Bounded Point Evaluations For Rationally Multicyclic Subnormal Operators

TL;DR

This paper characterizes the set of bounded point evaluations for rationally multicyclic subnormal operators, proving a key equality and deriving spectral properties related to the operator's cyclic vectors and spectrum.

Contribution

It establishes that the analytic bounded point evaluations equal the intersection of bounded point evaluations with the interior of the spectrum, answering a question by J. B. Conway.

Findings

Proves $abpe(S) = bpe(S) igcap ext{Int}(\sigma(S))$.

Shows the range of $S - \lambda_0$ is closed under certain conditions.

Identifies spectral properties related to cyclic vectors and the spectrum.

Abstract

Let $S$ be a pure bounded rationally multicyclic subnormal operator on a separable complex Hilbert space $\mathcal H$ and let $M_z$ be the minimal normal extension on a separable complex Hilbert space $\mathcal K$ containing $\mathcal H.$ Let $bpe(S)$ be the set of bounded point evaluations and let $abpe(S)$ be the set of analytic bounded point evaluations. We show $abpe(S) = bpe(S) \cap Int(\sigma (S)).$ The result affirmatively answers a question asked by J. B. Conway concerning the equality of the interior of $bpe(S)$ and $abpe(S)$ for a rationally multicyclic subnormal operator $S.$ As a result, if $\lambda_0\in Int(\sigma (S))$ and $dim(ker(S-\lambda_0)^*) = N,$ where $N$ is the minimal number of cyclic vectors for $S,$ then the range of $S-\lambda_0$ is closed, hence, $\lambda_0 \in \sigma (S) \setminus \sigma_e (S).$