Open Access

# Extended Weyl theorems and perturbations

### Abstract

In this paper we study the properties $( \rm{gaw}), (aw), ( \rm{gab})$ and $(ab)$, a variant of Weyl's type theorems introduced by Berkani. We established for a bounded linear operator defined on a Banach space several sufficient and necessary conditions for which the properties $(\rm{gaw}), (aw), ( \rm{gab})$ and $(ab)$ hold. Among other things, we study the stability of the properties $( \rm{gaw}), (aw), ( \rm{gab})$ and $(ab)$ for a polaroid operator $T$ acting on a Banach space, under perturbations by finite rank operators, by nilpotent operators and, more generally, by algebraic operators commuting with $T$.

Key words: Generalized Weyl’s theorem, generalized a-Weyl’s theorem, property (gb), property (gw), polaroid operators, perturbation theory.

### Article Information

 Title Extended Weyl theorems and perturbations Source Methods Funct. Anal. Topology, Vol. 19 (2013), no. 1, 80-96 MathSciNet MR3088321 zbMATH 1289.47027 Milestones Received 26/08/2012 Copyright The Author(s) 2013 (CC BY-SA)

### Authors Information

M. H. M. Rashid
Department of Mathematics and Statistics, Faculty of Science, P.O.Box(7), Mu'tah University, l-karak-Jordan

### Citation Example

M. H. M. Rashid, Extended Weyl theorems and perturbations, Methods Funct. Anal. Topology 19 (2013), no. 1, 80-96.

### BibTex

@article {MFAT664,
AUTHOR = {Rashid, M. H. M.},
TITLE = {Extended Weyl theorems and perturbations},
JOURNAL = {Methods Funct. Anal. Topology},
FJOURNAL = {Methods of Functional Analysis and Topology},
VOLUME = {19},
YEAR = {2013},
NUMBER = {1},
PAGES = {80-96},
ISSN = {1029-3531},
MRNUMBER = {MR3088321},
ZBLNUMBER = {1289.47027},
URL = {http://mfat.imath.kiev.ua/article/?id=664},
}