WEYL TYPE THEOREMS FOR POSINORMAL OPERATORS

Abstract

Let A be a bounded linear operator acting on infinite dimensional separable Hilbert space H. The study of operators sat- isfying Weyl's theorem, Browder's theorm, 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 which implies Weyl's theorem for dominant oper- ators. In Ho Jeon etat () proved that if A is dominant and satisfies ((A I )|M) = {0} ) (A I )|M for every M 2 Lat(A), then Weyl's theorem holds for A. Recently X.Cao () showed that gen- eralized a-Weyl's theorem holds for f(A), where f is an analytic function defined on an open neighborhood of (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 pa- per we prove that if A is conditionally totally posinormal (with certain condition) or totally posinormal, then generalized a-Weyl's theorem holds for A and for f(A), where f is an analytic function defined on an open neighborhood of (A).