Transference of metrics induced by unitary couplings, a Sarason theorem for the bidimensional torus, and a Sz.-Nagy-Foias theorem for two pairs of dilations
type: Item
instance of: scholarly article
title: Transference of metrics induced by unitary couplings, a Sarason theorem for the bidimensional torus, and a Sz.-Nagy-Foias theorem for two pairs of dilations
zbMATH Open document ID: 0781.47017
author: Mischa Cotlar
author: Cora Sadosky
published in: Journal of Functional Analysis
publication date: 1993-04-28T00:00:00Z
review text: The authors prove the following transference result: Let \(H_ 1\) and \(H_ 2\) be two Hilbert spaces, \(U_ 1\in L(H_ 1)\) and \(U_ 2\in(H_ 2)\), two unitary operators, and \(N_ 1\subset N_ 2\) and \(N_ 2\subset H_ 2\), two invariant subspaces, i.e., \[ U_ 1N_ 1\subset N_ 1,\quad U_ 2N_ 2\subset N_ 2, \] so that \(T_ 1=U_ 1|_{N_ 1}\) and \(T_ 2=U_ 2|_{N_ 2}\) are semiunitary operators in \(N_ 1\) and \(N_ 2\), respectively. Given a unitary coupling \(T\in L(N)\), of \(T_ 1\) and \(T_ 2\), there exists a unitary coupling \(U\in L(H)\) of \(U_ 1\) and \(U_ 2\) such that \[ \langle f_ 1,f_ 2\rangle_ H=\langle f_ 1,f_ 2\rangle_ N,\quad\forall f_ 1\in N_ 1,\;f_ 2\in N_ 2. \] This result allows them to prove a lifting theorem for invariant forms acting in scattering systems with one or two evolution groups.
Mathematics Subject Classification ID: 47A20
Mathematics Subject Classification ID: 47A40
zbMATH DE Number: 152051
zbMATH Keywords: Wold-Kolmogorov decomposition
zbMATH Keywords: transference
zbMATH Keywords: unitary operators
zbMATH Keywords: invariant subspaces
zbMATH Keywords: semiunitary operators
zbMATH Keywords: unitary coupling
zbMATH Keywords: lifting theorem for invariant forms
zbMATH Keywords: scattering systems with one or two evolution groups
reviewed by: D. C. Struppa
MaRDI profile type: Publication
full work available at URL: jfan.1993.1022
OpenAlex ID: W2020510631
DOI: 10.1006/JFAN.1993.1022
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description: scientific article; zbMATH DE number 152051