Judgement of two Weyl type theorems for bounded linear operators

Abstract

Let H be an infinite dimensional separable complex Hilbert space and B(H) the algebra of all bounded linear operators on H. T ? B(H) is said to satisfy property (UW?) if ?a(T)\?ea(T) = ?(T), where ?a(T) and ?ea(T) denote the approximate point spectrum and the essential approximate point spectrum of T respectively, ?(T) denotes the set of all poles of T. T ? B(H) satisfiesa-Weyl?s theorem if ?a(T)\?ea(T) = ?a 00(T), where ?a 00(T) = {? ? iso?a(T) : 0 < n(T ??I) < ?}. In this paper, we give necessary and sufficient conditions for a bounded linear operator and its function calculus to satisfy both property (UW?) and a-Weyl?s theorem by topological uniform descent. In addition, the property (UW?) and a-Weyl?s theorem under perturbations are also discussed.