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Approximate point spectrum and commuting compact perturbations

Published online by Cambridge University Press:  18 May 2009

Vladimir Rakočević
Vladimir Rakočević
Affiliation:
University of Nish, Faculty of Philosophy, Department of Mathematics, Cirila and Metodija 2, 18000 Nish, Yugolsavia
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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 σ(A) and σa(A) denote, respectively, the spectrum and approximate point spectrum of an element A of B(X). Set

σem(A)and σeb(A) are respectively Schechter's and Browder's essential spectrum of A ([16], [9]). σea (A) is a non-empty compact subset of the set of complex numbers ℂ and it is called the essential approximate point spectrum of A ([13], [14]). In this note we characterize σab(A) and show that if f is a function analytic in a neighborhood of σ(A), then σab(f(A)) = f(σab(A)). The relation between σa(A) and σeb(A), that is exhibited in this paper, resembles the relation between the σ(A) and the σeb(A), and it is reasonable to call σab(A) Browder's essential approximate point spectrum of A.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 28 , Issue 2 , July 1986 , pp. 193 - 198
Copyright
Copyright © Glasgow Mathematical Journal Trust 1986

References

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