Spectra originating from semi-B-Fredholm theory and commuting perturbations
Volume 219 / 2013
Abstract
Burgos, Kaidi, Mbekhta and Oudghiri [J. Operator Theory 56 (2006)] provided an affirmative answer to a question of Kaashoek and Lay and proved that an operator $F$ is of power finite rank if and only if $\sigma _{\rm dsc}(T+F) =\sigma _{\rm dsc}(T)$ for every operator $T$ commuting with $F$. Later, several authors extended this result to the essential descent spectrum, left Drazin spectrum and left essential Drazin spectrum. In this paper, using the theory of operators with eventual topological uniform descent and the technique used by Burgos et al., we generalize these results to various spectra originating from semi-B-Fredholm theory. As immediate consequences, we give affirmative answers to several questions posed by Berkani, Amouch and Zariouh. Moreover, we provide a general framework which allows us to derive in a unified way perturbation results for Weyl–Browder type theorems and properties (generalized or not). Our results improve many recent results by removing certain extra assumptions.
