Azad Rohilla, Harsh Trivedi, Shankar Veerabathiran
{"title":"Beurling quotient subspaces for covariant representations of product systems","authors":"Azad Rohilla, Harsh Trivedi, Shankar Veerabathiran","doi":"10.1007/s43034-023-00301-0","DOIUrl":null,"url":null,"abstract":"<div><p>Let <span>\\((\\sigma , V^{(1)}, \\dots , V^{(k)})\\)</span> be a pure doubly commuting isometric representation of the product system <span>\\({\\mathbb {E}}\\)</span> on a Hilbert space <span>\\({\\mathcal {H}}_{V}.\\)</span> A <span>\\(\\sigma \\)</span>-invariant subspace <span>\\({\\mathcal {K}}\\)</span> is said to be <i>Beurling quotient subspace</i> of <span>\\({\\mathcal {H}}_{V}\\)</span> if there exist a Hilbert space <span>\\({\\mathcal {H}}_W,\\)</span> a pure doubly commuting isometric representation <span>\\((\\pi , W^{(1)}, \\dots , W^{(k)})\\)</span> of <span>\\({\\mathbb {E}}\\)</span> on <span>\\({\\mathcal {H}}_W\\)</span> and an isometric multi-analytic operator <span>\\(M_\\Theta :{{\\mathcal {H}}_W} \\rightarrow {\\mathcal {H}}_{V}\\)</span>, such that </p><div><div><span>$$\\begin{aligned} {\\mathcal {K}}={\\mathcal {H}}_{V}\\ominus M_{\\Theta }{\\mathcal {H}}_W, \\end{aligned}$$</span></div></div><p>where <span>\\(\\Theta : {\\mathcal {W}}_{{\\mathcal {H}}_W} \\rightarrow {\\mathcal {H}}_{V} \\)</span> is an inner operator and <span>\\({\\mathcal {W}}_{{\\mathcal {H}}_W}\\)</span> is the generating wandering subspace for <span>\\((\\pi , W^{(1)}, \\dots , W^{(k)}).\\)</span> In this article, we prove the following characterization of the Beurling quotient subspaces: A subspace <span>\\({\\mathcal {K}}\\)</span> of <span>\\({\\mathcal {H}}_{V}\\)</span> is a Beurling quotient subspace if and only if </p><div><div><span>$$\\begin{aligned}&(I_{E_{j}}\\otimes ( (I_{E_{i}}\\otimes P_{{\\mathcal {K}}}) - \\widetilde{T}^{(i) *}\\widetilde{T}^{(i)}))(t_{i,j} \\otimes I_{{\\mathcal {H}}_{V}})\\\\&(I_{E_{i}}\\otimes ( (I_{E_{j}}\\otimes P_{{\\mathcal {K}}})- \\widetilde{T}^{(j) *}\\widetilde{T}^{(j)}))=0, \\end{aligned}$$</span></div></div><p>where <span>\\(\\widetilde{T}^{(i)}:=P_{{\\mathcal {K}}}\\widetilde{V}^{(i)} (I_{E_{i}} \\otimes P_{{\\mathcal {K}}})\\)</span> and <span>\\( 1 \\le i,j\\le k.\\)</span> As a consequence, we derive a concrete regular dilation theorem for a pure, completely contractive covariant representation <span>\\((\\sigma , V^{(1)}, \\dots , V^{(k)})\\)</span> of <span>\\({\\mathbb {E}}\\)</span> on a Hilbert space <span>\\({\\mathcal {H}}_{V}\\)</span> which satisfies Brehmer–Solel condition and using it and the above characterization, we provide a necessary and sufficient condition that when a completely contractive covariant representation is unitarily equivalent to the compression of the induced representation on the Beurling quotient subspace. Further, we study the relation between Sz. Nagy–Foias-type factorization of isometric multi-analytic operators and joint invariant subspaces of the compression of the induced representation on the Beurling quotient subspace.</p></div>","PeriodicalId":48858,"journal":{"name":"Annals of Functional Analysis","volume":null,"pages":null},"PeriodicalIF":1.2000,"publicationDate":"2023-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s43034-023-00301-0.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annals of Functional Analysis","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s43034-023-00301-0","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Let \((\sigma , V^{(1)}, \dots , V^{(k)})\) be a pure doubly commuting isometric representation of the product system \({\mathbb {E}}\) on a Hilbert space \({\mathcal {H}}_{V}.\) A \(\sigma \)-invariant subspace \({\mathcal {K}}\) is said to be Beurling quotient subspace of \({\mathcal {H}}_{V}\) if there exist a Hilbert space \({\mathcal {H}}_W,\) a pure doubly commuting isometric representation \((\pi , W^{(1)}, \dots , W^{(k)})\) of \({\mathbb {E}}\) on \({\mathcal {H}}_W\) and an isometric multi-analytic operator \(M_\Theta :{{\mathcal {H}}_W} \rightarrow {\mathcal {H}}_{V}\), such that
where \(\Theta : {\mathcal {W}}_{{\mathcal {H}}_W} \rightarrow {\mathcal {H}}_{V} \) is an inner operator and \({\mathcal {W}}_{{\mathcal {H}}_W}\) is the generating wandering subspace for \((\pi , W^{(1)}, \dots , W^{(k)}).\) In this article, we prove the following characterization of the Beurling quotient subspaces: A subspace \({\mathcal {K}}\) of \({\mathcal {H}}_{V}\) is a Beurling quotient subspace if and only if
where \(\widetilde{T}^{(i)}:=P_{{\mathcal {K}}}\widetilde{V}^{(i)} (I_{E_{i}} \otimes P_{{\mathcal {K}}})\) and \( 1 \le i,j\le k.\) As a consequence, we derive a concrete regular dilation theorem for a pure, completely contractive covariant representation \((\sigma , V^{(1)}, \dots , V^{(k)})\) of \({\mathbb {E}}\) on a Hilbert space \({\mathcal {H}}_{V}\) which satisfies Brehmer–Solel condition and using it and the above characterization, we provide a necessary and sufficient condition that when a completely contractive covariant representation is unitarily equivalent to the compression of the induced representation on the Beurling quotient subspace. Further, we study the relation between Sz. Nagy–Foias-type factorization of isometric multi-analytic operators and joint invariant subspaces of the compression of the induced representation on the Beurling quotient subspace.
期刊介绍:
Annals of Functional Analysis is published by Birkhäuser on behalf of the Tusi Mathematical Research Group.
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