transverse emittance of the \\ensuremath{\\text{H}^-\\ } beam. In our setup we conducted measurements only the vertical plane. The horizontal plane could be obtained likewise with the laser and detector setup 90$^{\\circ}$ rotated.\\\\\n\n\\begin{figure} [!hb]\n\\begin{center}\n\\includegraphics{Fig01_LW_Principle.eps}\n\\caption{Schematic principle of a laserwire emittance scanner. A fraction of the \\ensuremath{\\text{H}^-\\ } particle beam collides with photons of a focused laser beam. The neutralized \\ensuremath{\\text{H}^0\\ } are detected by a downstream detector after being separated from the main beam by a dipole magnet. By measuring the \\ensuremath{\\text{H}^0\\ } profile at the detector plane for different laser y-positions, the angular spread and ultimately the transverse phase-space can be reconstructed.}\n\\label{fig:concept}\n\\end{center}\n\\end{figure}\nThe dependence of the photo-detachment cross-section on the wavelength of the incident photon~\\citep{detach} is shown in Fig.~\\ref{fig:cross}. The probability of ejecting one electron by interacting with a laser photon in the visible or near infrared region is generally high so the signal produced by the laserwire will be robust even when relatively low power laser pulses are used. However, because of the very low binding energy of the outer electron in the \\ensuremath{\\text{H}^-\\ }, the particles can be easily neutralized also by means of other processes such as black body radiation, magnetic fields or collisions with residual gas atoms, the latter being the dominant source of background \\cite{cheymol2011}. In the next subsection, the processes of photo-detachment and collisions with the residual gases are considered quantitatively to predict the expected ratio between signal and background.\n\\begin{figure}[!htb]\n\\begin{center}\n\\includegraphics{Fig02_Cross_Section.eps}\n\\caption{Cross-section of the photo-detachment process of a non-relativistic \\ensuremath{\\text{H}^-\\ } ion. The dashed line indicates the selected laser wavelength for the emittance measurements at 3\\,MeV.}\n\\label{fig:cross}\n\\end{center}\n\\end{figure}\n\\subsection{Theoretical models and simulations}\nA theoretical framework of the laserwire interaction was developed to evaluate the expected performance of the laserwire instrument and establish the range of key parameters required for the laser source and detection system. The theoretical framework consists of a model of the laser photo-detachment signal, a model of the residual gas background and a comparative simulation.\n\\subsubsection{Photo-detachment model}\nThe probability that the outer electron is stripped from an \\ensuremath{\\text{H}^-\\ } ion during exposure to laser radiation is modeled by~\\cite{sns2002}:\n\\begin{equation}\\label{eq:stripping_probability}\n\\mathbb{P}_{Laser}=1-\\exp\\left(-\\sigma(E_{CM})\\rho(x,y,z)t\\right),\n\\end{equation}\nwhere $\\sigma(E_{CM})$ is the photo-detachment cross-section, $\\rho(x,y,z)$ is the laser photon density and $t$ is the time of interaction. The cross-section in~Eq.~\\ref{eq:stripping_probability} depends on the center-of-mass energy, \\begin{equation}\\label{eq:cm_energy}\nE_{CM}=\\gamma E_L\\left(1-\\beta cos\\Theta_L\\right),\n\\end{equation}\nwhich is a function of the laser energy $E_L$, the Lorentz factor  $\\gamma=1\/\\sqrt{1-\\beta^2}$, the relativistic velocity ratio of the \\ensuremath{\\text{H}^-\\ } ions $\\beta=v\/c$, and the angle $\\Theta_L$ between the laser and particle beams. In the case of the 3\\,MeV \\ensuremath{\\text{H}^-\\ } beam at LINAC4, the Lorentz factor is small enough that relativistic corrections have negligible impact on the cross-section. For the final laserwire system that will measure a 160\\,MeV beam, the effective value of the selected laser wavelength will be relativistically shifted close to the peak in the photo-detachment cross-section of Fig.~\\ref{fig:cross}.\n\nThe model simulated the stripping interactions that occur when an input distribution of \\ensuremath{\\text{H}^-\\ } ions cross the laser beam. Equation~\\ref{eq:stripping_probability} was applied to each \\ensuremath{\\text{H}^-\\ } ion, taking into account the time spent by the ion as it traverses the varying photon density of the laser beam. Thus the portion of ions stripped by the laser was determined, as detailed in~\\cite{Hofmann2013}. The simulations were performed for one laser pulse travelling in the $x$-direction that is geometrically centered in $y$ on the \\ensuremath{\\text{H}^-\\ } beam. The trajectories of \\ensuremath{\\text{H}^0\\ } particles produced in this overlap region were propagated 3\\,m downstream to the detector plane and integrated, such that the expected measurable profile of particles could be plotted.\n\\subsubsection{Neutralization due to collisions with residual gas}\nThe probability of ejecting one electron from an \\ensuremath{\\text{H}^-\\ } ion by collision with a stray particle present in the vacuum pipe is\n\\begin{equation} \n\\mathbb{P}_{Gas}=1-e^{-l\/\\lambda}\\ {\\rm with}\\ \\lambda=\\frac{kT}{\\sigma P};\\,\\sigma\\sim\\frac{1}{\\beta^2_{rel}}.\n\\label{eq:gas_strip}\n\\end{equation}\nThis stripping probability $\\mathbb{P}_{Gas}$ is calculated from the interaction length $l$, the beam pipe temperature $T$, and the pressure $P$ of the residual gas, which consists mainly of hydrogen. The dependence of the collisional detachment cross-section, $\\sigma$, on the beam energy was derived from the literature~\\citep{gas_lebt,gas_mev}.\n\\subsubsection{Signal and background simulations}\nThe parameters used to calculate the distribution of photo-detached ions at the detector plane are listed in Tab.~\\ref{tab:sim_laser}.\nOf particular interest was the calculation of the signal produced by a laser with a relatively low peak power (in the kW range) that could be transported in optical fibre. The laser pulse duration was also chosen in a way that the produced signal could be realistically detected and digitized without bandwidth restrictions. The value for the \\ensuremath{\\text{H}^-\\ } beam current and the beam size were set to reflect real conditions of the LINAC4 machine during commissioning.\\\\\nThe simulations for the background distribution in the plane of the \\ensuremath{\\text{H}^0\\ } detector were performed using gas pressures and beam dynamics data from the \\ensuremath{\\text{H}^-\\ } source to the bending magnet, through the accelerator components described in Sec.~\\ref{sec:linac4},\nTab.~\\ref{tab:conditions_backg} summarizes the parameters for the background simulation. $\\mathbb{P}_{Gas}$ was calculated in 2\\,cm intervals along the beam axis and the stripped portion was propagated to the \\ensuremath{\\text{H}^0\\ } detector. \n\\begin{table}[h]\n\\begin{center}\n\\caption{Photo-detachment signal simulation parameters.}\n\\label{tab:sim_laser}\n\\begin{tabular}{l c r}\n\\hline\\hline\nParameter & Value & Units\\\\\n\\hline\nLaser pulse energy          &67          &$\\mu$J \\\\\nLaser pulse length (2$\\sigma$)          &106         &ns \\\\\nLaser waist radius          &76          &$\\mu$m \\\\\nIon beam current            &8.5        &mA \\\\ \nVertical ion beam size      &1.01       &mm\\\\\nDistance from the IP to the detector\t\t&3.4    &m\\\\ \n\\hline\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\\begin{table}[htbp]\n  \\centering\n  \\caption{Residual gas background simulation parameters.}\n    \\begin{tabular}{lcccr}\n    \\hline\n    \\hline\n    Parameter               &LEBT   &RFQ    &Chopper    &Units\\\\\n    \\hline\n\tLength                  & 2.0   & 3.0   & 3.7   & m  \\\\\n    Residual gas pressure   & 20    & 10    & 1     & 10$^{-10}$\\,bar  \\\\\n    Min. aperture radius    & 50    & 3.54  & 6.2   & mm  \\\\\n    Mean pulse current      & 13    & 11    & 8.5   & mA  \\\\\n    \\hline\n    \\hline\n    \\end{tabular}%\n  \\label{tab:conditions_backg}%\n\\end{table}\nThe results of both the simulations of signal and background are summarized in Fig.~\\ref{fig:laser_backg}. \n\\begin{figure}[!b]\n\\begin{center}\n\\includegraphics{Fig03_Signal_Bkg_Simulation.eps}\n\\caption{Flux of \\ensuremath{\\text{H}^0\\ }  arriving at the detection plane from laser stripping in the center of the \\ensuremath{\\text{H}^-\\ } beam - in average while laser pulse duration - (dots); \\ensuremath{\\text{H}^0\\ } flux from the residual gas background - permanently while macro-pulse - (dashed line).}\n\\label{fig:laser_backg}\n\\end{center}\n\\end{figure}\nThe dotted curve in the plot shows the resulting \\ensuremath{\\text{H}^0\\ } density in the detector plane produced by a laser pulse crossing the  3\\,MeV \\ensuremath{\\text{H}^-\\ } beam center - in average while laser pulse duration - and the dashed curve corresponds to the flux of \\ensuremath{\\text{H}^0\\ } produced by residual gas interaction. Due to the time structure of both signals being very different, they can be separated in frequency domain without difficulty. In fact, being the signal produced only during the interaction with the pulsed laser, it will have a similar duration, whereas the background, as it is generated all along the beam-pipe, will be composed of \\ensuremath{\\text{H}^0\\ } s with kinetic energies ranging from 45\\,keV to 3\\,MeV and it arrives at the detector plane unbunched and with a duration comparable with the particle macro-pulse. \n\n\\subsection{\\label{sec:requirements}System requirements}\nAnalyzing the simulation result, summarized in Fig~\\ref{fig:laser_backg}, one can derive requirements for the laser and detector system.\\\\ Tab.~\\ref{tab:laser_req} gives an overview of the requirements for the laser system. The wavelength is for our system not very critical, since the cross-section is very broad (see Fig.~\\ref{fig:cross}). Therefore we aimed for 1064\\,nm laser where a wide choice of lasers are commercial available. To reach a signal level which is significant above the background level one needs a laser with peak power in the kilowatt range. To relax the bandwidth requirements for the detector, a laser pulse-length of at least 10\\,ns is required. With an repetition rate of more than 10\\,kHz, synchronized to the LINAC4 timing one can sample the 400\\,$\\mu$s \\ensuremath{\\text{H}^-\\ } pulse multiple times. \\\\\nTo realize a laserwire scanner, a nearly constant diameter of the laser beam which is a factor of 10 smaller than the \\ensuremath{\\text{H}^-\\ } beam diameter is the pre-condition. Taking the \\ensuremath{\\text{H}^-\\ } beam $\\sigma$ of 1...3\\,mm, a laser beam with a beam quality factor ($M^2$) smaller than 3 is needed. E.g. in our simulation we used an laser with $M^2$ = 1.8 and achieved a waist radius of 76\\,$\\mu$m with a Rayleigh length of 6.5\\,mm. \nThis will be discussed more in detail in section~\\ref{sec:inst_setup}\\\\\n\n\\begin{table}[h]\n\\begin{center}\n\\caption{Requirements for the laser system}\n\\label{tab:laser_req}\n\\begin{tabular}{lcr}\n\\hline\\hline\nParameter                           &Requirement        &Units\\\\\n\\hline\nWavelength                          & 900 $\\pm$ 300     &nm\\\\\nPulse peak power                    & 1...10            &kW\\\\\nPulse length                        & 10...100          &ns\\\\\nRepetition rate                     & $>$ 10            &kHz\\\\\nSynchronisation to LINAC4           & Required          &--\\\\\t\nBeam quality factor ($M^2$)         & $<$ 3             &--\\\\\n\\hline\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\nRegarding the \\ensuremath{\\text{H}^0\\ } detector the time resolution is a key-factor. It needs to resolve a laser pulse so that the signal is not dominated by residual gas background but laser stripped \\ensuremath{\\text{H}^0\\ }. A bandwidth of min. $20$\\,MHz is required to record the beam pulse with negligible distortion \\cite{hofmann15}. To achieve a reasonable electrical signal the detector also needs to have internal gain. This requirement follows from simulation results shown in Fig.~\\ref{fig:laser_backg}. As the \\ensuremath{\\text{H}^0\\ } density at the detector will be in the order of 10~$\\ensuremath{\\text{H}^0\\ } ns^{-1}\\,mm^{-2}$, it corresponds to only nanoampere currents.\\\\\nOn the same time, the background flux of approx. 1~$\\ensuremath{\\text{H}^0\\ } ns^{-1}\\,mm^{-2}$ can create significant damage in detector materials integrated over the LINAC4 pulse length of 400\\,$\\mu$s. For this reason the radiation resistance of the detector is also an important factor.\n\\section{\\label{sec:exp_setup}Experimental setup}\n\\subsection{\\label{sec:linac4}Overview of LINAC4}\n\nThe final layout of the LINAC4  will consist of a 45\\,keV \\ensuremath{\\text{H}^-\\ } ion source, a radio frequency quadrupole (RFQ), which accelerates the beam to 3\\,MeV, and a chopper. A drift tube linac (DTL), a cell-coupled drift tube linac (CCDTL) and a Pi-mode structure (PIMS) will be then used to accelerate the beam to its final energy of 160\\,MeV\\,\\cite{l4}. During the LINAC4 construction, the beam commissioning is taking place in stages and at every stage the beam has to be fully characterized. A schematic block diagram of the machine is shown in Fig.~\\ref{fig:l4} and the design parameters of the LINAC4 are summarized in Table\\,\\ref{tab:l4params}.\n\\begin{figure}[!hb]\n\\begin{center}\n\\includegraphics{Fig04_L4_Block_Diagram.eps}\n\\caption{LINAC4 block diagram.}\n\\label{fig:l4}\n\\end{center}\n\\end{figure}\n\\begin{table}[htb!]\n\\caption{\\label{tab:l4params} LINAC4 Parameters}\n\\begin{tabular}{l c c r}\n\\hline \\hline\nParameter             & Symbol             & Value               & Units\\\\\n\\hline\nOverall linac length        \t\t\t& $\\text{L}$            \t\t\t& 90    & m \\\\\nOutput energy               \t\t& $\\text{E}$            \t\t\t& 160   & MeV \\\\       \nBunch frequency       \t\t& f$_{\\text{bunch}}$    \t& 352.2 & MHz \\\\\nPulse length           \t\t& t$_{\\text{pulse}}$    \t\t& 400   & $\\mu$s\\\\\nPulse repetition rate          & t$_{\\text{rep}}$       & 1.2   & s\\\\\nAverage pulse current       \t\t& I$_{\\text{pulse}}$    \t\t& 40    & mA\\\\\nBeam transverse emittance\t\t& $\\varepsilon$         \t\t& 0.4   & $\\pi$~mm~mrad\\\\\n\\hline\\hline\n\\end{tabular}\n\\end{table}\n\\subsection{\\label{sec:3mev_setup}Diagnostic setup at 3\\,MeV}\nThe laserwire system was integrated into a movable test bench that was used to measure the beam parameters after the chopper, during the commissioning phase at 3\\,MeV. The test bench also contained various diagnostic tools such as beam position monitors, beam current transformers (BCT), slit and grid emittance scanner etc. for full characterisation of the beam at different commissioning stages. The schematic diagram of the test bench is illustrated in Fig.\\,\\ref{fig:test_bench} (right hand side).\\\\\n\\begin{figure*}[!tbh]\n\\centering\n\\includegraphics*[width=2\\columnwidth]{Fig05_L4LW_Test_Bench.eps}\n\\caption{Layout of the \\SI{3}{MeV} diagnostics test bench as it was installed after the Chopper, including the operational emittance meter (slit and grids) and the laserwire emittance scanner prototype.}\n\\label{fig:test_bench}\n\\end{figure*}\nThe laser and \\ensuremath{\\text{H}^0\\ } monitoring subsystems, which will be described in detail in the next sections, have been installed as close as possible to the slit and wire grids respectively in order to facilitate the comparison between the two methods. The spectrometer dipole in between the slit (and laser) and the wire grids (and \\ensuremath{\\text{H}^0\\ } detector) was designed to steer the beam (during periods not used for slit-grid measurements) into the spectrometer line for absolute and relative energy measurements via horizontal profile measurements at the end of this line. Powering such a dipole allows separating the stripped \\ensuremath{\\text{H}^0\\ } (to be measured in the straight line by the diamond detector) from the un-stripped \\ensuremath{\\text{H}^-\\ } particles and thus performing the laserwire measurements.\\\\\nSimulations of the beam envelope with nominal quadrupole settings resulted in an expected beam size in the plane of the laser of $\\sigma_x = 2.6$\\,mm and $\\sigma_y = 1.4$\\,mm.\n\\subsection{\\label{sec:laser_setup}Laser system and optical layout}\nThe laser is a Q-switched, diode pumped, Yb doped fibre Master Oscillator Power Amplifier (MOPA) manufactured by ``Manlight S.A.S'' (Lannion, France), model: ML-30-PL-R-TKS. The oscillator generates pulses with approximately 80~ns width (FWHM) at wavelength of 1080~nm and a repetition rate selectable between 30 and 100~kHz.\n\nThe fibre amplifier is pumped by a laser diode that can run either in continuous wave (CW) or can be driven by an external signal with a repetition rate of up to 5~kHz and an adjustable duty cycle, which enables the train of amplified pulses to be synchronized to an external source. The maximum average power of the laser system is approximately 28~W when pumped in continuous wave (CW) regime and the maximum pulse peak power about 8.5~kW.\n\nIn order to optimize the duty cycle of the laser, and therefore use a lower average power, the pump diode was synchronized to the LINAC4 macropulse. Also, the pulse duration was set to 1 ms in order to comfortably overlap the laser amplified pulses with the 400 $\\mu$s long macropulses. A detailed overview of synchronization between the laser pulses and the accelerator bunches is shown in Fig.~\\ref{fig:timing}, including a zoomed view of the overlap between the single laser pulse and the ions micro-bunches.\n\n\\begin{figure}[!htb]\n\\begin{center}\n\\includegraphics{Fig06_Timing_Diagram.eps}\n\\caption{Timing diagram of the laserwire system.}\n\\label{fig:timing}\n\\end{center}\n\\end{figure}\nThe transport system designed to convey the laser beam to the interaction point is based on a large mode area (LMA) optical fibre. The schematic of the optical system is shown in Fig.\\,\\ref{fig:test_bench} (left hand side). The coupling and focusing optics were mounted on breadboards and enclosed in two separate interlocked boxes. In the coupling box, the output from the Q-switched MOPA fibre-laser is coupled into a 5~m long optical LMA fibre with a core diameter of 20~$\\mu$m. In order to continuously monitor the laser power participating to the photo-neutralization a fast photodiode with sub-nanosecond risetime is set inside the coupling box to pick up a parasitic signal. The laser delivery system is setup just in front of the beam-pipe, which is accessible to the laser via  a anti-reflexion coated vacuum window. The output face of the fibre together with the focus optics are set on a small plate and mounted on a stack of two automated translation stages with 50 mm range and $\\sim$1~$\\mu$m step resolution, one set to scan the laser beam vertically (scanning stage or Y-stage) and a second one to allow longitudinal positioning of the laser focus within the vacuum pipe (longitudinal stage or X-stage). The focus optics consist of collimation lens with 6.24~mm focal length, a remotely controllable variable beam expander that has a range of magnification adjustable between 1x and 8x and a focusing lens with focal length of 500 mm that focuses the laser radius to $\\sim$75~$\\mu$m at the interaction region, which is much smaller than the particle beam size to be scanned.\n\nA secondary optical path is also installed within the focusing box. It is setup in a way that when the Y-stage is set at its lower position, the laser beam is directed towards an optical window that reflects a small portion of light into a CCD camera set on a translation stage so that the laser spatial characteristics can be measured on demand. The larger part of light that is transmitted through this optical window is detected by a photodiode for peak power and pulse duration measurements.\n\n\\subsection{\\label{sec:diamond_setup} Diamond detector system}\nIn section~\\ref{sec:requirements} the requirements for a suitable \\ensuremath{\\text{H}^0\\ } detector were determined. After analysis of different types of detectors we choose a  polycrystalline (pCVD) diamond strip detector for the measurement campaign. This kind of detector is able to measure even single particle events due to its internal gain of about $10^4$ electrons per penetrating \\ensuremath{\\text{H}^0\\ }. The response time in the sub-nanosecond range permits to resolve an arriving laserpulse (FWHM~=~80~ns) with sufficient time resolution. Finally, the radiation hardness of approx. $10^{15}\\,proton\/cm^2$ comparing to $10^{14}\\,proton\/cm^2$ for silicon \\cite{si_vs_dia} is essential.\\\\\nA photo of the detector is shown in Fig.~\\ref{fig:diamond}. The 20~x~20\\,mm, 500\\,$\\mu$m thick pCVD diamond disc is mounted on a ceramics printed circuit board (PCB) while the five 0.2\\,$\\mu$m thick aluminium electrodes on the front side of the detector are bonded to the circuit paths. On the PCBs backside a bias voltage of 500\\,V is applied to readout the charge created in the diamond bulk. Capacitors (1\\,$\\mu$F) in parallel to the detector rapidly recharge the diamond when electrons are read out.\\\\\nThe PCB with the detector is fixed to a fork-shaped support-frame situated inside a 4-way crosschamber. The setup is mounted on a stepping motor which provides vertical movement in a range of 80\\,mm  with 150\\,$\\mu$m resolution.\\\\\nThe signals need to be pre-amplified directly after the vacuum feed-through due to an unexpected low sensitivity of the detector. The effect causing this will be discussed quantitatively in chapter~\\ref{dia_sig}. We were using AC-coupled linear amplifiers with 46\\,dB gain and 100\\,MHz bandwidth. Thereafter the signal from the diamond channels were digitized with a 1 GS\/s oscilloscope.\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width = 7cm]{Fig07_Diamond_Detector.png}\n\\caption{pCVD diamond detector with 5 strip electrodes \\cite{cividec}.}\n\\label{fig:diamond}\n\\end{center}\n\\end{figure}\n\\section{\\label{sec:calibration}System characterisation and measurements}\n\\subsection{\\label{sec:inst_setup}Laser characterisation}\nAs the laser beam is used as a probe to scan the particle beam, a full characterisation of its properties is required.\nTo measure the spatial characteristics of the laser beam we used a CCD camera which was set on a motorized translation stage in our lower level diagnostic optical path (see laser delivery system in Fig.~\\ref{fig:test_bench}). We moved the camera in steps of 1~mm and recorded the beam spot size variation along the longitudinal axis. Fig.~\\ref{fig:m2} shows the laser spot size at the focal plane of the focusing lens. The horizontal and vertical laser diameters extracted from the images are plotted along the direction of propagation in Fig.~\\ref{fig:m2}. The two sets of data were fitted with the laser propagation formula for quasi-Gaussian beams,\n\\begin{equation}\n\\label{eq:m2} \nW(z)=W_{0}~\\sqrt{1+(z \\frac{\\lambda~M^{2}}{\\pi~W_{0}^{2}})^{2}}\n\\end{equation}\nwhere W(z) is the laser spot radius (distance from the center of the distribution to the position where the intensity drops by a factor $e^{2}$), $W_{0}$ is the minimum laser waist, $\\lambda$ is the laser wavelength and $M^{2}$ is a factor which describes the quality of the real beam compared to an ideal $TEM_{00}$ Gaussian beam (for which $M^{2}=1$) \\cite{laser_phy}.\nFor our application the results in the vertical plane are by far more important. They define the resolution for the sampling (in analogy to the slit size). The resulting value for $M^{2}_y$ is 2.0, the laser waist $W_{0y}$ at the focal plane of the lens is approximately 75$\\mu$m (corresponding to a $\\sigma$ of 37.5$\\mu$m) and the Rayleigh length is about 8\\,mm. Summarizing the characteristics and comparing to the \\ensuremath{\\text{H}^-\\ } beam the laser beam can be approximated as a thin cylinder with constant diameter ($\\diameter = 150 \\mu$m).\\\\\nRegarding the characterisation of the longitudinal parameters of the laser pulse we measured the pulse shape before and after the fibre propagation with fast photodiodes. Furthermore we measured the average power before and after the fibre to check the coupling and transport efficiency of the fibre transport. The efficiency of the fibre delivery was found to be about 70\\,\\%\\,$\\pm\\,5\\%$ with higher efficiency at lower laser power.\\\\\nFor the emittance measurements the laser beam pulse energy interacting with the \\ensuremath{\\text{H}^-\\ } beam was measured to be 154\\,$\\pm\\,6\\,\\mu$J.\\\\\n\\begin{figure}[!htb]\n\\begin{center}\n\\includegraphics[width=1\\columnwidth]{Fig08_M2_Measurements.eps}\n\\caption{Measured 2$\\sigma$ width of the laser beam focused by a lens with a focal length $f$ = 500~mm. At the bottom left is shown the laser spot image taken at a position of -3mm, where the minimum vertical size is obtained.}\n\\label{fig:m2}\n\\end{center}\n\\end{figure}\n\\subsection{\\label{sec:diamond_signal}Diamond detector signal examination}\n\\label{dia_sig}\nTo test the signal response of the diamond detector, laser and detector were positioned in order to maximize the signal in one specific channel of the diamond detector.\\\\\nFig.~\\ref{fig:detector_pd_signal_trace} shows in the upper plot the amplified laser pulses, interacting with the \\ensuremath{\\text{H}^-\\ } beam, recorded with a photodiode in the coupling box. In the lower plot, the resulting signal of an amplified channel of the diamond detector during a \\ensuremath{\\text{H}^-\\ } beam pulse is displayed. This signal corresponds to one laser position and one diamond position and each segment represents a 1~$\\mu$s time-interval.\\\\\nComparing the continuous laser pulses with the signal on the diamond detector, one finds a clear correlation just between the segments 4 and 24 where the 400~$\\mu$s long \\ensuremath{\\text{H}^-\\ } macropulse can interact with the laserpulses. In this timeframe the \\ensuremath{\\text{H}^0\\ } created by the laser interaction form sharp peaks in the diamond signal. Furthermore the \\ensuremath{\\text{H}^0\\ } background originating by collisions with the residual gas molecules cause an offset on the diamond.\n\n\\begin{figure}[!t]\n\\begin{center}\n\\includegraphics{Fig09_Detector_Pd_Signal.eps}\n\\caption{Comparison of laser pulses recorded with a photodiode and the amplified signal at the diamond detector.}\n\\label{fig:detector_pd_signal_trace}\n\\end{center}\n\\end{figure}\n\\begin{figure}[!b]\n\\begin{center}\n\\includegraphics{Fig10_Detector_Pd_Signal_Peaks.eps}\n\\caption{Zoom on one of the diamond signals of Fig.~\\ref{fig:detector_pd_signal_trace} (solid - inverted), compared to the corresponding laser pulse (dashed). Dotted trace -- linear fit of the background.}\n\\label{fig:pulses}\n\\end{center}\n\\end{figure}\n\nFig.~\\ref{fig:pulses} shows a zoom to one of the pulses where the diamond signal (solid) and the laser pulse (dashed) are overlaid. The distinct agreement in the shape of the signals demonstrates the linear relation between laser-power and output signal of the diamond detector.\\\\\nTo characterize further the detector response, a comparison was done between the \\ensuremath{\\text{H}^-\\ } beam current recorded shortly behind the laser interaction and the unamplified diamond detector raw signal. Fig.~\\ref{fig:diamond-vs-bct} shows the relation of both signals. The slight increase of the floor of the diamond signal can be explained by the \\ensuremath{\\text{H}^0\\ } atoms created by residual gas collisions. The ratio between the peak signal and baseline value is fairly constant (between 10 and 11). Given the uncertainties of the simulation input (e.g. gas pressure, detector time response etc...), this value compares well to the simulations (see Fig.~\\ref{fig:laser_backg}).\n\n\\begin{figure}[!htb]\n\\begin{center}\n\\includegraphics{Fig11_Det_BCT_Signal.eps}\n\\caption{Unamplified signal from a diamond detector channel sampling the \\ensuremath{\\text{H}^0\\ } particles for a given laser position ($E_{pulse}~=~67~\\mu J; f_{pulse} = 30~kHz$), as function of time along the 3\\,MeV \\ensuremath{\\text{H}^-\\ } beam pulse. This is compared to the beam current measured with a beam current transformer (BCT) downstream the laser station.}\n\\label{fig:diamond-vs-bct}\n\\end{center}\n\\end{figure}\nIn addition the time response of the diamond signal was compared with the BCT signal all along the LINAC4 macropulse. Even though in some cases the signal from the diamond detector reproduced well the beam current evolution~\\cite{linac14}, in most cases, as presented here, the agreement was poor in the second part of the beam pulse.\\\\\nEven though such an occurrence, obviously affecting the overall measurement accuracy, has not been fully understood yet, it can be well related to the implantation of \\SI{3}{\\mega\\eV} protons into the diamond bulk, after approx. 50\\,$\\mu$m of detector material. Simulation results give more than $10^{7}$ protons implanted by the background for one LINAC-pulse. The variation of the electrostatic field inside the diamond could thus lead to a decreased Charge Collection Efficiency (CCE) during the LINAC macropulse. This might also explain the signal amplitude of just a few mV, which is two orders of magnitude lower than expected by calculations not considering the effects of the proton implantation \\cite{hofmann15}. This disturbance is supposed to be negligible for higher beam energies (starting from \\SI{12}{\\mega\\eV}) for which the protons have high probability to traverse and escape the diamond bulk without being stopped. First measurement results at the 12\\,MeV beam show first evidence confirming this hypothesis.\\\\\n\n\n\\section{\\label{sec:results}Profile and emittance measurement results}\n\\subsection{\\label{sec:data}Analysis method}\nTo extract the transverse emittance value from the raw data recorded with the diamond detector an analysis routine was developed and applied to the gathered data. The first action is to define the segments with useful data in the digitized signal (e.g. Seg. 6-23 in Fig.~\\ref{fig:detector_pd_signal_trace}). Hereafter each segment is treated individually. At first a finite impulse response (FIR) low-pass filter (3\\,dB cutoff at 22\\,MHz) is used to suppress high-frequency noise. Now a time-interval is defined, which separates the laserpulse from the background (in Fig.~\\ref{fig:pulses} e.g. 300...700 ns). By fitting linearly the data outside of this window a subtraction of the diamond signal from the background can be conducted. Finally the data in the time-interval of the laserpulse can be integrated.\\\\ \nTo get a mean emittance value along the duration of the LINAC4 macropulse, the integrated values for the selected segments are averaged. The resulting value corresponds to one sample in the transverse phasespace. The referring $y\/y'$ values are calculated by the current detector and laser position.\\\\\nThe RMS transverse emittance can finally be calculated by using these samples in the phasespace as shown in Eq.~\\ref{eq:em_formula}.\n\n\\begin{equation}\\label{eq:em_formula}\n\\epsilon_{y} = \\sqrt{\\langle y^2 \\rangle \\langle y'^2 \\rangle -\\langle yy' \\rangle^2}\n\\end{equation}\nTo extract the vertical beam profile from this data, the values at different $y'$ positions can be integrated for each $y$ coordinate. \n\n\\subsection{\\label{sec:emittance}Profile and emittance reconstruction and comparison with slit~\\&~grid system}\nIn order to measure vertical transverse emittance the following technique has been used. The laser focus was moved vertically across the ion beam with variable step size (0.5~mm at the tails and 0.25~mm around the peak) in order to accurately sample the shape of the distribution. At each laser position, the diamond detector was scanned across the beamlet of neutral particles with steps of 1.81~mm in order to increase the angular resolution. Each detector scan corresponding to certain laser focus position was stored in separate data file. The data from these files then were analysed using the same analysis method as described above. The integrated and averaged signal then was used to plot the phase-space distribution.\\\\\nThe laserwire results were verified with the independent measurements from conventional slit and grid emittance meter. The slit, installed at the same point as the laser IP was scanned across the ion beam in vertical direction. At each vertical position, well defined with respect to the ion beam center, the slit selects a narrow slice of the ion beam. The distribution of the particles transmitted through the slit was measured after 3.4~m drift space by two wire grids~\\cite{slitgrid}. Same as in case of laserwire, by scanning the slit vertically across the ion beam and measuring the profiles of particle beamlets passed through the slit the whole phase-space was reconstructed. Measured phase-space distributions for 3~MeV setup obtained with laserwire and slit and grid methods are presented in Fig.~\\ref{fig:phase_space}.\n\n\\begin{figure}[!htb]\n\\begin{center}\n\\includegraphics{Fig12_Phase_Space_3_Mev.eps}\n\\caption{Vertical transverse phase-space of the 3 MeV \\ensuremath{\\text{H}^-\\ } beam measured by laserwire (top) and slit and grid method (bottom).}\n\\label{fig:phase_space}\n\\end{center}\n\\end{figure}\n\\begin{table}[htbp]\n  \\centering\n  \\caption{Comparison of normalized emittance and Twiss parameters obtained with both methods}\n    \\begin{tabular}{lcc}\n    \\hline\n    \\hline\n                                   &Slit-Grid   &Laser-Diamond\\\\\n    \\hline\n\tNorm. emittance [$\\pi$ mm mrad]   & 0.242   & 0.215\\\\\n\tBeta [m]                        & 0.906   & 0.848\\\\\n\tGamma [$m^{-1}$]                & 1.16    & 1.328\\\\\n\tAlpha                           & -0.225  & -0.348\\\\\n    \\hline\n    \\hline\n    \\end{tabular}%\n  \\label{tab:twiss}%\n\\end{table}\n\\begin{figure}[!htb]\n\\begin{center}\n\\includegraphics{Fig13_LW_PhaseSpace_Profiles.eps}\n\\caption{Vertical profile of the \\ensuremath{\\text{H}^-\\ } beam measured by laserwire (dotted curve) and slit and grid (dashed curve) methods.}\n\\label{fig:profiles}\n\\end{center}\n\\end{figure}\nAs one can see from the plot, despite the lower spatial resolution of the laserwire data in this first test, the ellipse size and orientation in both pictures is in a good agreement. Vertical position of centroids on both pictures are slightly different. It can be explained by the fact that the exact position of the laser focus and the detector center was not perfectly aligned with respect to the center of the ion beam.\\\\\nTable~\\ref{tab:twiss} shows a comparison of the obtained values for the normalized RMS emittance and the twiss parameters. Both methods show reasonable agreement. The differences can be explained by several effects. The implantation effects of the protons in the diamond bulk might have caused some non-linearities in the detector response. Furthermore for the quadrupol settings used, a small percentage of particles might have missed the diamond detector, as it is smaller than the wire-grid. Finally, the noise suppression excluded all values in phase space which are below a certain threshold corresponding to a percentage of the maximum amplitude. Some dependency on the applied threshold also has an influence on the results shown in Table~\\ref{tab:twiss}.\\\\\nIn order to compare as well the vertical beam profiles, projections of the phase-space distributions were calculated. The vertical profile of the ion beam for both laserwire and slit and grid measurements is presented in Fig.~\\ref{fig:profiles}.\nAs one can see, the beam core is in a very good agreement. Values at the tails are slightly bigger when measured with the laserwire system. Also, it should be noted that the beam profiles obtained with two different methods have a non-Gaussian shape. The fact that the vertical profile of the ion beam has a non-Gaussian shape regardless of the measurement methods indicates that such shape of the beam is caused by the processes of beam generation at the source rather than a measurement error.\n\n\\section{\\label{sec:conclusions}Summary and outlook}\n\nIn the last 2 years the development of a laserwire emittance scanner was accomplished. In a first step simulations were conducted to determine the key parameters for laser and detector subsystems. After characterisation of the identified components, a prototype system was designed and installed at the LINAC4 3\\,MeV testbench.\nIn comparison to conventional systems for emittance measurement, the proposed laserwire system has major advantages. Because of its principle, space charge perturbations are excluded. Its range of application starts at beam energies of MeV and reaches beyond Multi-GeV. Since no mechanical parts intercept the beam, it is a reliable and fully non-destructive method.\\\\\nThe chosen laser system with kilohertz pulse repetition rate allows to to probe the emittance value of the \\ensuremath{\\text{H}^-\\ } beam pulse with microsecond resolution. The fibre based laser system increases the reliability of the laser transport system and reduces greatly its complexity.\\\\\nThanks to its diamond-based detector system with high bandwidth and sensitivity the presented system works with 3 orders of magnitude less laser peak-power than previous laser-based  systems, which marks a rise of the state of the art in this field.\\\\\n\nThe prototype system was successfully commissioned at the LINAC4 3\\,MeV beam. A comparison of phase-space and beam profile with the conventional slit and grid technique showed very good agreement. The twiss parameters and normalized emittance showed reasonable agreement taking into account the systematic error sources for this first prototype. \\\\\nThe following commissioning stages at 50~MeV and 100~MeV at LINAC4 will be used to test a modified system for beam profile measurements by collecting the detached electrons. Subsequently further improvements for the final system design will be conducted. Main targets will be the detector resolution and performance as well as the range of the fibre-based laser transport. Finally an operational system is planned to be installed in the 160\\,MeV area of LINAC4 to monitor constantly the transverse emittance of the \\ensuremath{\\text{H}^-\\ } beam at its future operation.\n\n\\section{Acknowledgements}\nWe want to thank the LINAC4 operations team to give us the opportunity to benchmark our system, the FETS collaboration for lending us their laser, Peter Savage and Richard Epsom for support with design and manufacture of final focus assembly, the BI-group at CERN for their engineering support and Benjamin Cheymol and Francesca Zocca for the slit and grid reference measurements. Moreover we acknowledge the support of the Marie Curie Networks LA3NET and oPAC which are funded by the European Commission under Grant Agreement Number 289191 and 289485.\\\\\nEspecially we would like to acknowledge the contribution of Christoph Gabor to this work and the development of laserwire technologies generally, who sadly passed away before this paper could be published. \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec1}\nMany of the results obtained for the blocks of  group algebras are extended to the general situation of $p$-permutation algebras, for instance this is the case of the Brauer pairs that where introduced in this general situation in  \\cite{BrPu}. M. Harris obtained in \\cite{Ha} two category isomorphisms involving generalized Brauer pairs  determined by a $G$-invariant block of  $kN,$ where $N$ is normal in $G.$ We  extend these results to a general class of $p$-permutation algebras. \n\nThroughout the paper $k$ is a field of characteristic $p,$ not necessarily algebraically closed and $N$ is a normal subgroup of the finite group $G.$  Section \\ref{sec2} refers to the definition of saturated triples, while Section \\ref{sec3} offers an extension of a useful result. Section \\ref{sec4} proves the first main result of the paper, that is Theorem \\ref{theorem39}. Sections \\ref{sec5} and \\ref{sec6} gather information on covering points on permutation algebras, that were introduced in \\cite{Co}. Finally, in Section \\ref{sec7} we prove the second main result of the paper, namely Theorem \\ref{theorem63}.\n\nFor basic facts regarding points on $G$-algebras we follow \\cite{Th} and for the language of Brauer pairs in the context of $p$-permutation algebras we refer to \\cite{AsKeOl} and \\cite{BrPu}. We use  methods on lifting idempotents as in \\cite[ Section 3]{Pu}. \n\n\\section{On the definition of saturated triples}\\label{sec2}\nFirst  we collect  without proof some properties about idempotents   of a  $k$-algebra $A,$ as consequences of \\cite[Proposition 3.19]{Pu}.\n\\begin{lem}\\label{Lemma71} Let $A$ be a unital $k$-algebra. Let $e,f$ be two idempotents of $A$.\n\\begin{itemize}\n\\item[(i)] If $e$ is in $Z(A)$ or $f$ is in $Z(A)$  then $ef=fe$  is also an idempotent;\n\\item[(ii)] If $e$ is primitive then $ef=0$ or  there is $a\\in A^*$ with $ef^a=f^ae=e;$\n\\item[(iii)] If $e$ is a primitive idempotent of $A$ and $e\\in Z(A)$ or $f\\in Z(A)$ then either $ef=0$ or $ef=e=fe.$\n\\end{itemize}\n\\end{lem}\n\nThe purpose of the next proposition is to show that  the definition of saturated triple $(A,c,G)$ (see\\cite[Definition 3.2, Part IV]{AsKeOl}) does not require $c$ to be a central idempotent.\n\\begin{prop}\\label{prop22} Let $A$ be a $p$-permutation $G$-algebra and $c$ a primitive idempotent of $A^G$. Let $Q$ be a $p$-subgroup of $G$ and $e$ a block of $A(Q)$. If $e$ is a primitive idempotent of $(A(Q))^{C_G(Q,e)}$ then $(Q,e)$ is a $(A,c,G)$-Brauer pair if and only if $\\operatorname{Br}_Q(b)e=e.$\n\\end{prop}\n\n\\begin{proof}\nIf $(Q,e)$ is a $(A,c,G)$-Brauer pair then $\\operatorname{Br}_Q(b)\\neq 0.$ The $N_G(Q)$-algebra epimorphism $A^Q\\to A(Q)$ gives $\\operatorname{Br}_Q(A^G)=A(Q)^{N_G(Q)},$ hence $\\operatorname{Br}_Q(c)\\in (A(Q))^{C_G(Q,e)}.$ At this point Lemma \\ref{Lemma71} establishes the result.\n\\end{proof}\n\\begin{rem} Recall from  \\cite[Introduction p.92]{KeKuMi} that a triple $(A,c,G)$ is saturated if: $c$ is a central, primitive idempotent of $A^G$ (where $A$ is  a $p$-permutation $G$-algebra); and, for any $(A,c,G)$-Brauer pair $(Q,e)$, the idempotent $e$ is primitive in $A(Q)^{C_G(Q,e)}$. Following carefully the proof of \\cite[Theorem 1.6]{KeKuMi} we noticed that the condition (i) of this theorem, the fact that $c$ is a central idempotent of $A$, is needed in \\cite[Lemma 3.2]{KeKuMi} and \\cite[Lemma 3.4]{KeKuMi}. As a matter of fact in \\cite[Lemma 3.4]{KeKuMi} the authors missed this assumption.  However, by Proposition \\ref{prop22} we can show now that the conclusions of  \\cite[Lemma 3.2]{KeKuMi} and \\cite[Lemma 3.4]{KeKuMi} are true assuming condition (ii) of  \\cite[Theorem 1.6]{KeKuMi}, without assuming that $c$ is central in $A$.\n\\end{rem}\n\\section{Generalized Brauer pairs and $p$-permutation $N$-interior $G$-algebras}\\label{sec3}\nLet $A$ be a $p$-permutation $N$-interior $G$-algebra, where $N\\trianglelefteq G$. Also consider $c\\in A^G$, a primitive idempotent of that algebra.\n\nSimilar to \\cite[Definition 1.6]{BrPu} we recall that\n$(Q,f_Q)$ is an $(A,c,G)$- Brauer pair if $Q$ is a $p$-subgroup of $G$ and $f_Q$ is a block of $A(Q)$ with $\\operatorname{Br}_Q(c)f_Q\\neq 0$. We follow the notations of \\cite{KeKuMi} and \\cite[Part IV]{AsKeOl}.\nNotice  that in an $(A,c,G)$-Brauer pair,  $Q$ is chosen such that $\\operatorname{Br}_Q(c)\\neq 0$. Also $\\operatorname{Br}_Q(c)f_Q$ is an idempotent of $A(Q)^{C_N(Q)}$.\nThe next proposition is an extension of \\cite[Lemma 8.6.4]{Link} and of \\cite[IV, Lemma 3.5]{KeKuMi}, although the proof is similar to those of the mentioned lemmas, for completeness we present it here.\n\\begin{prop}\\label{prop23}\nLet $(Q,f_Q)$ denote an $(A,c,G)$-Brauer pair verifying that $f_Q$ is primitive in $A(Q)^{C_N(Q)}$. Consider a $p$-subgroup $S$ of $G$ and a subgroup $H$ of $G$ with  $C_N(Q)\\leq H\\leq N_G(Q, f_Q)$, such that $Q\\leq S\\leq H.$ Also let $f_S$ be a block of $A(S)$. The following are equivalent:\n\\begin{itemize}\n    \\item[i)] $(S,f_S)$ is an $(A,c,G)$- Brauer pair with $(Q,f_Q)\\leq(S,f_S)$;\n    \\item[ii)] $(S,f_S)$ is an $(A(Q),f_Q,H)$-Brauer pair.\n\\end{itemize}\n\n\\end{prop}\n\\begin{proof}\nSince $A(Q)^H\\subseteq A(Q)^{C_N(Q)}$ it follows that $f_Q$ remains a primitive idempotent in $A(Q)^H$. In \\cite[Theorem 1.8]{BrPu} is stated that $(Q,f_Q)$ is the unique Brauer pair included in $(S,f_S)$ and that $f_Q$ is invariant with respect to the $S$-action. This means $\\operatorname{Br}_S(f_Q)f_S=f_S$.\n\nConversely, by our assumptions we have  $\\operatorname{Br}_Q (c)f_Q=f_Q$ and $\\operatorname{Br}_S (f_Q) f_S =f_S$, see Proposition \\ref{prop22}. Indeed, $\\operatorname{Br}_S:A(Q)^S\\rightarrow A(S)$ is an epimorphism of algebras verifying $\\operatorname{Br}_S((Z(A(Q)^S))\\subseteq Z(A(S))$.\n\nThe commutativity of the diagram\n$$\\begin{xy} \\xymatrix{ A^S\\ar@{->}[rr]^{\\operatorname{Br}_Q}\\ar@{->}[dr]_{\\operatorname{Br}_S} &&A(Q)^S\\ar@{->}[dl]^{\\operatorname{Br}_S}\\\\ &A(S)    } \\end{xy}$$\ngives\n$$\\operatorname{Br}_S(c)f_S=\\operatorname{Br}_S(c)\\operatorname{Br}_S(f_Q)f_S=\\operatorname{Br}_S(\\operatorname{Br}_Q(c))\\cdot \\operatorname{Br}_S(f_Q)f_S = $$\n$$ = \\operatorname{Br}_S(\\operatorname{Br}_Q(c)\\cdot f_Q)f_S = \\operatorname{Br}_S(f_Q)f_S  = f_S$$\nThis shows that $(S,f_S)$ is a $(A,c,G)$-Brauer pair.\n\nApplying again \\cite[Theorem 1.8]{BrPu}, it provides an $S$-invariant block of $A(Q)$, denoted $f_Q'$, which satisfy $(Q, f_Q')\\leq (S,f_S)$ and $\\operatorname{Br}_S(f_Q')f_S = f_S$. If $f_Q\\neq f_Q'$ then $$f_S = f_S \\operatorname{Br}_S(f_Q) = f_S \\operatorname{Br}_S(f_Q)\\operatorname{Br}_S(f_Q')=0,$$ a contradiction.\n\\end {proof}\n\n\\section{Isomorphisms of categories of Brauer pairs} \\label{sec4}\nIn this section we assume that $c$ is a $G$-invariant primitive idempotent of $A^N$ and that $Q$ is a defect group of $c$ in $N$. \n\nFor any defect group $Q$ of $c$ in $N$ there is a defect group $P$ of $c$ in $G$ such that $Q=P\\cap N$. This claim is well known but, for consistency,  we give the justifying details  in the next lines. Indeed, if $Q$ is a defect group in $N$ of $c$, then $c\\in A_Q^N$ and $Q$ is minimal with this property. Similarly, if $R$ is a defect group of $c$ in $G$ then $R$ is minimal with the property $c\\in A_R^G$. Given the decomposition:\n$$A_R^G\\subseteq \\underset{x\\in [N\\setminus {G\/R}]}{\\displaystyle\\sum} A_{N\\cap\\  R^x  }^N,$$\nwe obtain $c\\in A^N_{N\\cap R^x}$, for some $x\\in G$. Since $\\operatorname{Br}_R(c)\\neq 0$ it follows $\\operatorname{Br}_{R^x}(c)\\neq 0$, hence $\\operatorname{Br}_{N\\cap\\ R^x}(c)\\neq 0$. So, there is $y\\in N$ with $Q^y=N\\cap\\ R^x$. It follows $Q=N\\cap\\ R^z,\\ z=xy^{-1}$. We set\n$P =R^z$ and the claim is proved.\n\\begin{rem}\\label{rem32}\nIf we fix a $(A,c,G)$- Brauer pair $(Q,f_Q)$ such that $Q$ is a defect group of $c$ in $N$, the above discussion provides a maximal $(A,c,G)$-Brauer pair $(P,f_P)$ such that\n$$(Q,f_Q)\\leq (P,f_P)^x\\text{ for some } x\\in N.$$\n\\end{rem}\n\\begin{defn}\\label{definition33} Let $(Q,f_Q)$ be a pair as in Remark \\ref{rem32}, such that $f_Q$ remains primitive in $A(Q)^{C_N(Q)}$. Denote $H=N_G(Q,f_Q)$ and define the sets:\n$$F_1:=\\{(R,f_R)|(R,f_R)\\text{ is a }(A,c,G)\\text{-Brauer pair such that}$$\n$$(Q,f_Q)\\leq(R,f_R)\\text{ as }(A,c,G)\\text{-Brauer pairs }\\}$$\n$$F_2:=\\{(R,f_R)|(R,f_R)\\text{ is a }(A(Q),f_Q,H)\\text{-Brauer pair such }$$\n$$\\text{that }(Q,f_Q)\\leq(R,f_R)\\text{ as }(A(Q),f_Q, H)\\text{-Brauer pairs}\\}$$\n\n\\end{defn}\n\\begin{cor}\\label{corollary34}\nAssume $(Q,f_Q)$ is as in Definition \\ref{definition33}. Let $S$ be a $p$-subgroup of $G$ such that $Q\\leq S$ and let $f_S$ be a block of $A(S)$. The following are equivalent:\n\\begin{itemize}\n    \\item[a)] $(S,f_S)\\in F_1;$\n    \\item[b)] $(S,f_S)\\in F_2.$\n\\end{itemize}\n\\end{cor}\n\\begin{proof}\nRemark \\ref{rem32} states that $Q\\trianglelefteq S$. The rest of the proof follows from Proposition \\ref{prop23}.\n\\end{proof}\n\\begin{prop}\n\\label{Proposition35}\nLet $S,T$ be two $p$-subgroups of $G$ and let $f_S,f_T$ denote blocks of $A(S),$ $A(T)$, respectively.\n\nThe following are equivalent:\n\\begin{itemize}\n    \\item[1)] $(S,f_S)\\leq(T,f_T)$ as elements of $F_1$;\n    \\item[2)] $(S,f_S)\\leq(T,f_T)$ as elements of $F_2$.\n\\end{itemize}\n\\end{prop}\n\\begin{proof}\nLet $e$ denote a primitive idempotent  appearing in a pairwise orthogonal  decomposition of $f_T$ into primitive idempotents in $A(T).$ By lifting idempotents, there exists a primitive idempotent $i\\in A^T$  such that $e=\\operatorname{Br}_T(i)$, hence $\\operatorname{Br}_T(i)f_T=\\operatorname{Br}_T(i)$. By our assumption we also have $\\operatorname{Br}_S(i)f_S\\neq 0$. Since $i$ is  primitive in $A^T$, we obtain that $j:=\\operatorname{Br}_Q(i)$ is also a primitive idempotent of $A(Q)^T$. We have\n$$0\\neq \\operatorname{Br}_T(i)f_T=\\operatorname{Br}_T^{A(Q)}(j)f_T.$$\nSimilarly  we obtain $$0\\neq \\operatorname{Br}_S(i)f_S=\\operatorname{Br}_S^{A(Q)}(\\operatorname{Br}_Q(i))f_S.$$\nThis proves that $(S,f_S)\\leq (T,f_T)$ as elements of $F_2$ according to \\cite[Corollary 1.9]{BrPu}.\n\nConversely, considering a similar decomposition of $f_T$ in $A(T)$, we find a primitive idempotent $e$ with $f_Te=f_T$. Let $j$ be such that $\\operatorname{Br}_T^{A(Q)}(j)=e$ and let $i$ be such that $\\operatorname{Br}_T(i)=e$. Then we may assume $i$ and $j$ to be primitive idempotents in $A^T,A(Q)^T$, respectively. Further we get $\\operatorname{Br}_Q(i)=j^a,$ for some unit $a$ in $A(Q)^T.$\nWe obtain:\n$$\\operatorname{Br}_T(i)f_T = \\operatorname{Br}_T^{A(Q)}(j^a)f_T = f_T\\neq 0.$$\nAgain $\\operatorname{Br}_T(i)\\neq 0$ implies $\\operatorname{Br}_S(i)\\neq 0$ and then\n$$0\\neq \\operatorname{Br}_S^{A(Q)}(j^a)f_S=\\operatorname{Br}_S(i)f_S.$$\nThis concludes the proof.\n\\end{proof}\n\\begin{cor}\n\\label{corollary36} Let  $(U,f_U)$ be an $(A,c,G)$-Brauer pair such that $(Q,f_Q)\\leq (U,f_U)$. Then $(U,f_U)$ is a maximal $(A,c,G)$-Brauer pair if and only if it is a maximal $(A(Q),f_Q,H)$-Brauer pair.\n\\end{cor}\n\n\n\nIf $(U,f_U)$ is a maximal $(A,c,G)$-Brauer pair, then $\\mathcal{F}_{(U,f_U)}(A,c,G)$ is a finite category, called fusion system,  introduced in \\cite[Definition 2.3]{KeKuMi}.\nWe will use the same notations for the homomorphisms in  $\\mathcal{F}_{(U,f_U)}(A,c,G)$. If $S,T\\leq U$  then\n$$\\mathrm{Hom}_{\\mathcal{F}_{(U,f_U)}(A,c,G)}(S,T)=\\{c_x: S\\rightarrow T| x\\in G,\\  {}^x(S,f_S)\\leq (T,f_T) \\},$$\nwith $c_x(u)=xux^{-1}$ for any $u\\in S$.\n  The similar notations are used for $\\mathcal{F}_{(U,f_U)}(A(Q),f_Q,H)$. In this case (that is $(U,f_U)$ is maximal), the full subcategory of $\\mathcal{F}_{(U,f_U)}(A,c,G)$  whose elements are those of $F_1$ is denoted by $\\mathcal{C}$. The full subcategory of $\\mathcal{F}_{(U,f_U)}(A(Q),f_Q,H)$  whose elements are those of $F_2$ is is denoted by $\\mathcal{D}$.\n\n\\begin{prop}\n\\label{proposition38}\nLet  $(U,f_U)$ be a maximal  $(A,c,G)$-Brauer pair such that $(Q,f_Q)\\leq (U,f_U)$. If $(S,f_S)\\in F_1$ and $(T,f_T)$ is a Brauer pair of  $\\mathcal{F}_{(U,f_U)}(A,c,G)$ then\nthe set $\\mathrm{Hom}_{\\mathcal{F}_{(U,f_U)}(A,c,G)}(S,T)$ is included in $ \\mathrm{Hom}_{\\mathcal{F}_{(U,f_U)}(A(Q), f_Q,H)}(S,T)$ . \n\\end{prop}\n\\begin{proof}\nSimilarly to the proof of \\cite[Proposition 5]{Ha}.\n\\end{proof}\nBy collecting the results of this section we can prove now the following theorem.\n\\begin{thm}\n\\label{theorem39} The identity functor from $\\mathcal{D}$ to the category $\\mathcal{C}$ provides an isomorphism.\n\\end{thm}\n\n\n\\section{Covering points revisited} \\label{sec5}\nIn this section we assume that our $p$-permutation $G$-algebra $A$ contains $C$ as a direct sumand of $k$-modules, where $C$ is an $N$-interior $G$-algebra, for some normal subgroup $N$ of $G$. We recall the definition of covering idempotents in the general situation, see \\cite[Definition 3.2]{Co}.\n\\begin{defn}\n\\label{definition41} Let $j\\in C^N$ denote a primitive idempotent with defect group $Q$, a $p$-subgroup of $N$. The primitive idempotent $i\\in A^G$ covers $j$ if:\n\\begin{itemize}\n    \\item[$a)$] $i$ admits a defect group $P$ such that $P\\cap N=Q$;\n    \\item[$b)$] there is a primitive idempotent $f$ of $A^N$ with defect group $Q$ such that\n    $$if^a=f^ai=f^a\\text{ and }jf=fj=fi\\text{ for some } a\\in(A^N)^*.$$\n\\end{itemize}\n\\end{defn}\n\\begin{rem}\n\\label{remark42}\nIf $i\\in A^G$ covers $j$ than the entire $(A^G)^*$-conjugacy class of $i$ covers $j$ as in \\cite[Definition 3.2]{Co}. Consequently a point $\\alpha\\subseteq A^G$ covers $j$, if $\\alpha$ contains an idempotent that verifies $a)$ and $b)$ in Definition \\ref{definition41}. Also note that the entire $(A^N)^*$-conjugacy class of $j$ is covered by $\\alpha$.\n\\end{rem}\nOur situation requires  a particular definition of covering idempotents, that is also equivalent to the block covering situation when working with group algebras.\n\\begin{defn}\n\\label{definition43}\nLet $j\\in C^N$ be a primitive idempotent with defect group $Q$ in $N$ and  assume $j$ is $G$-invariant. A point $\\alpha\\in A^G$ covers $j$ if there is $i\\in\\alpha$ and a point $\\epsilon\\subseteq A^N$ with  defect group $Q$ verifying the conditions:\n\\begin{itemize}\n    \\item[$a')$] $ji = ij$;\n    \\item[$b')$] $if = fi = f$ and $jf_1=f_1j=f_1$ for some $f$ and $f_1$ belonging to $\\epsilon$.\n\\end{itemize}\n\\end{defn}\n\\begin{prop}\n\\label{proposition44} Fix a primitive idempotent $j\\in C^N$, admitting a defect group $Q$ and assume $j\\in C^G$. If the point $\\alpha\\subseteq A^G$ covers $j$ as in Definition \\ref{definition43} then $\\alpha$ covers $j$ as in Definition \\ref{definition41}.\n\\end{prop}\n\\begin{proof}\nCondition $b')$ of  Definition \\ref{definition43} implies condition $b)$ of Definition  \\ref{definition41}. We only need to prove that if $a')$ holds then a defect group $P$ of $\\alpha$ gives $P\\cap N = Q$.\\\\\nLet $R$ be a defect group of $\\alpha$. There is $a\\in A^R$ such that\n$$i=Tr_R^G(a)=\\underset{x\\in [N\\setminus G\/R]}{\\sum}Tr_{N\\cap R^x}^N(a^x)\\in\\underset{x\\in[N\\setminus G\/R]}{\\sum}A_{N\\cap R^x}^N$$\nSince $if=fi=f$ the idempotent $f$ belongs to some ideal $A_{N\\cap R^x}^N$, implying $Q^y\\leq N\\cap R^x$ for some $y\\in N$. Then $Q\\leq N\\cap R^g$ for some $g\\in G$. Let $T$ denote a defect group of $j$ as an idempotent of $A^G$, hence $j\\in C_T^G\\subseteq A_T^G$. We also have $ij=ji=i$ in $A^G$, hence  we may assume $R\\leq T$. Next, there is $b\\in C^T$ with\n$$j=Tr_T^G(b)=\\underset{y\\in [N\\setminus G\/T]}{\\sum}Tr_{N\\cap T^y}^N(b^y)\\in\\underset{y\\in[N\\setminus G\/T]}{\\sum}C_{N\\cap T^y}^N$$\nBy our assumption $j\\in C_{N\\cap T^y}^N$ for some $y\\in G$. Since $\\operatorname{Br}_T(j)\\neq 0$ we get $\\operatorname{Br}_{T^y}(j)\\neq 0$ and then $\\operatorname{Br}_{N\\cap T^y}(j)\\neq 0 $. This means $Q^t=N\\cap T^y$ for some $t\\in N$, equivalently $Q = N\\cap T^z$ for some $z\\in G$. At last $$Q\\leq N\\cap R^g\\leq N\\cap T^g = Q^{z^{-1}g},$$ forcing $z^{-1}g\\in N_G(Q)$ and $$Q=N\\cap R^g = N\\cap T^g.$$\n\\end{proof}\n\\begin{cor}\n\\label{corollary45}\nIf $j$ is a primitive idempotent of $C^N$ with defect group $Q$ and $G_j$ is the stabilizer of $j$ in $G$ then there is a defect group $T$ of $j$ in $G_j$ such that $T\\cap N=Q$.\n\\end{cor}\n\\begin{rem}\n\\label{remark46}\nAs it follows from \\cite[Remark 3.1]{Co} there is a point $\\epsilon\\in A^N$ with defect group $Q$ such that $jf = fj = j$ in $A^N$ for some $f\\in \\epsilon$.\nThe justification for introducing  Definition \\ref{definition43} is explained in the next lines. Since $j$ is $G$-fixed we obtain $j\\in C^G\\leq A^G$. The points covering $j$ are some of the $(A^G)^*$-conjugacy classes of primitive idempotents of $A^G$ appearing in a decomposition $j=\\underset{i\\in I}{\\sum}i$ of $j$ into pairwise orthogonal idempotents. All idempotents $i\\in I$ give $ij=ji=i$,  but only  part of them verify $if_1=f_1i=f_1$, for some $f_1\\in\\epsilon$.\n\\end{rem}\n\\begin{cor}\n\\label{corollary47}\nLet $j$ be a primitive idempotent in  $C^N$, admitting a defect group $Q$ and assume that $j$ is $G$-invariant. The point $\\alpha\\subseteq A^G$ covers $j$ if and only if  there exists a point $\\epsilon$ of $N$ on $jAj,$ adimitting with defect group $Q,$    and a primitive idempotent $i\\in\\alpha$ satisfying $i\\in(jAj)^G$ and $if=fi=f$ for some $f\\in\\epsilon$.\n\\end{cor}\n\\begin{proof}\nIf $\\alpha\\subseteq A^G$ covers $j$,  Definition \\ref{definition43} states that there is $i\\in\\alpha$ such that $ij=ji=i$ or, equivalently, $i\\in(jAj)^G$. There is also $\\epsilon'\\subseteq A^N$ with defect group $Q$ such that $if=fi=f$ and $jf_1=f_1j=f_1$ where $f,f_1\\in\\epsilon'$. Then $$jf=jif=if=fj=f,$$ where $f$ and $f_1$ are $(A^N)^*$-conjugates, primitive idempotents of $(jAj)^N$. Let $\\epsilon$ be the point of $(jAj)^N$ which contains $f$. Since $\\epsilon'$ is the unique point of $A^N$ corresponding to $\\epsilon$ via the embedding $$(jAj)^N\\rightarrow A^N$$ and since $f,f_1\\in\\epsilon'$ we must have $f_1\\in\\epsilon$. Clearly $\\epsilon$ has defect group $Q$.\n\nThe converse is immediate by noticing that for any $a\\in((jAj)^N)^*$ the element $b:=a+i-j$ is in $(A^N)^*$ and for any $f\\in\\epsilon$ we have $f^a=f^b$.\n\\end{proof}\n\n\\section{On pairs determined by covering points} \\label{sec6}\nAs in Section \\ref{sec4} we continue with a $p$-permutation $G$-algebra $A$, which contains $C$ as direct summand of $k$-modules. The first term $C$ is an $N$-interior $G$-algebra, for a normal subgroup $N$ of $G$. Let $c$ be a $G$-invariant primitive idempotent of $C^N$. Recall that  a  $( C, c, G)$- Brauer pair $(R,f_R)$ consists of a $p$-subgroup $R$ in $G$ and a block $f_R$ of $C(R)$ such that $\\operatorname{Br}_R(c)f_R\\neq 0$.\n\n\\begin{rem}\n\\label{remark52}\nIf the $p$-subgroup $Q$ of $N$ is a defect group of $c$ and $(Q,f_Q)$ is a $(C,c,G)$- Brauer pair such that $f_Q$ is primitive in $C(Q)^{C_N(Q)}$ then we may also consider the sets $F_1$ and $F_2$ as in Section \\ref{sec3} such that Theorem \\ref{theorem39} still applies.\n\\end{rem}\n\\begin{prop}\n\\label{proposition53}\nLet $\\alpha\\subseteq A^G$ be a point covering $c$. There is a defect group $P$ of $\\alpha$ such that $P\\cap N=Q$ and a block $f_P$ in $C(P)$ such that $(P,f_P)\\in F_1$.\n\\end{prop}\n\\begin{proof}\nBy Proposition \\ref{proposition44} we have a defect group $R$ of $\\alpha$ with $R\\cap N=Q$. We obtain $\\operatorname{Br}_Q(c)f_Q=f_Q$ in $C(Q)^{C_N(R)}$ and then, for any $x\\in N_G(Q)$, we still have $\\operatorname{Br}_Q(c)f_Q^x=f_Q^x$ in $C(Q)$. If we set $H=N_G(Q,f_Q)$ then $s:=\\underset{x\\in N_G(Q)\/H}{\\sum}f_Q^x$ is an idempotent lying in $$C(Q)^{N_G(Q)}\\cap Z(C(Q))$$  and satisfying $$\\operatorname{Br}_Q(c)=\\operatorname{Br}_Q(c)s=s\\  \\operatorname{Br}_Q(c)=s$$ in $C(Q)^{N_G(Q)}$, since $\\operatorname{Br}_Q(c)$ is a primitive idempotent in this algebra. In particular $\\operatorname{Br}_Q(c)$ lies in $$Z(C(Q))\\cap C(Q)^R\\subseteq Z(C(Q)^R),$$ since $Q$ is normal in $R$.\n\n Since $\\alpha$ covers $c$ there is $i\\in\\alpha$ with $ic=i=ci$. Hence $\\operatorname{Br}_R(c)\\neq 0.$ Let $\\operatorname{Br}_R(c)=\\sum f_R$ be a decomposition of $\\operatorname{Br}_R(c)$ into blocks of  $Z(C(R))$. Also consider  $c=\\underset{j\\in J}{\\sum}j,$  a decomposition of $c$ into  pairwise orthogonal primitive idempotents of $C^R$. There is $j\\in J$ giving   $\\operatorname{Br}_R(j)\\neq 0.$  The idempotent $\\operatorname{Br}_R(j)$ is primitive in $C(R)$ and then there is a unique block of $C(R)$, denoted $\\overline{f_R}$, with the properties $0\\neq \\operatorname{Br}_R(j)\\overline{f_R}$ and $0\\neq \\operatorname{Br}_R(c)\\overline{f_R}.$\n\nWe also have $\\operatorname{Br}_Q(j)\\operatorname{Br}_Q(c)\\neq 0$, a relation which determines at least one block, say $f_Q^x$, with $\\operatorname{Br}_Q(j)f_Q^x\\neq 0$. According to \\cite[Corollary 1.9]{BrPu} the blocks $\\overline{f_R}$ of $C(R)$ and $f_Q^x$ of $C(Q)$ must verify $(Q,f_Q^x)\\leq (R,\\overline{f_R})$. Since $x\\in N_G(Q)$ this last relation becomes $(Q,f_Q)\\leq (R^{x^{-1}},\\overline{f_R}^{x^{-1}})$. We set $P:=R^{x^{-1}}$ and $f_P:=\\overline{f_R}^{x^{-1}}$ to conclude the proof.\n\\end{proof}\n\n\n\n\\section{Isomorphism of fusion subcategories} \\label{sec7}\nIn this section $A$ is an interior $G$-algebra  which admits a $G\\times G$-stable $k$-basis.  As before,  $C$ is an $N$-interior $G$-algebra and a direct summand of $A$ as $k$-module. We also assume that $A$ is projective as an $k[N\\times 1]$-module and as an $k[1\\times N]$-module. We keep the assumptions on the block $f_Q.$\n \nLet $X:=\\overline{N}^{\\operatorname{Aut}(Q)}_A(Q)$  denote the extended Brauer quotient, the $N_N(Q)$-interior $N_G(Q)$-algebra constructed in \\cite{PuZh}. Note that, according to \\cite[Proposition 2.2]{CoTo}, there is an action of $N_G(Q)$ on $\\operatorname{Aut}(Q)$. Also let $Y:=\\overline{N}^{\\operatorname{Aut}(Q)}_C(Q)$ denote the extended Brauer quotient constructed in \\cite{CoTo}, associated with the   $N$-interior $G$-algebra $C.$ Notice that $Y$ is an $N_N(Q)$-interior $N_G(Q)$-subalgebra of $X$, in fact $Y$ is a direct summand of $X$ as $k$-modules. Furthermore if $K\\leq \\operatorname{Aut}(Q)$ with $K=N_N(Q)\/C_N(Q)$, then the map\n$$ C(Q)\\otimes_{C_N(Q)}N_N(Q)\\ni \\bar{c}\\otimes n\\mapsto \\overline{cn}\\in \\overline{N}_C^K(Q)$$ is an $N_N(Q)$-interior $N_G(Q)$-algebra isomorphism, making $C(Q)\\otimes_{C_N(Q)}N_N(Q)$ into an $N_N(Q)$-interior $N_G(Q)$-subalgebra of $Y.$ For details see \\cite[Section 2]{CoMa}. Finally if $A=kG=kN\\otimes_NG$ then \\cite[Remark 4.5) 2]{CoMa} states $X=kN_G(Q).$\n\n\\begin{lem}\n\\label{lemma61}\nLet $H = N_G(Q,f_Q)$. Any point of $H$ on $X$ that covers $f_Q$ has a defect group $P$ such that $P\\cap N = Q$ and, there is a block $f_P$ of $C(P)$ such that $(Q,f_Q)\\leq (P,f_P)$ as $(C(Q),f_Q, H)$-Brauer pairs.\n\\end{lem}\n\\begin{proof}\nThe block $f_Q$ of $C(Q)$ remains a primitive idempotent of $C(Q)^L$, for any subgroup $L$ of $H$ containing $QC_N(Q)$. The algebra $C(Q)\\otimes_{C_N(Q)}QC_N(Q)$ is an $H$-acted subalgebra of $Y$, and then $f_Q$ is a primitive idempotent of $(C(Q)\\otimes_{C_N(Q)}QC_N(Q))^{QC_N(Q)}$, according to \\cite[Theorem 3.1]{CoTo}.\n\nWe have $$f_Q\\in C(Q)^{C_N(Q)}=C(Q)_Q^{QC_N(Q)}\\subseteq (C(Q)\\otimes_{C_N(Q)}QC_N(Q))^{QC_N(Q)}_Q$$ and $\\operatorname{Br}_Q^{C(Q)\\otimes_{C_N(Q)}QC_N(Q)}(f_Q)\\neq 0$, implying that $Q$ is a defect group of $f_Q$ in $QC_N(Q)$. Indeed, in fact $\\operatorname{Br}_Q^{C(Q)\\otimes_{C_N(Q)}QC_N(Q)}(f_Q)=f_Q$, since $(C(Q)\\otimes u)(Q)=0$ for any $u\\in Q\\setminus Z(Q)$.\n\nAssuming that for some $u\\in Q\\setminus Z(Q)$ there is an element $\\overline{c}$ belonging to a $N_G(Q)$-stable basis of $C(Q)$ that verifies $(\\overline{c}\\otimes u)^y = \\overline{c}\\otimes u$ for any $y\\in Q$, we obtain $y^{-1}uy=u$ for any $y\\in Q$, a contradiction. Then $$(C(Q)\\otimes_{C_N(Q)}QC_N(Q))(Q)=C(Q),$$ equality which clarifies the last statement. We apply Proposition \\ref{proposition53} for $H,QC_N(Q), X,$ $C(Q)\\otimes_{C_N(Q)}QC_N(Q) $ and $f_Q$, in place of $G,N,A,C$ and $c$ respectively, concluding that for a $p$-subgroup $P$ in $H$  with $P\\cap QC_N(Q)=Q$ and a block $f_P\\in(C(Q)\\otimes_{C_N(Q)}QC_N(Q))(P)$, we obtain $(Q,f_Q)\\leq (P,f_P)$ as\n$(C(Q)\\otimes_{C_N(Q)}QC_N(Q), f_Q,H)$- Brauer pairs. Notice that\n$$(C(Q)\\otimes_{C_N(Q)}QC_N(Q))(P)=(C(Q)\\otimes_{C_N(Q)}QC_N(Q)(Q))(P)=(C(Q))(P)=C(P)$$\nand then\n$(Q,f_Q)\\leq(P,f_P)$ as $(C(Q),f_Q,H)$-Brauer pairs.\n\\end{proof}\n\\begin{prop}\n\\label{proposition62} Any point of $H$ on $X$ with defect group $P$ that covers $f_Q$ determines a point of $G$ on $A$ with defect group $P$ that covers $c$. Moreover, there is a block $f_P$ of $C(P)$ such that $(Q,f_Q)\\leq(P,f_P)$ as $(C(Q), f_Q,H)$-Brauer pairs and as $(C,c,G)$-Brauer pairs.\n\\end{prop}\n\\begin{proof}\nLemma \\ref{lemma61} applies in this situation and yields the $(C(Q),f_Q, H)$- Brauer pair denoted $(P,f_P)$. Theorem \\ref{theorem39} applied for the $G$-algebra $C$ states that $(P,f_P)$ is also a $(C,c,G)$-Brauer pair containing $(Q,f_Q)$. All that remains to prove is the first assertion  of the proposition.\n\nConsider the map\n$$Tr_H^{N_G(Q)}:(f_QX f_Q)^H\\rightarrow(f_Q X f_Q)_H^{N_G(Q)},\\quad a\\mapsto Tr_H^{N_G(Q)}(a).$$\nIt is easily shown that $Tr_H^{N_G(Q)}$ is an algebra isomorphism. Let $\\gamma\\subseteq X^H$ be a point that covers $f_Q$. There is $l\\in (f_QX f_Q)^H\\cap \\gamma$ with defect group $P$ and $P\\cap N=Q$. There is also a primitive idempotent $f$, with defect group $Q$, in $(f_QX f_Q)^{QC_N(Q)}$, giving $lf=fl=f$. Now, the element\n$Tr_H^{N_G(Q)}(l)$ is a primitive idempotent with defect group $P$. Indeed,  otherwise $$Tr_H^{N_G(Q)}(l)\\in\\underset{R\\leq P}{\\sum}(f_QX f_Q)_R^P,$$ and since $f_Q$ is $P$-fixed we get $$l=f_Q\\cdot Tr_H^{N_G(Q)}(l)\\in\\underset{R\\leq P}{\\sum}(f_QX f_Q)_R^P.$$\nThere is $f_1$, a primitive idempotent of  $(f_QX f_Q)^{N_N(Q,f_Q)},$ such that $lf_1=f_1l=f_1$ and $f_1f^a=f^af_1=f^a$ for some $a\\in ((f_QX f_Q)^{QC_N(Q)})^*$. The block $f_Q$ is also a primitive idempotent of $$(C(Q)\\otimes_{C_N(Q)}N_N(Q,f_Q))^{N_N(Q,f_Q)},$$ actually a block of $$C(Q)\\otimes_{C_N(Q)}N_N(Q,f_Q),$$ with defect group $Q$. Since $\\operatorname{Br}_Q(f^a)\\neq 0$ we must have $\\operatorname{Br}_Q(f_1)\\neq 0$. Also $f_1f_Q=f_1$, forcing $f_1$ to be projective relative to $Q$. This shows that $\\gamma$ covers $f_Q$, seen as an $H$-invariant, primitive idempotent of $N_N(Q,f_Q)$ on $C(Q)\\otimes_{C_N(Q)}N_N(Q,f_Q)$.\nNext, we have $f_1Tr_H^{N_G(Q)}(l)=Tr_H^{N_G(Q)}(l)f_1=f_1$. The idempotent\n$Tr_{N_N(Q,f_Q)}^{N_N(Q)}(f_1)$ lies in $(f_QX f_Q)_{N_N(Q,f_Q)}^{N_N(Q)},$\nbeing primitive  with defect group $Q$.\n\nNext, let $T$ denote a set of pairwise orthogonal primitive idempotents of the algebra $(\\operatorname{Br}_Q(c)X \\operatorname{Br}_Q(c))^{N_N(Q)}$ with\n$$Tr_{N_N(Q,f_Q)}^{N_N(Q)}(f_1)=\\sum_{t\\in T}t.$$\nLet $W$ denote a set of pairwise orthogonal primitive idempotents of $(\\operatorname{Br}_Q(c)X \\operatorname{Br}_Q(c))^{N_G(Q)}$ such that $$Tr_{H}^{N_G(Q)}(l)=\\sum_{w\\in W}w.$$\nThere is at least one idempotent in $T$, say $t$, belonging to a point with defect group $Q$. Then, since\n$$Tr_{H}^{N_G(Q)}(l)Tr_{N_N(Q,f_Q)}^{N_N(Q)}(f_1)=Tr_{N_N(Q,f_Q)}^{N_N(Q)}(f_1)$$\nwe obtain $Tr_{H}^{N_G(Q)}(l)t=t$, equality that assures the existence of an invertible element $a\\in((\\operatorname{Br}_Q(c)X \\operatorname{Br}_Q(c))^{N_N(Q)})^*$ such that $t^aw=wt^a=t^a$ for some $w\\in W$.\\\\\nIf $\\operatorname{Br}_P(w)=0$ then $\\operatorname{Br}_Q(w)=0$, forcing $\\operatorname{Br}_Q(t^a)=0$. This contradiction proves that $w$ belongs to a point of $N_G(Q)$ on $X$ that covers $\\operatorname{Br}_Q(c)$.\nThe last statement says that the point of $N_G(Q)$ on $X$ that contains $w$ covers $\\operatorname{Br}_Q(c)$, viewed as a $N_G(Q)$-invariant primitive idempotent of $Y^{N_N(Q)}$. Since $ \\operatorname{Br}_Q(c)\\in C(Q)^{N_N(Q)}$ being also $N_G(Q)$-invariant, $w$ determines a point of $N_G(Q)$ on $A(Q)$ that covers $\\operatorname{Br}_Q(c)$. We have\n$$(\\operatorname{Br}_Q(c) A(Q) \\operatorname{Br}_Q(c))^{N_G(Q)}\\leq (\\operatorname{Br}_Q(c)X \\operatorname{Br}_Q(c))^{N_G(Q)},$$\nand according to \\cite[Proposition 3.3]{PuZh} there is $w_1\\in (\\operatorname{Br}_Q(c)A(Q)\\operatorname{Br}_Q(c))^{N_G(Q)}$ such that $w_1=w^{a_1}$ for some unit $a_1$ in $(\\operatorname{Br}_Q(c)X \\operatorname{Br}_Q(c))^{N_G(Q)}$.\nThere is also a primitive idempotent $t_1\\in (\\operatorname{Br}_Q(c)A(Q)\\operatorname{Br}_Q(c))^{N_N(Q)}$ such that $$w_1t_1=t_1w_1=t_1, \\quad t_1\\cdot t^{aa_1b}=t^{aa_1b}t_1=t^{aa_1b}$$ for some unit $b$ of $(\\operatorname{Br}_Q(c)X \\operatorname{Br}_Q(c))^{N_N(Q)}$, where $w_1$ has defect group $P$, while $t_1$ has defect group $Q$.\nThe point containing $w_1$ covers $\\operatorname{Br}_Q(c)$ and lifts to $\\alpha, $ a point of $G$ on $A$ that covers $c.$ This is a consequence of \\cite[Theorem 3.5]{Co}. Notice that $\\alpha$ verifies the axioms of Definition \\ref{definition41}.\n\nAt last, the  algebra epimorphism\n$$\\operatorname{Br}_Q:(cAc)^G\\rightarrow (\\operatorname{Br}_Q(c) A(Q) \\operatorname{Br}_Q(c))^{N_G(Q)}$$\nproves that $w_1$ lifts to an idempotent of $(cAc)^G$, hence $\\alpha$ also verifies the axioms of Definition \\ref{definition43}.\n\\end{proof}\nLet $\\mathcal{C}(P,f_P)$ denote the full subcategory of $\\mathcal{C}$  consisting of Brauer pairs $(T,f_T)$ such that\n$(T,f_T)\\leq(P,f_P).$ Let $\\mathcal{D}(P,f_P)$ denote the full subcategory of $\\mathcal{D}$ consisting of Brauer pairs $(T,f_T)$ with $(T,f_T)\\leq(P,f_P)$.\nAs a conclusion of the above results we can show the second main result of this paper.\n\\begin{thm}\\label{theorem63}\nThe category $\\mathcal{C}(P,f_P)$ is isomorphic to $\\mathcal{D}(P,f_P)$.\n\\end{thm}\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}