Berezin Ranges of Composition Operators on $H^2$

Abstract

We explore properties of the Berezin range $B(C_{\varphi })$ of a composition operator $C_{\varphi }$ on the Hardy space $H^2$, including constraints on its location in the complex plane $\mathbb{C}$, topological properties, shape, symmetry, and convexity. For example, we show that $B(C_{\varphi })\subseteq \{z\in \mathbb{C}:\Re (z)>0\text{and}|z|\le 1+|\varphi (0)|\}$, that $B(C_{\varphi })$ is closed if and only if $C_{\varphi }$ is the identity operator, and that $B(C_{\varphi })$ containa all its boundary points except for $0$ if $C_{\varphi }$ is compact. Addressing questions raised by Cowen and Felder, we show that $C_{\varphi }$ can be a (nondegenerate) disk and characterize when $B(C_{\varphi })$ is convex if $\varphi$ is linear.

arXiv comment27 pages, 8 figures