Bounded Point Evaluations For Rationally Multicyclic Subnormal Operators

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, $\lambda_0 \in \sigma (S) \setminus \sigma_e (S).$