The Bergman Shift Operator on Polynomial Lemniscates

Abstract

We investigate the relationship between the Bergman shift operator and the support of the corresponding measure. We pay special attention to the situation when the measure of orthogonality is concentrated on a polynomial lemniscate. As an application of our new results, we obtain a ratio asymptotic result for a wide variety of measures supported on polynomial lemniscates.Comment: 15 pages; version 2: small correction to Lemma 4.1, extended bibliograph