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Generalized spectrum and commuting compact perturbations

Published online by Cambridge University Press:  20 January 2009

Vladimir Rakočević
Vladimir Rakočević
Affiliation:
University of Niš, Faculty of Philosophy, Department of Mathematics, Ćirila and Metodija 2, 18000 Niš, Yugoslavia
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Abstract

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Let X be an infinite-dimensional complex Banach space and denote the set of bounded (compact) linear operators on X by B(X) (K(X)). Let N(A) and R(A) denote, respectively, the null space and the range space of an element A of B(X). Set R(A∞)=∩nR(An) and k(A)=dim N(A)/(N(A)∩R(A∞)). Let σg(A) = ℂ\{λ∈ℂ:R(A−λ) is closed and k(A−λ)=0} denote the generalized (regular) spectrum of A. In this paper we study the subset σgb(A) of σg(A) defined by σgb(A) = ℂ\{λ∈ℂ:R(A−λ) is closed and k(A−λ)<∞}. Among other things, we prove that if f is a function analytic in a neighborhood of σ(A), then σgb(f(A)) = f(σgb(A)).

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 36 , Issue 2 , June 1993 , pp. 197 - 209
Copyright
Copyright © Edinburgh Mathematical Society 1993

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