Riesz points and Weyl's theorem
type: Item
instance of: scholarly article
title: Riesz points and Weyl's theorem
zbMATH Open document ID: 0948.47002
DOI: 10.1007/BF01236471
author: Bruce Barnes
published in: Integral Equations and Operator Theory
publication date: 2000-05-10T00:00:00Z
review text: A Riesz point for a bounded linear operator \(T\) on a complex Banach space is simply an isolated point of the spectrum \(\sigma(T)\) such that the corresponding spectral projection has finite rank. The Weyl spectrum \(\sigma_W(T)\) is that part of \(\sigma(T)\) that remains unchanged under every compact perturbation of \(T\). In the paper under review, the following two conditions on \(T\) are considered: I. \(\sigma_W(T)\) equals the complement in \(\sigma(T)\) of the set of isolated eigenvalues having finite-dimensional eigenspace (``Weyl's theorem''). II. \(\sigma_W(T)\) equals the complement in \(\sigma(T)\) of the set of Riesz points of \(T\) (``Version II of Weyl's theorem''). One first establishes some connections between the these two conditions on \(T\). Several closure properties are then pointed out for the class of operators enjoying II. After that, one proves that, if \(\sigma_W(T)\) is simply connected, then Version II of Weyl's theorem holds for every compact perturbation of \(T\). The converse assertion is also considered. At the final of paper, an example is presented in order to ``draw some conclusions which contrast Weyl's theorem and Version II'' (in the author's words).
Mathematics Subject Classification ID: 47A10
Mathematics Subject Classification ID: 47A15
Mathematics Subject Classification ID: 47A53
Mathematics Subject Classification ID: 47A55
zbMATH DE Number: 1333092
zbMATH Keywords: Riesz point
zbMATH Keywords: Weyl's theorem
zbMATH Keywords: Fredholm operator
zbMATH Keywords: Browder spectrum
zbMATH Keywords: compact perturbation
zbMATH Keywords: weighted shift
zbMATH Keywords: spectral projection
zbMATH Keywords: isolated eigenvalues
zbMATH Keywords: finite-dimensional eigenspace
reviewed by: Daniel Beltiţă
MaRDI profile type: Publication
cites work: Q3317595
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cites work: Compact Perturbations, Normal Eigenvalues and a Problem of Salinas
description: scientific article; zbMATH DE number 1333092