The Kato-type spectrum and local spectral theory |
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| Authors: | T L Miller V G Miller M M Neumann |
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| Institution: | (1) Dept. of Mathematics and Statistics, Mississippi State University, Drawer MA, Mississippi State, MS 39762 |
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| Abstract: | Let T ∈ ℒ(X) be a bounded operator on a complex Banach space X. If V is an open subset of the complex plane such that λ-T is of Kato-type for each λ ∈ V, then the induced mapping f(z) ↦ (z-T)f(z) has closed range in the Fréchet space of analytic X-valued functions on V. Since semi-Fredholm operators are of Kato-type, this generalizes a result of Eschmeier on Fredholm operators and leads to
a sharper estimate of Nagy’s spectral residuum of T. Our proof is elementary; in particular, we avoid the sheaf model of Eschmeier and Putinar and the theory of coherent analytic
sheaves. |
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| Keywords: | decomposable operator semi-Fredholm operator semi-regular operator Kato decomposition Bishop’ s property (β ) property (δ ) |
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