Weyl’s theorem for algebraically totally hereditarily normaloid operators

A Banach space operator T ∈ B(X) is said to be totally hereditarily normaloid, T ∈ THN, if every part of T is normaloid and every invertible part of T has a normaloid inverse. The operator T is said to be an H(q) operator for some integer q 1, T ∈ H(q), if the quasi-nilpotent part H0(T − λ) = (T − λ)−q (0) for every complex number λ. 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 σ(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 σ(T ) on which it is defined.  2004 Elsevier Inc. All rights reserved