Spectral picture for rationally multicyclic subnormal operators

Abstract

For a pure bounded rationally cyclic subnormal operator $S$ on a separable complex Hilbert space $\mathcal{H}$, Conway and Elias showed that $clos\left(\sigma \right(S)\setminus {\sigma }_e(S\left)\right)=clos(Int(\sigma \left(S\right)\left)\right)$. This article examines the property for rationally multicyclic ($N$-cyclic) subnormal operators. We show that there exists a $2$-cyclic irreducible subnormal operator $S$ with $clos\left(\sigma \right(S)\setminus {\sigma }_e(S\left)\right)\ne clos(Int(\sigma \left(S\right)\left)\right)$. We also show the following. For a pure rationally $N$-cyclic subnormal operator $S$ on $\mathcal{H}$ with the minimal normal extension $M$ on $\mathcal{K}\supset \mathcal{H}$, let ${\mathcal{K}}_m=clos(span\{(M^{\ast }{)}^{k}x:x\in \mathcal{H},0\le k\le m\}$. Suppose that $M{|}_{{\mathcal{K}}_{N-1}}$ is pure. Then $clos\left(\sigma \right(S)\setminus {\sigma }_e(S\left)\right)=clos(Int(\sigma \left(S\right)\left)\right)$.