Contractions with a unilateral shift summand are reflexive
type: Item
instance of: scholarly article
title: Contractions with a unilateral shift summand are reflexive
zbMATH Open document ID: 0582.47014
DOI: 10.1007/BF01195874
author: Pei Yuan Wu
published in: Integral Equations and Operator Theory
publication date: 1984-01-01T00:00:00Z
review text: A bounded linear operator T on a complex, separable Hilbert space H is reflexive if Alg T\(=Alg Lat T\), where Alg T and Alg Lat T denote, respectively, the weakly closed algebra generated by T and I, and the weakly closed algebra consisting of those operators which leave invariant all the invariant subspaces of T. Generalizing Deddens' result that isometries are reflexive, the author shows that the direct sum of a unilateral shift and any contraction is reflexive. Some further generalizations are also obtained. This paper concludes with a conjecture that if T is a contraction and there exists an operator \(W\neq 0\) such that \(WT=SW\) for some unilateral shift S, then T is reflexive. It has since been verified by \textit{H. Bercovici} and \textit{K. Takahashi} [J. Lond. Math. Soc., II. Ser. 32, 149-156 (1985; Zbl 0536.47009)].
Mathematics Subject Classification ID: 47A45
Mathematics Subject Classification ID: 47A15
Mathematics Subject Classification ID: 47L30
zbMATH DE Number: 3931816
zbMATH Keywords: reflexive operator
zbMATH Keywords: weakly closed algebra consisting of those operators which leave invariant all the invariant subspaces
zbMATH Keywords: direct sum of a unilateral shift and any contraction is reflexive
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3931816