# Composition operators, convexity of their Berezin range and related questions

## TL;DR

This paper investigates the convexity of Berezin ranges for operators, especially composition operators on Hardy and Bergman spaces, and establishes a Berezin set mapping theorem for superquadratic functions and positive operators.

## Contribution

It characterizes the convexity of Berezin ranges for certain composition operators and proves a new Berezin set mapping theorem for superquadratic functions and positive operators.

## Key findings

- Convexity of Berezin range characterized for specific composition operators.
- Established Berezin set mapping theorem for superquadratic functions.
- Analyzed the structure of Berezin ranges in Hardy and Bergman spaces.

## Abstract

The Berezin range of a bounded operator $T$ acting on a reproducing kernel Hilbert space $\mathcal{H}$ is the set $\text{Ber}(T)$ := $\{\langle T\hat{k}_{x},\hat{k}_{x} \rangle_{\mathcal{H}} : x \in X\}$, where $\hat{k}_{x}$ is the normalized reproducing kernel for $\mathcal{H}$ at $x \in X$. In general, the Berezin range of an operator is not convex. In this paper, we discuss the convexity of range of the Berezin transforms. We characterize the convexity of the Berezin range for a class of composition operators acting on the Hardy space and the Bergman space of the unit disk. Also for so-called superquadratic functions, we prove the Berezin set mapping theorem for positive self-adjoint operators $A$ on the reproducing kernel Hilbert space $\mathcal{H}(\Omega)$, namely we prove that $f(\mathrm{Ber}(\Phi(A)))=\mathrm{Ber}(\Phi(f(A)))$, where $\Phi:\mathcal{B}%\left( \mathcal{H}\left( \Omega\right) \right) \mathcal{\rightarrow}\mathcal{B}\left( \mathcal{K(}Q\mathcal{)}\right) $ is a normalized positive linear map.