Bounded point evaluations for cyclic Hilbert space operators

Abstract

Dedicated to Professor S. Naimpally on the occasion of his 70 th birthday. Abstract.In this talk, to be given at a conference at Seconda Università degli Studi di Napoli in September 2001, we shall describe the set of analytic bounded point evaluations for an arbitrary cyclic bounded linear operator T on a Hilbert space H and shall answer some questions due to L. R. Williams. 2000

On the largest analytic set for cyclic operators

“…Let T ∈ ᏸ(Ᏼ) be a cyclic operator on a Hilbert space Ᏼ. It is shown in [3] that if T possesses Bishop's property (β), then B a (T ) = Γ (T )\σ ap (T ) if and only if B a (T ) ∩ σ p (T ) = ∅ and was derived from this result that if T is hyponormal, M-hyponormal, or p-hyponormal operator, then B a (T ) = Γ (T )\σ ap (T ) (see also [2]). However, using generalized spectral theory, it is proved in [8] that B a (T )\σ lr e (T ) = Γ (T )\σ g (T ), where σ g (T ) denotes the generalized spectrum of T .…”

Local Spectra of Unilateral Operator Weighted Shifts

“…It is shown in Theorem 2.5 of [28] that a nonnormal hyponormal scalar (unilateral or bilateral) weighted shift has fat local spectra (see also [7,Theorem 3.7]). The next example shows that this result is not valid for hyponormal operator weighted shifts.…”