A note on property (gb) and perturbations

TL;DR

This paper investigates property (gb) of bounded linear operators on Banach spaces, examining its stability under various perturbations and providing counterexamples where property (gb) is not preserved.

Contribution

It extends the study of property (gb) by analyzing its stability under nilpotent, finite rank, and quasi-nilpotent perturbations, including counterexamples.

Findings

Property (gb) is not always preserved under commuting quasi-nilpotent perturbations.

Property (gb) is not always preserved under commuting finite rank perturbations.

Counterexamples demonstrate the limits of stability for property (gb).

Abstract

An operator $T \in \mathcal{B}(X)$ 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 $\sigma_{SBF_{+}^{-}}(T)$ coincides with the set $\Pi(T)$ of all poles of the resolvent of $T$. In this note 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, by quasi-nilpotent 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.