Property \((w)\) and perturbations. III
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.
spectrum and resolvent of linear operators
scientific article; zbMATH DE number 5549419
scientific article