with Paul McGuire

**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 $z \in K_e-K_a$.