On the Spectral Picture of an Irreducible Subnormal Operator II

In this paper the we construct irreducible subnormal operators with a prescribed spectral picture, subject to natural necessary conditions.  In 1988 P. McGuire did a significant part of this, but left one important case open.  That is, the case of prescribing the spectral picture when the index was to be -1.

Main Theorem:  Suppose that K, K_e, and K_a are compact sets in the complex plane such that:
1) $\partial{K} \subseteq K_a \subseteq K_e \subseteq K$ and $\partial{K_a} \subseteq K_e$;
2) R(K) has only one non-trivial Gleason part G and clG = K;
3) for each component V_n of K-K_e an integer a_n \leq -1 has been chosen,

then there exists an irreducible subnormal operator S for which:

1) $\sigma(S) = K$, $\sigma_{ap}(S) = K_a$, $\sigma_e(S) = K_e$;
2) ind(S-z) = a_n for all $z \in V_n$;
3) ind(S-z) = -\infty for all $z \in K_e-K_a$.