On the spectral picture of an irreducible subnormal operator

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 result of this paper is that the only additional requirement needed for the pair $\left\{ {K,{K_e}} \right\}$ to be the spectrum and essential spectrum, respectively, is that ${K_e}$ 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 ${K_e}$.