A note on property (gb) and perturbations

Details A note on property (gb) and perturbations A note on property (gb) and perturbations

2012-03-06

arxiv.org/abs/1203.1134v1

Mathematics

Functional Analysis

10 pages

Scientific paper

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.

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