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 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.
Bourhim, Abdellatif, "Bounded Point Evaluations and Local Spectral Theory" (2000). Mathematics Faculty Scholarship. Paper 85.
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