Bounded Point Evaluations and Local Spectral Theory

TL;DR

This paper investigates bounded point evaluations for cyclic operators, providing a negative answer to a previous question, and generalizes results on local spectra of certain operators, with implications for spectral theory.

Contribution

It offers a negative answer to a question about bounded point evaluations and generalizes existing results on local spectra of hyponormal weighted shifts.

Findings

Negative answer to Williams' question on bounded point evaluations

Generalization of results on local spectra of hyponormal weighted shifts

Simplified proof of local spectral properties of pure quasinormal operators

Abstract

We study in this paper the concept of bounded point evaluations for cyclic operators. We give a negative answer to a question of L.R. Williams {\it Dynamic Systems and Apllications} 3(1994) 103-112. Furthermore, we generalize some results of Williams and give a simple proof of theorem 2.5 of L.R. Williams (The Local Spectra of Pure Quasinormal Operators J. Math. anal. Appl. 187(1994) 842-850) that non normal hyponormal weighted shifts have fat local spectra.