Spectra originating from semi-B-Fredholm theory and commuting perturbations

Qingping Zeng ¹   ; Qiaofen Jiang ²   ; Huaijie Zhong ²

¹ College of Computer and Information Fujian Agriculture and Forestry University 350002 Fuzhou, P.R. China and School of Mathematics and Computer Science Fujian Normal University 350007 Fuzhou, P.R. China
² School of Mathematics and Computer Science Fujian Normal University 350007 Fuzhou, P.R. China

Studia Mathematica, Tome 219 (2013) no. 1, pp. 1-18

Résumé

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.

Détail

DOI : 10.4064/sm219-1-1

Keywords: burgos kaidi mbekhta oudghiri nbsp operator theory provided affirmative answer question kaashoek lay proved operator power finite rank only sigma dsc sigma dsc every operator commuting later several authors extended result essential descent spectrum drazin spectrum essential drazin spectrum paper using theory operators eventual topological uniform descent technique burgos generalize these results various spectra originating semi b fredholm theory immediate consequences affirmative answers several questions posed berkani amouch zariouh moreover provide general framework which allows derive unified perturbation results weyl browder type theorems properties generalized results improve many recent results removing certain extra assumptions