Spectral Picture For Rationally Multicyclic Subnormal Operators

TL;DR

This paper investigates the spectral properties of rationally multicyclic subnormal operators, revealing that certain spectral equalities hold under specific conditions but not universally, thus extending previous results on cyclic operators.

Contribution

It demonstrates the existence of 2-cyclic subnormal operators where spectral set closures differ and identifies conditions under which the spectral equality holds for N-cyclic operators.

Findings

Existence of 2-cyclic irreducible subnormal operator with spectral closure discrepancy.

Spectral equality holds for pure rationally N-cyclic subnormal operators under certain purity conditions.

Provides conditions ensuring the spectral set closure equality for 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 $clos(\sigma (S) \setminus \sigma_e (S)) = clos(Int (\sigma (S))).$ 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 $clos(\sigma (S) \setminus \sigma_e (S)) \neq clos(Int (\sigma (S))).$ (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 $\mathcal K_m = clos (span\{(M^*)^kx: ~x\in\mathcal H,~0\le k \le m\}.$ Suppose $M |_{\mathcal K_{N-1}}$ is pure, then $clos(\sigma (S) \setminus \sigma_e (S)) = clos(Int (\sigma (S))).$