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 $B(T) : = \{ < Tk_x, k_x >_H : x \in X \}$, where $k_x$ is the normalized reproducing kernel for $H$ at $x \in X$. Primarily, we focus on characterizing convexity of this range for a class of composition operators acting on the Hardy space of the unit disk.