Year: 2026
Type: preprint
Abstract: We explore properties of the Berezin range $B(C_ϕ)$ of a composition operator $C_ϕ$ 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 $B(C_ϕ) \subseteq \{z\in \mathbb{C}: \Re(z) > 0 \ \text{and}\ |z| \le 1 + |ϕ(0)|\}$, that $B(C_ϕ)$ is closed if and only if $C_ϕ$ is the identity operator, and that $B(C_ϕ)$ containa all its boundary points except for $0$ if $C_ϕ$ is compact. Addressing questions raised by Cowen and Felder, we show that $C_ϕ$ can be a (nondegenerate) disk and characterize when $B(C_ϕ)$ is convex if $ϕ$ is linear.
Source: arXiv (Cornell University)
Authors: Bourdon, Paul, Le, Trieu, Thompson, Derek
Institutions: University of Virginia, Taylor University, University of Toledo
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7221803698