Spectral Picture For Rationally Multicyclic Subnormal Operators

Abstract

For a pure bounded rationally cyclic subnormal operator $S$ on a separable complex Hilbert space $\mathcal{H},$ J. B. Conway and N. Elias (Analytic bounded point evaluations for spaces of rational functions, J. Functional Analysis, 117:1{24, 1993) showed that $clos(\ddot{I}ƒ(S)\setminus \ddot{I}{ƒ}_e(S))=clos(Int(\ddot{I}ƒ(S))).$ This paper examines the property for rationally multicyclic (N-cyclic) subnormal operators. We show: (1) There exists a 2-cyclic irreducible subnormal operator $S$ with $clos(\ddot{I}ƒ(S)\setminus \ddot{I}{ƒ}_e(S))\ne clos(Int(\ddot{I}ƒ(S))).$ (2) 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 $M{|}_{{\mathcal{K}}_{N-1}}$ is pure, then $clos(\ddot{I}ƒ(S)\setminus \ddot{I}{ƒ}_e(S))=clos(Int(\ddot{I}ƒ(S))).$

17 pages. arXiv admin note: text overlap with arXiv:1710.11265