Weyl's theorem for algebraically totally hereditarily normaloid operators

Abstract

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 H 0 ( 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.