MATHEMATICA BOHEMICA, Vol. 131, No. 1, pp. 29-38, 2006

Single valued extension property and generalized Weyl's theorem

M. Berkani, N. Castro, S. V. Djordjevic

Mohammed Berkani, Universite Mohammed I, Faculte des Sciences, Departement de Mathematiques, Oujda, Morocco, e-mail: berkani@sciences.univ-oujda.ac.ma; Nieves Castro Gonzalez, Facultad de Informatica, Campus de Montegancedo, Boadilla del Monte, 28660 Madrid Spain, e-mail: nieves@fi.upm.es; Slavisa V. Djordjevic, Facultad de Ciencias Fisico-Matematicas, BUAP, Puebla, Mexico, e-mail: slavdj@fcfm.buap.mx

Abstract: Let $T$ be an operator acting on a Banach space $X$, let $\sigma(T)$ and $\sigma_{BW}(T)$ be respectively the spectrum and the B-Weyl spectrum of $T$. We say that $T$ satisfies the generalized Weyl's theorem if $\sigma_{BW}(T)= \sigma(T) \setminus E(T)$, where $E(T)$ is the set of all isolated eigenvalues of $T$. The first goal of this paper is to show that if $T$ is an operator of topological uniform descent and $0$ is an accumulation point of the point spectrum of $T,$ then $T$ does not have the single valued extension property at $0$, extending an earlier result of J. K. Finch and a recent result of Aiena and Monsalve. Our second goal is to give necessary and sufficient conditions under which an operator having the single valued extension property satisfies the generalized Weyl's theorem.

Keywords: single valued extension property, B-Weyl spectrum, generalized Weyl's theorem

Classification (MSC 2000): 47A53, 47A55

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