Property $(\rm{gw})$ and perturbations

Abstract

The property $(\rm{gw})$ is a variant of generalized Weyl's theorem, for a bounded operator $T$ acting on a Banach space. In this note we consider the preservation of property $(\rm{gw})$ under a finite rank perturbation commuting with $T$, whenever $T$ is isoloid, polaroid, or $T$ has analytical core $K(\lamda_0 I -T ) = \set{0}$ for some $\lamda_0\in\mathbb{C}$. The preservation of property $(\rm{gw})$ is also studied under commuting nilpotent or under algebraic perturbations. The theory is exemplified in the case of some special classes of operators.