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.


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