On the convexity of the Berezin range of composition operators and related questions

Abstract

The Berezin range of a bounded operator $T$ acting on a reproducing kernel Hilbert space $\mathcal{H}$ is the set $B(T)$ := $\{⟨T{\hat{k}}_x,{\hat{k}}_x{⟩}_{\mathcal{H}}:x\in X\}$, 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. Primarily, we focus on characterizing the convexity of the Berezin range for a class of composition operators acting on the Fock space on $\mathbb{C}$ and the Dirichlet space of the unit disc $\mathbb{D}$. We prove an analogue of the elliptic range theorem for the unitarily equivalent Berezin range of an operator on a two-dimensional reproducing kernel Hilbert space and characterize the convexity of the unitarily equivalent Berezin range for a bounded operator $T$ on a reproducing kernel Hilbert space $\mathcal{H}$.

20 Pages