On the Spectral Picture of an Irreducible Subnormal Operator
Paul McGuire
Department of Mathematics, Bucknell University, Lewisburg, Pennsylvania 17837, USA
This paper extends the following result of R.F. Olin and J.E.Thomson: A compact subset $K$ of the plane is the spectrum of an irreducible subnormal operator if and only if $\mathcal{R}(K)$ has exactly one nontrivial Gleason part $G$ such that $K$ is the closure of $G$. The main reuslt of this paper is that the only additional requirement needed for the pair $\{K,K_e\}$ be a compact subset of $K$ which contains the boundary of $K$. Additionally, results are obtained on the question of which index values can be specified on the various components of the complement of $K_e$.