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 σ . (T +Q) = σ . (T ) where σ . ∈ {σ D , σ 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 σ D (T ) is the Drazin spectrum of T and σ LD (T ) its left Drazin spectrum. However, under the additional hypothesis iso σ ub (T ) = ., 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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