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 ${\sigma }_{\ast }(T+Q)={\sigma }_{\ast }(T)$where ${\sigma }_{\ast }ın\{{\sigma }_D,{\sigma }_{LD}\}$, 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$\mathrm{iso}{\sigma }_{ub}(T)=\mathrm{\varnothing }$, the mentioned equality holds.Moreover, we study the preservation of various spectra originatingfrom B-Fredholm theory under perturbations by Riesz operators.