A Note on Property ( g b ) and Perturbations

Abstract

An operator $T\in ℬ(X)$ defined on a Banach space $X$ satisfies property $(gb)$ if the complement in the approximate point spectrum ${\sigma }_a(T)$ of the upper semi-B-Weyl spectrum ${\sigma }_{SB{F}_{+}^{-}}(T)$ coincides with the set $\Pi (T)$ of all poles of the resolvent of $T$. In this paper, we continue to study property $(gb)$ and the stability of it, for a bounded linear operator $T$ acting on a Banach space, under perturbations by nilpotent operators, by finite rank operators, and by quasinilpotent operators commuting with $T$. Two counterexamples show that property $(gb)$ in general is not preserved under commuting quasi-nilpotent perturbations or commuting finite rank perturbations.