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New approach to a-Weyl’s theorem through localized SVEP and Riesz-type perturbations. (English) Zbl 1511.47013

Summary: In this paper, we study the properties \((bz)\) and \((w_{\pi_{00}^a})\) which we had introduced in [K. Ben Ouidren and H. Zariouh, Rend. Circ. Mat. Palermo (2) 70, No. 2, 819–833 (2021; Zbl 07383957)], for an operator having the SVEP on the complementary of distinguished parts of its spectrum. We prove in particular, that a bounded linear operator \(T\) acting on a Banach space has the SVEP on the complementary of \(\sigma_{uf}(T)\) if and only if \(T\) possesses property \((bz)\). We also study the stability of these properties under several commuting perturbations, and we prove that if \(T\) possesses property \((bz)\) then \(f(T)+R\) possesses property \((bz)\) for every Riesz operator \(R\) commuting with \(T\) and \(f \in \operatorname{Hol}(\sigma(T))\). We also give an example which shows that the property \((w_{\pi_{00}^a})\) is generally unstable under this type of perturbations, and we prove that if \(T\) possesses property \((w_{\pi_{00}^a})\) then \(T + R\) possesses property \((w_{\pi_{00}^a}) \Leftrightarrow {\pi_{00}^a}(T+R) \cap \sigma_a (T) \subset {\pi_{00}^a}(T)\). Analogous results are proved for the property \((gw_{\pi_{00}^a})\) and some applications to the class of a-isoloid-type operators are given.

MSC:

47A53 (Semi-) Fredholm operators; index theories
47A10 Spectrum, resolvent
47A11 Local spectral properties of linear operators
47A55 Perturbation theory of linear operators

Citations:

Zbl 07383957
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References:

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