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Generalized Weyl’s theorem and hyponormal operators. (English) Zbl 1061.47021
A Banach space operator $$T\in B(X)$$ is ‘B-Fredholm’ if there exists a natural number for which the induced operator $$T_n:T^n(X)\longrightarrow T^n(X)$$ is Fredholm (in the usual sense); $$T$$ is ‘B-Weyl’ if $$T_n$$ has index $$0$$, and $$T$$ satisfies the generalized Weyl’s theorem if the complement in $$\sigma(T)$$ of the set of $$\lambda$$ for which $$T-\lambda$$ fails to be B-Weyl consists of the set of isolated points of $$\sigma(T)$$ which are eigenvalues of $$T$$ (with no restriction on multiplicity). The authors prove that hyponormal (Hilbert space) operators satisfy the generalized Weyl’s theorem, and that the B-Weyl spectrum of such an operator satisfies the spectral mapping theorem.

##### MSC:
 47B20 Subnormal operators, hyponormal operators, etc. 47A53 (Semi-) Fredholm operators; index theories 47A55 Perturbation theory of linear operators
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