Single valued extension property and generalized Weyl’s theorem. (English) Zbl 1114.47015

Summary: Let \(T\) be an operator acting on a Banach space \(X\), let \(\sigma (T)\) and \( \sigma _{BW}(T) \) be the spectrum and the B-Weyl spectrum of \(T\), respectively. 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 [Pac.J.Math.58, 61–69 (1975; Zbl 0315.47002)] and a recent result of P. Aiena and O. Monsalve [J. Math.Anal.Appl.250, No.2, 435–448 (2000; Zbl 0978.47002 )]. 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.


47A53 (Semi-) Fredholm operators; index theories
47A55 Perturbation theory of linear operators
47A10 Spectrum, resolvent
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