Convexity of Berezin Range and Berezin Radius Inequalities via a class of Seminorm

TL;DR

This paper introduces a new family of seminorms called the $\sigma_t$-Berezin norm on bounded operators in reproducing kernel Hilbert spaces, providing new inequalities and characterizations of convexity of the Berezin range.

Contribution

It defines the $\sigma_t$-Berezin norm, establishes its properties, and applies it to characterize convexity of Berezin ranges for various operators on Hardy and Fock spaces.

Findings

Refined upper bounds for Berezin radius of operators.

Characterization of invertible operators that are unitary.

Convexity conditions for Berezin range of specific operators.

Abstract

Let $B(\mathcal{H})$ denote the $C^*$-algebra of all bounded linear operators acting on a reproducing kernel Hilbert space $\mathcal{H}(\Omega).$ In this paper, we introduce a new family of seminorms on $B(\mathcal{H})$, called the $\sigma_t$-Berezin norm, defined as $$ \|A\|_{{ber}_{\sigma_t}} = \sup_{\lambda,\mu\in \Omega} \left\{ \left( \left|\left\langle A\hat{k}_\lambda,\hat{k}_\mu\right\rangle\right|^p \, \sigma_t \, \left|\left\langle A^*\hat{k}_\lambda,\hat{k}_\mu\right\rangle\right|^p \right)^{\frac{1}{p}} \right\}, $$ where $A\in B(\mathcal{H}), ~p \geq 1, ~t \in [0,1]$ and ~$\sigma_t$ denotes an interpolation path of a symmetric mean $\sigma$. We show that this family of seminorms characterizes invertible operators that are unitary. Several fundamental properties of the $\sigma_t$-Berezin norm are established, along with a collection of new inequalities that yield refined upper bounds for the Berezin radius of bounded linear operators, thereby improving existing results in the literature. Furthermore, we investigate the convexity of the Berezin range of operators acting on weighted Hardy space and Fock space over $\mathbb{C}^n$. We characterised the convexity of the Berezin range of composition operator with elliptic automorphism and finite rank operators with different weights on the weighted Hardy space. We also characterized convexity of the Berezin range of composition operator on Fock space over $\mathbb{C}^n$ with symbol $\phi(z)=Az$, where $A$ is a scalar matrix of order $n$.