Analytic Bounded Point Evaluations for Rationally Cyclic Operators on Banach Spaces

Abstract

In the general context of rationally cyclic operators on Banach spaces, this article centers around descriptions of the set of analytic bounded point evaluations in the spirit of function theoretic operator theory. In particular, a classical formula due to Trent for subnormal cyclic operators is extended to the case of subdecomposable operators, based on tools from local spectral theory and the Kato spectrum. Particular emphasis is placed on the connection to localized versions of the single-valued extension property and Bishop’s property (β). The results are exemplified in the case of hyponormal operators, unilateral and bilateral weighted shifts, and the Cesaro operator on the Bergman space.