The commutant of rationally cyclic subnormal operators and rational approximation

Let denote the set of analytic bounded point evaluations forR ^q (K, ). Assume that . In this paper, we first show that if is a finitely connected domain and if the evaluation map fromR ^q (K, )L () toH () is surjective, then | is absolutely continuous with respect to harmonic measure for . This generalizes Olin and Yang's corresponding result for polynomials and the proof we present here is simpler. We also provide an example that shows this absolute continuity property fails in general when is an infinitely connected domain. In the second part, we then offer a solution to a problem of Conway and Elias.