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 $abpe(S)=bpe(S)\cap Int(\ddot{I}ƒ(S)).$ 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 $\hat{I}{»}_0\in Int(\ddot{I}ƒ(S))$ and $dim(ker(S-\hat{I}{»}_0{)}^{\ast })=N,$ where $N$ is the minimal number of cyclic vectors for $S,$ then the range of $S-\hat{I}{»}_0$ is closed, hence, $\hat{I}{»}_0\in \ddot{I}ƒ(S)\setminus \ddot{I}{ƒ}_e(S).$

11 pages