An operator inequality for weighted Bergman shift operators

Abstract

We prove an operator inequality for the Bergman shift operator on weighted Bergman spaces of analytic functions in the unit disc with weight function controlled by a curvature parameter $\alpha$ assuming nonnegative integer values. This generalizes results by Shimorin, Hedenmalm and Jakobsson concerning the cases $\alpha =0$ and $\alpha =1$. A naturally derived scale of Hilbert space operator inequalities is studied and shown to be relaxing as the parameter $\alpha >-1$ increases. Additional examples are provided in the form of weighted shift operators.