Berezin Ranges of Composition Operators on $H^2$

We explore properties of the Berezin range $B(C_{\phi })$ of a composition operator $C_{\phi }$ 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 $B(C_{\phi })\subseteq \{z\in \mathbb{C}:\mathrm{\Re }(z)>0\text{ and }\mathrm{\mid }z\mathrm{\mid }\le 1+\mathrm{\mid }\phi (0)\mathrm{\mid }\}$, that $B(C_{\phi })$ is closed if and only if $C_{\phi }$ is the identity operator, and that $B(C_{\phi })$ containa all its boundary points except for $0$ if $C_{\phi }$ is compact. Addressing questions raised by Cowen and Felder, we show that $C_{\phi }$ can be a (nondegenerate) disk and characterize when $B(C_{\phi })$ is convex if $\phi$ is linear.