Extended Weyl theorems and perturbations

Abstract

In this paper we study the properties $( \rm{gaw}), (aw), ( \rm{gab})$ and $(ab)$, a variant of Weyl's type theorems introduced by Berkani. We established for a bounded linear operator defined on a Banach space several sufficient and necessary conditions for which the properties $(\rm{gaw}), (aw), ( \rm{gab})$ and $(ab)$ hold. Among other things, we study the stability of the properties $( \rm{gaw}), (aw), ( \rm{gab})$ and $(ab)$ for a polaroid operator $T$ acting on a Banach space, under perturbations by finite rank operators, by nilpotent operators and, more generally, by algebraic operators commuting with $T$.