Property \((w)\) and perturbations (Q2644043)

A bounded linear operator \(T\in L(X)\), defined on a Banach space \(X\), satisfies property (\(w\)), a variant of Weyl's theorem, if the complement in the approximate point spectrum \(\sigma_a(T)\) of the Weyl essential approximate spectrum \(\sigma_{wa}(T)\) coincides with the set of all isolated points of the spectrum which are eigenvalues of finite multiplicity. In this note, we study the stability of property (\(w\)) for a bounded operator \(T\) acting on a Banach space under perturbations by finite rank operators, by nilpotent operators, and quasi-nilpotent operators commuting with \(T\). Editorial remark: This paper is the first of a series. Parts II and III have been reviewed, see [\textit{P.\,Aiena}, J.\ Math.\ Anal.\ Appl.\ 342, No.\,2, 830--837 (2008; Zbl 1171.47010); \textit{P.\,Aiena, M.\,T.\thinspace Biondi} and \textit{F.\,Villafañe}, J.~Math.\ Anal.\ Appl.\ 353, No.\,1, 205--214 (2009; Zbl 1171.47011)].