B-Weyl spectrum and poles of the resolvent
type: Item
instance of: scholarly article
title: B-Weyl spectrum and poles of the resolvent
zbMATH Open document ID: 1043.47004
DOI: 10.1016/S0022-247X(02)00179-8
author: M. Berkani
published in: Journal of Mathematical Analysis and Applications
publication date: 2002-12-16T00:00:00Z
review text: A bounded linear operator \(T\) on a Banach space \(X\) is called a \(B\)-Fredholm operator if for some integer \(n\), the range space \(R(T^n)\) is closed and \(T: R(T^n) \rightarrow R(T^n)\) is a Fredholm operator. When the index is \(0\), \(T\) is called a \(B\)-Weyl operator. In this case, the \(B\)-Weyl spectrum is defined by \(\sigma_{BW}(T) = \{ \lambda : T - \lambda I\) is not a \(B\)-Weyl operator\}. Let \(E(T)\) denote the set of isolated eigenvalues of \(\sigma(T)\). In [Proc. Am. Math. Soc. 130, 1717--1723 (2002; Zbl 0996.47015)], the author has shown that \(\sigma_{BW}(T) = \sigma(T)-E(T)\) for a normal operator \(T\) on a Hilbert space. In the present paper, the author shows that in the context of Banach spaces, \(\sigma_{BW} \subset \sigma(T) - E(T)\) if and only if \(E(T)\) coincides with the set of all the poles of the resolvent of \(T\).
reviewed by: T.S.S.R.K. Rao
Mathematics Subject Classification ID: 47A10
Mathematics Subject Classification ID: 47B07
zbMATH DE Number: 1846009
zbMATH Keywords: B-Weyl spectrum
zbMATH Keywords: B-Fredholm operator
zbMATH Keywords: poles of the resolvent
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1846009