Analytic structures for subnormal operators

For μ a compactly supported measure on ℂ, we construct a mutually absolutely continuous measure ν so that P²(ν) has analytic bounded point evaluations, and the operator of multiplication by z on P²(ν) has every invariant subspace hyperinvariant. We also construct an equivalent measure σ so that R²(K, σ) has as analytic bounded point evaluations precisely the interior of the set of weak-star continuous point evaluations of R^∞(K, μ). In the course of the proof, we classify weak-star closed super-algebras of R^∞(K, μ) when R(K) is hypo-Dirichlet.