Convexity of the Berezin Range

Abstract

This paper discusses the convexity of the range of the Berezin transform. For a bounded operator T acting on a reproducing kernel Hilbert space H (on a set X), this is the set B(T):={〈Tkˆx,kˆx〉H:x∈X}, where kˆx is the normalized reproducing kernel for H at x∈X. Primarily, we focus on characterizing convexity of this range for a class of composition operators acting on the Hardy space of the unit disk.