WEYL TYPE THEOREMS FOR POSINORMAL OPERATORS

Abstract

Let be a bounded linear operator acting on infinite dimensional separable Hubert space H. The study of operators satisfying Weyl's theorem, Browder's theorem, the SVEP and Bishop's property is of significant interest and is currently being done by a number of mathematicians around the world. It is known that Weyl's theorem holds for M-hyponormal operators, but does not hold for dominant operators. Hence it is an interesting problem to seek a condition that implies Weyl's theorem for dominant operators. Ho Jeon et al. proved that if A is dominant and satisfies σ((A - λI)|M) = {0} ⇒ (A -λI)\M for every M ∈ Lat(A), then Weyl's theorem holds for A. Recently Cao showed that the generalized a-Weyl's theorem holds for ƒ(A), where ƒ is an analytic function defined in an open neighbourhood of σ(A) in the case where A* is p-hyponormal or M-hyponormal. Also Aiena showed that a-Weyl's theorem holds for some classes of operators. In this paper we prove that if A* is conditionally totally posinormal (with certain condition) or totally posinormal, then the generalized a-Weyl's theorem holds for A and for ƒ(A), where ƒ is an analytic function defined in an open neighbourhood of σ(A).