The Berezin transform and 𝑚th-order Bergman metric

Abstract

We improve upon recent directional derivative estimates for Berezin’s operator calculus, and consider the relation between the m m th-order Bergman metric of Burbea and the classical Bergman metric in the analysis of higher order directional derivative estimates of the Berezin symbols of general bounded operators. A new metric, naturally arising in our analysis, is introduced and certain comparison theorems are established among this metric, the m m th-order Bergman metric and the classical Bergman metric on the unit ball, the polydisk and C n \mathbb C^n .