Convexity of the Berezin Range

TL;DR

This paper investigates the convexity of the Berezin transform range for operators on reproducing kernel Hilbert spaces, with a focus on composition operators on the Hardy space of the unit disk.

Contribution

It characterizes the convexity of the Berezin range specifically for a class of composition operators on the Hardy space.

Findings

Identifies conditions under which the Berezin range is convex.

Provides a characterization for composition operators on the Hardy space.

Enhances understanding of the geometric properties of Berezin transforms.

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.