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On Weyl’s theorem for functions of operators. (English) Zbl 1427.47004

Summary: Let \(H\) be a complex separable infinite dimensional Hilbert space. In this paper, a variant of the Weyl spectrum is discussed. Using the new spectrum, we characterize the necessary and sufficient conditions for both \(T\) and \(f(T)\) satisfying Weyl’s theorem, where \(f \in\) Hol\(({\sigma}(T))\) and Hol\(({\sigma}(T))\) is defined by the set of all functions \(f\) which are analytic on a neighbourhood of \({\sigma}(T)\) and are not constant on any component of \({\sigma}(T)\). Also we consider the perturbations of Weyl’s theorem for \(f(T)\).

MSC:

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