On the stability of the spectral properties under commuting perturbations

Abstract

In this paper, we examine the stability of several spectral properties under commuting perturbations. In particular, we show that if $T\in L(X)$ is an isoloid operator satisfying generalized Weyl's theorem and if $F\in L(X)$ is a power finite rank operator that commutes with $T$, then generalized Weyl's theorem holds for $T+F$. In addition, we consider the permanence of Bishop's property $(\beta)$, at a point, under commuting perturbation that is an algebraic operator.