Weyl Type Theorems for Functions of Operators

Abstract

Abstract.A-Weyl's theorem and property (ω), as two variations of Weyl's theorem, were introduced by Rakočević. In this paper, we study a-Weyl's theorem and property (ω) for functions of bounded linear operators. A necessary and sufficient condition is given for an operator T to satisfy that f (T) obeys a-Weyl's theorem (property (ω)) for all f ∈ Hol(σ (T)). Also we investigate the small-compact perturbations of operators satisfying a-Weyl's theorem (property (ω)) in the setting of separable Hilbert spaces.2000 Mathematics Subject Classification. Primary 47A53, 47A60; Secondary 47A10.1. Introduction. This paper is a continuation of a previous paper of the authors and Feng [17], where the stability of Weyl's theorem under holomorphic functional calculus is studied. A-Weyl's theorem and property (ω) as two variations of Weyl's theorem, which have been recently studied in [3, 4, 5], were introduced by Rakočević [21,22]. The purpose of this paper is to investigate a-Weyl's theorem and property (ω) for functions of operators on Banach spaces. A necessary and sufficient condition is given for an operator T to satisfy that f (T) obeys a-Weyl's theorem (property (ω)) for each function f analytic on some neighbourhood of σ (T).We first give some notations and terminologies. Throughout this paper, ‫ރ‬ and ‫ގ‬ denote the set of complex numbers and the set of natural numbers respectively. X will always denote a complex infinite dimensional Banach space. We let B(X ) denote the algebra of all bounded linear operators on X , and let K(X ) denote the ideal of compact operators in B(X ).Let T ∈ B(X ). We denote by σ (T) and σ p (T) the spectrum of T and the point spectrum of T respectively. Denote by n(T) and R(T) the kernel of T and the range