POLAROID OPERATORS AND GENERALIZED BROWDER-WEYL THEOREMS

Abstract

A Banach space operator T ∈ B(X ) is polaroid (left polaroid) if isolated points of the spectrum (resp., isolated points λ of the approximate point spectrum) of T are poles of the resolvent of T (resp., are such that (T − λI) has finite ascent ≤ d and (T − λI)X is closed). Necessary and sufficient conditions for operators T ∈ B(X ) to satisfy generalized and a-generalized Browder and Weyl theorems are given. In the case of polaroid (resp., left polaroid) operators T , it is proved that T satisfies generalized Weyl’s theorem (resp., generalized a–Weyl’s theorem) if and only if T satisfies Weyl’s theorem (resp., a–Weyl’s theorem).