GENERALIZED WEYL'S THEOREM FOR FUNCTIONS OF OPERATORS AND COMPACT PERTURBATIONS

Abstract

Let H be a complex separable infinite dimensional Hilbert space. In this paper, a necessary and sufficient condition is g for an operator T on H to satisfy that f(T) obeys generalized Weyl's the- orem for each function f analytic on some neighborhood of �(T). Also we investigate the stability of generalized Weyl's theorem under (small) compact perturbations. Throughout this paper, H will always denote a complex separable infinite dimensional Hilbert space. We denote by B(H) the algebra of all bounded linear operators on H, and by K(H) the ideal of compact operators in B(H). Let T 2 B (H). We denote by �(T) andp(T) the spectrum of T and the point spectrum of T respectively. Denote by kerT and ranT the kernel of T and the range of T respectively. T is called a semi-Fredholm operator, if ranT is closed and either dimkerT or dimkerTis finite; in this case, indT := dimkerT −dimkerTis called the index of T. In particular, if −1 < indT <