Weyl type theorems for functions of operators
type: Item
instance of: scholarly article
zbMATH Open document ID: 1256.47014
DOI: 10.1017/S0017089512000092
published in: Glasgow Mathematical Journal
publication date: 2012-08-22T00:00:00Z
Mathematics Subject Classification ID: 47A53
Mathematics Subject Classification ID: 47A60
Mathematics Subject Classification ID: 47A10
zbMATH DE Number: 6069856
zbMATH Keywords: Weyl's theorem
zbMATH Keywords: a-Weyl's theorem
MaRDI profile type: Publication
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full work available at URL: s0017089512000092
OpenAlex ID: W2012684167
title: Weyl type theorems for functions of operators
author: Sen Zhu
author: Chunguang Li
author: Ting Ting Zhou
review text: Let \(X\) be a complex Banach space. For a bounded linear operator \(T\) on \(X\), let \(\sigma(T)\) be the spectrum, \(\sigma_a(T)\) be the approximate point spectrum, and \(\sigma_{ea}(T)=\cap_{K}\sigma_a(T+K)\), where the intersection is taken over all compact operators \(K\) on \(X\), the essential approximate point spectrum. Let \(\pi_{00}(T)\) be the set of isolated eigenvalues in \(\sigma(T)\) such that the kernel of \(\lambda I-T\) is finite dimensional and let \(\pi_{00}^{a}(T)\) be the set of all eigenvalues which are isolated points in \(\sigma_{a}(T)\) and such that the kernel of \(\lambda I-T\) is finite dimensional. It is said that a-Weyl's theorem holds for \(T\), briefly denoted as \(T\in (a-W)\), if \(\sigma_a(T)\setminus \sigma_{ea}(T)=\pi_{00}^{a}(T)\). If \(\sigma_a(T)\setminus \sigma_{ea}(T)=\pi_{00}(T)\), then \(T\in (\omega)\). In this paper, there are established necessary and sufficient conditions for \(T\) such that \(f(T)\in (a-W)\), respectively \(f(T)\in (\omega)\), for every function \(f\) which is holomorphic on a neighborhood of \(\sigma(T)\).
reviewed by: Janko Bračič
description: scientific article; zbMATH DE number 6069856