Triangulations of operators with two-isometric liftings
type: Item
instance of: scholarly article
title: Triangulations of operators with two-isometric liftings
zbMATH Open document ID: 1486.47003
author: Witold Majdak
author: Laurian Suciu
published in: Integral Equations and Operator Theory
publication date: 2021-05-06T00:00:00Z
review text: The paper deals with bounded linear operators on a complex Hilbert space which have 2-isometric liftings. Recall that a bounded linear operator \(T\) on \(H\) is called a 2-isometry if \(I-2T^*T+T^{*2}T^2=0\). There is a special class of \(2\)-isometric liftings, namely, those liftings which satisfy the range inclusion of the covariance operators. Here, the covariance operator for \(T\) is given by \(\Delta_T=T^*T-I\). If \(T\) is a \(2\)-isometry, then \(\Delta_T\) is a positive operator, \(\ker \Delta_T\) is an invariant subspace for \(T\) and \(T|_{\ker \Delta_T}\) is an isometry. One of the main results of the paper under review classifies 2-isometric liftings \(S\) on \(K\) of a \(2\)-isometry \(T\) on \(H\) which satisfy the condition \[ \operatorname{Range}(\Delta_{S|_{K \ominus H}}) \subseteq \overline{\operatorname{Range}(\Delta_{S})} \] (see Theorem 2.3 for a precise statement). The authors also discuss various types of \(2\)-isometric liftings and state some open problems pertaining to classifications of these liftings.
reviewed by: Sameer Chavan
Mathematics Subject Classification ID: 47A05
Mathematics Subject Classification ID: 47A15
Mathematics Subject Classification ID: 47A20
Mathematics Subject Classification ID: 47A63
Mathematics Subject Classification ID: 00A27
zbMATH DE Number: 7345255
zbMATH Keywords: Brownian unitary (isometric) operator
zbMATH Keywords: 2-isometric lifting
zbMATH Keywords: concave (expansive) operator
zbMATH Keywords: $A$-contraction
zbMATH Keywords: covariance operator
MaRDI profile type: Publication
full work available at URL: s00020-021-02625-9
OpenAlex ID: W3128286221
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DOI: 10.1007/S00020-021-02625-9
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description: scientific article; zbMATH DE number 7345255