Bishop's property (β) and an elementary operator

Abstract

A Banach space operator T ∈ B(¥cal{X}) is hereditarily polaroid, T ∈ (¥cal{HP}), if the isolated points of the spectrum of every part T_p of the operator are poles of the resolvent of T_p; T is hereditarly normaloid, T ∈ (¥cal{HN}), if every part T_p of T is normaloid. Let (¥cal{HNP}) denote the class of operators T ∈ B(¥cal{X}) such that T ∈ (¥cal{HP}) ∩ (¥cal{HN}). (¥cal{HNP}) operators such that the Berberian-Quigley extension T° of T is also in (¥cal{HNP}) satisfy Bishop's property (β). Given Hilbert space operators A, B^* ∈ B(¥cal{H}), let d_AB ∈ B(B(¥cal{H})) stands for either of the elementary operators δ_AB(X) = AX - XB and Δ_AB(X) = AXB - X. If A, B^* ∈ (¥cal{HP}) satisfy property (β), and the eigenspaces corresponding to distinct eigenvalues of A (resp., B^*) are orthogonal, then f(d_AB) satisfies Weyl's theorem and f(d_AB)^* satisfies a-Weyl's theorem for every function f which is analytic on a neighbourhood of σ(d_AB). Finally, we characterize perturbations of d_AB by quasinilpotent and algebraic operators A, B ∈ B(¥cal{H}).