Evaluation points for cyclic operators with Bishop's property (\(\beta\))
type: Item
instance of: scholarly article
title: Evaluation points for cyclic operators with Bishop's property (\(\beta\))
zbMATH Open document ID: 1053.47004
DOI: 10.1016/S0022-1236(03)00127-7
author: Mostafa Mbekhta
author: Hassan Zerouali
published in: Journal of Functional Analysis
publication date: 2004-03-15T00:00:00Z
review text: Let \(T\) a bounded operator on a Hilbert space \(\mathcal H\). Let \(B(T)\) denote the set of bounded point evaluations, of \(T\) and \(B_a(T)\) the set of analytic point evaluations, of \(T\). \textit{L. R. Williams} [Dyn. Syst. Appl. 3, 103-112 (1994; Zbl 0805.47019)] showed that for any cyclic operator, \(B_a(T) \subset B(T) \setminus \sigma_{ap}(T)\), where \(\sigma_{ap}(T)\) denotes the approximate spectrum. The authors of the present work prove the following precise result: If \(T\) is cyclic and satisfies the Bishop property \(\beta\), then \(B_a(T)=B(T)\setminus \sigma_{ap}(T)\). As a corollary, they also obtain a local spectral version of their theorem. The proof, which is quite elegant, is carried out by relating the analytic point evaluations to a certain generalized resolvent set. Operators that satisfy the Bishop property (\(\beta\)) are, for instance, subnormal operators and decomposable operators. A particular case of the main result for the operator of multiplication by \(z\) in the Hardy space was proved by \textit{T. T. Trent} [Pac. J. Math. 80, 279-292 (1979; Zbl 0405.46043)]. The authors also give sufficient conditions on \(B_a(T)\) under which the spectrum of \(T\) and the local spectrum of \(T\) at any nonzero \(y\in \mathcal H\) coincide. Other interesting results in the paper under review are certain descriptions of the spectral pictures of quasi-similar cyclic operators. For instance, if \(T\) and \(S\) are quasi-similar cyclic operators satisfying Bishop's property (\(\beta\)), then \(T\) and \(S\) have the same approximate spectrum and the same essential spectrum.
Mathematics Subject Classification ID: 47A16
Mathematics Subject Classification ID: 47A10
Mathematics Subject Classification ID: 47A11
zbMATH DE Number: 2057775
zbMATH Keywords: cyclic operator
zbMATH Keywords: Bishop property
zbMATH Keywords: point evaluation
zbMATH Keywords: local spectral theory
reviewed by: Alfonso Montes-Rodriguez
MaRDI profile type: Publication
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full work available at URL: s0022-1236(03)00127-7
OpenAlex ID: W2044956623
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description: scientific article; zbMATH DE number 2057775