Glasgow Mathematical Journal

Research Article

Approximate point spectrum and commuting compact perturbations

Vladimir Rakočevića1

a1 University of Nish, Faculty of Philosophy, Department of Mathematics, Cirila and Metodija 2, 18000 Nish, Yugolsavia

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

S0017089500006509_eqnU1

S0017089500006509_eqnU2

S0017089500006509_eqnU3

S0017089500006509_eqnU4

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 xs2102 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)) = fab(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.

(Received June 26 1985)