Weyl's theorem for algebraically totally hereditarily normaloid operators

A Banach space operator T is an element of B(X) is said to be totally hereditarily normaloid, T is an element of THN, if every part of T is normaloid and every invertible part of T has it normaloid inverse. The operator T is said to be an H(q) operator for some integer q >= 1. T is an element of H(q), if the quasi-nilpotent part H-0(T - lambda) = (T - lambda)(-q(0)) for every complex number lambda. It is proved that if T is algebraically H(q), or T is algebraically THN and X is separable, then f(T) satisfies Weyl's theorem for every function f analytic in an open neighborhood of sigma(T), and T* satisfies a-Weyl's theorem. If also T* has the single valued extension property, then f(T) satisfies a-Weyl's theorem for every analytic function f which is non-constant on the connected components of the open neighborhood of a (T) on which it is defined. (c) 2004 Elsevier Inc. All rights reserved.