Vol. LXXX, 1 (2011)
p. 119 - 132

Perturbation results for Weyl type theorems

M. Berkani and H. Zariouh

Received: June 17, 2010;   Accepted: December 12, 2010

Abstract.   In [12] we introduced and studied properties (gab) and (gaw), which are extensions to the context of B-Fredholm theory, of properties (ab) and (aw) respectively introduced also in [12]. In this paper we continue the study of these properties and we consider their stability under commuting finite rank, compact and nilpotent perturbations. Among other results, we prove that if T is a bounded linear operator acting on a Banach space X, then T possesses property (gaw) if and only if T satisfies generalized Weyl's theorem and E(T) = Ea(T).
We prove also that if T possesses property ab or property (aw) or property (gaw) respectively, and N is a nilpotent operator commuting with T, then T+N possesses property ab or property aw or property (gaw) respectively. The same result holds for property (gab) in the case of a-polaroid operators.  

Keywords:  property ab, property (gab), property aw, property (gaw), B-Weyl operators  

AMS Subject classification: Primary:  47A53, 47A10, 47A11  

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Acta Mathematica Universitatis Comenianae
ISSN 0862-9544   (Printed edition)

Faculty of Mathematics, Physics and Informatics
Comenius University
842 48 Bratislava, Slovak Republic  

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