On the Berezin range and the Berezin radius of some operators

TL;DR

This paper investigates the convexity properties of the Berezin range and Berezin radius for finite rank operators on Hardy and Bergman spaces, providing new operator inequalities and characterizations related to the numerical range.

Contribution

It introduces new results on the convexity of the Berezin range for finite rank operators and characterizes the closure of the numerical range via the convex hull of the Berezin set.

Findings

Convexity of Berezin range for finite rank operators established.

Operator inequalities derived from scalar inequalities.

Characterization of numerical range closure using Berezin set convex hull.

Abstract

For a bounded linear operator $T$ acting on a reproducing kernel Hilbert space $\mathcal{H}(\Omega)$ over some non-empty set $\Omega$, the Berezin range and the Berezin radius of $T$ are defined respectively, by $\text{Ber}(T) := \{\langle T\hat{k}_{\lambda},\hat{k}_{\lambda} \rangle_{\mathcal{H}} : \lambda \in \Omega\}$ and $\text{ber}(T)$ := $\sup\{|\gamma|: \gamma \in \text{Ber}(T)\}$, where $\hat{k}_{\lambda}$ is the normalized reproducing kernel for $\mathcal{H}(\Omega)$ at $\lambda \in \Omega$. In this paper, we study the convexity of the Berezin range of finite rank operators on the Hardy space and the Bergman space over the unit disc $\mathbb{D}$. We present applications of some scalar inequalities to get some operator inequalities. A characterization of closure of the numerical range of reproducing kernel Hilbert space operator in terms of convex hull its Berezin set is discussed.