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Weyl’s theorem for operator matrices. (English) Zbl 0923.47001
Summary: “Weyl’s theorem holds” for an operator when the complement in the spectrum of the “Weyl spectrum” coincides with the isolated points of the spectrum which are eigenvalues of finite multiplicity. By comparison “Browder’s theorem holds” for an operator when the complement in the spectrum of the Weyl spectrum coincides with Riesz points. Weyl’s theorem and Browder’s theorem are liable to fail for \(2\times 2\) operator matrices. In this paper we explore how Weyl’s theorem and Browder’s theorem survive for \(2\times 2\) operator matrices in Hilbert space.

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