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Zbl 1050.47014
Berkani, M.; Koliha, J.J.
Weyl type theorems for bounded linear operators. (English)
[J] Acta Sci. Math. 69, No. 1-2, 359-376 (2003). ISSN 0001-6969

The aim of the paper under review is to show that, from the point of view of Weyl type theorems, the notion of B-Weyl spectrum (see {\it M. Berkani} [Integral Equations Oper. Theory 34, 244-249 (1999; Zbl 0939.47010)]) generalizes the notion of Weyl spectrum, as in the case of normal operators on Hilbert spaces. To explain this, we need some notation. \par Let $T$ be a bounded operator on a Banach space. One defines the following sets: the Weyl spectrum $\sigma_W(T)$ is the set of $\lambda\in {\Bbb C}$ such that $T-\lambda I$ is not a Fredholm operator of index $0$, the B-Weyl spectrum $\sigma_{BW}(T)$ is the set of $\lambda\in {\Bbb C}$ such that $T-\lambda I$ is not a B-Weyl operator of index $0$, $\sigma_{{SF_{+}^{-}}}(T)$ is the set of $\lambda\in {\Bbb C}$ such that $T-\lambda I$ is not an upper semi-Fredholm operator of negative index, and $\sigma_{{SBF_{+}^{-}}}(T)$ is the set of $\lambda\in {\Bbb C}$ such that $T-\lambda I$ is not an upper semi-B-Fredholm operator of negative index. We also need the following important sets of eigenvalues: $E_0(T)$ is the set of all eigenvalues of $T$ of finite multiplicity isolated in the spectrum of $T$, $E(T)$ is the set of all eigenvalues of $T$ isolated in the spectrum of $T$, and similarly, $E_0^a(T)$ is the set of all eigenvalues of $T$ of finite multiplicity isolated in the approximate point spectrum of $T$, and $E_0^a(T)$ is the set of all eigenvalues of $T$ isolated in the approximate point spectrum of $T$. \par Then one says that $T$ satisfies the generalized $a$-Weyl's theorem, the $a$-Weyl's theorem, the generalized Weyl's theorem, or Weyl's theorem, if $\sigma_{{SBF_{+}^{-}}}(T)=\sigma^a (T)\setminus E^a(T)$, $\sigma_{{SF_{+}^{-}}}(T)=\sigma^a (T)\setminus E^a_0(T)$, $\sigma_{BW}(T)=\sigma (T)\setminus E(T)$, or $\sigma_{W}(T)=\sigma (T)\setminus E_0(T)$, respectively. \par The author proves that if $T$ satisfies the generalized $a$-Weyl's theorem, then $T$ satisfies the $a$-Weyl's theorem and the generalized Weyl's theorem. Also, $T$ satisfies Weyl's theorem provided that $T$ obeys the generalized Weyl's theorem. \par Similar results are proved with respect to Browder's theorem.
[Ingrid Beltiţă (Bucureşti)]

MSC 2000:: *47A53 (Semi-)Fredholm operators; index theories
47A10 Spectrum and resolvent of linear operators
47A55 Perturbation theory of linear operators
47A25 Spectral sets

Keywords: semi-Fredholm operator; Weyl spectrum; Weyl type theorem; Drazin inverse; B-Weyl spectrum

Citations: Zbl 0939.47010

Cited in: Zbl 1097.47012

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