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Property (gb) and perturbations. (English) Zbl 1221.47011

J. Math. Anal. Appl. 383, No. 1, 82-94 (2011); corrigendum ibid. 385, No. 2, 1195 (2012).
Summary: An operator \(T\) acting on a Banach space \(X\) possesses property (gb) if \(\sigma_a(T)\setminus\sigma_{\text{SBF}^-_+}(T)=\pi(T)\), where \(\sigma _{a}(T)\) is the approximate point spectrum of \(T\), \(\sigma_{\text{SBF}^-_+}(T)\) is the essential semi-\(B\)-Fredholm spectrum of \(T\), and \(\pi (T)\) is the set of all poles of the resolvent of \(T\). In this paper, we study property (gb) in connection with Weyl type theorems, which is analogous to generalized Browder’s theorem. Several sufficient and necessary conditions for which property (gb) holds are given. We also study the stability of property (gb) for a polaroid operator \(T\) acting on a Banach space under perturbations by finite rank operators, by nilpotent operators and, more generally, by algebraic and Riesz operators commuting with \(T\).

MSC:

47A10 Spectrum, resolvent
47A55 Perturbation theory of linear operators

Citations:

Zbl 1226.47014
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References:

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