B-Fredholm spectra and Riesz perturbations

Abstract

Let $T$ be a bounded linear Banach space operator and let $Q$ be aquasinilpotent one commuting with $T$. The main purpose of thepaper is to show that we do not have $\sigma_{*}(T+Q)=\sigma_{*}(T)$where $\sigma_{*}ın\{\sigma_{D},\sigma_{LD}\}$, contrary to whathas been announced in the proof of Lemma 3.5 from M. Amouch,{Polaroid operators with SVEP and perturbations of property (gw)},Mediterr. J. Math. {6} (2009), 461–470, where$\sigma_{D}(T)$ is the Drazin spectrum of $T$ and $\sigma_{LD}(T)$its left Drazin spectrum. However, under the additional hypothesis$\operatorname{iso}\sigma_{ub}(T)=\emptyset$, the mentioned equality holds.Moreover, we study the preservation of various spectra originatingfrom B-Fredholm theory under perturbations by Riesz operators.