MATEMATIČKI VESNIK
МАТЕМАТИЧКИ ВЕСНИК



MATEMATIČKI VESNIK
B-Fredholm spectra and Riesz perturbations
M. Berkani and H. Zariouh

Abstract

Let $T$ be a bounded linear Banach space operator and let $Q$ be a quasinilpotent one commuting with $T$. The main purpose of the paper is to show that we do not have $\sigma_{*}(T+Q)=\sigma_{*}(T)$ where $\sigma_{*}\in\{\sigma_{D},\sigma_{LD}\}$, contrary to what has 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 originating from B-Fredholm theory under perturbations by Riesz operators.

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Keywords: B-Fredholm spectrum; Riesz perturbations.

MSC: 47A53, , 47A10, 47A11

Pages:  155--165     

Volume  67 ,  Issue  3 ,  2015