On the Weyl spectrum: Spectral mapping theorem and Weyl's theorem
type: Item
instance of: scholarly article
title: On the Weyl spectrum: Spectral mapping theorem and Weyl's theorem
zbMATH Open document ID: 0998.47012
author: Xiu-Ling Zhang
author: Jinchuan Hou
published in: Journal of Mathematical Analysis and Applications
publication date: 2002-08-08T00:00:00Z
review text: Let \({\mathcal B}({\mathcal H})\) be the set of all bounded linear operators on a Banach space \({\mathcal H}\) and \({\mathcal F}_0\subset {\mathcal B}({\mathcal H})\) be the set of all Fredholm operators of index \(0.\) The Weyl spectrum \(\sigma _w(T)\) of an operator \(T\in {\mathcal B}({\mathcal H})\) is defined by \(\sigma _w(T) =\{\lambda\in {\mathbb{C}}: T-\lambda I\notin {\mathcal F}_0\}.\) They say that \(T\) satisfies the Weyl spectral mapping theorem if \(\sigma _w(f(T))=f(\sigma _w(T))\) for any function \(f\) analytic on a neighborhood of the spectrum \(\sigma (T).\) They say that Weyl's theorem holds for \(T\) if \(\sigma _w(T)\) consists precisely of all points in \(\sigma(T)\) except the isolated eigenvalues of finite multiplicity. It is shown that if \(T\) is a dominant operator or an analytic quasi-hyponormal operator then \(T\) satisfies the Weyl spectral mapping theorem. Moreover, Weyl's theorem holds for \(f(T)+F\) if ``dominant'' is replaced by ``\(M\)-hyponormal'', where \(F\) is any finite rank operator commuting with \(T.\)
Mathematics Subject Classification ID: 47B20
Mathematics Subject Classification ID: 47A53
Mathematics Subject Classification ID: 47A10
Mathematics Subject Classification ID: 47A55
Mathematics Subject Classification ID: 47A60
zbMATH DE Number: 1218591
zbMATH Keywords: Weyl's theorem
zbMATH Keywords: Fredholm operators
zbMATH Keywords: quasi-hyponormal operators
zbMATH Keywords: Weyl spectrum
zbMATH Keywords: Weyl spectral mapping theorem
zbMATH Keywords: dominant operator
reviewed by: V. M. Deundjak
MaRDI profile type: Publication
full work available at URL: jmaa.1997.5897
OpenAlex ID: W2003529560
cites work: Q5601718
cites work: Q3223360
cites work: Spectral mapping theorems for essential spectra
cites work: Q3697643
cites work: Q3713003
cites work: A spectral mapping theorem for the Weyl spectrum
cites work: On the Weyl spectrum. II
cites work: M-hyponormal operators
cites work: Q3933486
DOI: 10.1006/JMAA.1997.5897
description: scientific article; zbMATH DE number 1218591