Property \((w)\) and perturbations. III
type: Item
instance of: scholarly article
title: Property \((w)\) and perturbations. III
zbMATH Open document ID: 1171.47011
author: Pietro Aiena
author: F. Villafañe
author: M. T. Biondi
published in: Journal of Mathematical Analysis and Applications
publication date: 2009-04-29T00:00:00Z
review text: The present article continues the authors' study of class of linear operators defined on Banach spaces that verify property \((w)\), a variant of Weyl's theorem introduced by \textit{V.\,Rakočević} [Mat.\ Vesn.\ 37, 423--426 (1985; Zbl 0596.47001); see also \textit{P.\,Aiena} and \textit{P.\,Pena}, J.~Math.\ Anal.\ Appl.\ 324, No.\,1, 566--579 (2006; Zbl 1101.47001)]. The preservation of property \((w)\) under certain classes of perturbations was studied in Parts~I and II of the present series of articles [\textit{P.\,Aiena} and \textit{M.\,T.\thinspace Biondi}, J.~Math.\ Anal.\ Appl.\ 336, No.\,1, 683--692 (2007; Zbl 1182.47012); \textit{P.\,Aiena}, J.\ Math.\ Anal.\ Appl.\ 342, No.\,2, 830--837 (2008; Zbl 1171.47010), reviewed above] and in [\textit{P.\,Aiena, J.\,R.\thinspace Guillen} and \textit{P.\,Pena}, Linear Algebra Appl.\ 428, No.\,8--9, 1791--1802 (2008; Zbl 1136.47003)]. In the present paper, the authors ``consider the preservation property \((w)\) under a finite rank perturbation commuting with \(T\), whenever \(T\) is polaroid or \(T\) has analytical core \(K(\lambda _{0}I - T)=\{0\}\) for some \(\lambda _0 \in \mathbb C\). The preservation of property \((w)\) is also studied under commuting nilpotent or under injective quasi-nilpotent perturbations. The theory is exemplified in the case of some special classes of operators.'' This article gives insight to further research in property \((w)\) and perturbations.
reviewed by: V. Lokesha
Mathematics Subject Classification ID: 47A53
Mathematics Subject Classification ID: 47A10
Mathematics Subject Classification ID: 47A55
zbMATH DE Number: 5549419
zbMATH Keywords: Weyl's theorems
zbMATH Keywords: property (w)
zbMATH Keywords: SVEP
zbMATH Keywords: polaroid operator
zbMATH Keywords: (semi-)Fredholm operators
zbMATH Keywords: spectrum and resolvent of linear operators
MaRDI profile type: Publication
full work available at URL: j.jmaa.2008.11.081
OpenAlex ID: W4213380685
cites work: Q3154380
cites work: Classes of operators satisfying a-Weyl's theorem
cites work: Property \((w)\) and perturbations. II
cites work: Property \((w)\) and perturbations
cites work: Some characterizations of operators satisfying \(a\)-Browder's theorem
cites work: Property \((w)\) for perturbations of polaroid operators
cites work: Variations on Weyl's theorem
cites work: Weyl's theorems for some classes of operators
cites work: Analytically class A operators and Weyl's theorem
cites work: Q5584224
cites work: Q4937978
cites work: Q3369257
cites work: Riesz projections for a class of Hilbert space operators
cites work: Q4257793
cites work: Weyl spectra and Weyl's theorem
cites work: Another note on Weyl’s theorem
cites work: Q3917042
cites work: On (p,k)-quasihyponormal operators
cites work: Asymptotic intertwining and spectral inclusions on Banach spaces
cites work: Weyl's and Browder's theorems for operators satisfying the SVEP
cites work: Q3728676
cites work: Q4204666
cites work: Semi-Browder operators and perturbations
cites work: Isolated points of spectrum of (\(p,k\))-quasihyponormal operators
DOI: 10.1016/J.JMAA.2008.11.081
description: scientific article; zbMATH DE number 5549419