On the convexity of Berezin range and Berezin radius inequalities via a class of semi-norms

TL;DR

This paper introduces a new semi-norm called the $\sigma_$-Berezin norm on operators in a reproducing kernel Hilbert space, explores its properties, and uses it to improve bounds on the Berezin radius and analyze convexity of Berezin ranges for certain operators.

Contribution

It defines a novel semi-norm on operators, investigates its properties, and applies it to derive improved Berezin radius bounds and analyze convexity of Berezin ranges.

Findings

Derived improved bounds for the Berezin radius.

Established convexity of Berezin range for specific operators.

Developed inequalities related to the $\sigma_$-Berezin norm.

Abstract

This paper introduces a new family of semi-norms, say $\sigma_\mu$-Berezin norm on the space of all bounded linear operators $B(\mathcal{H})$ defined on a reproducing kernel Hilbert space $\mathcal{H}$, namely, for each $\mu \in [0,1]$ and $p\geq 1$, $$\|T\|_{\sigma_{\mu}\text{-ber}}= \sup_{\lambda\in\Omega}\left\lbrace \left(|\langle T\hat{k}_\lambda,\hat{k}_\lambda\rangle |^p~ \sigma_{\mu}~ \|T\hat{k}_\lambda\|^p\right)^{\frac{1}{p}}\right\rbrace $$ where $T\in B(\mathcal{H})$ and $\sigma_{\mu}$ is an interpolation path of the symmetric mean $\sigma$. We investigate many fundamental properties of the $\sigma_\mu$-Berezin norm and develop several inequalities associated with it. Utilizing these inequalities, we derive improved bounds for the Berezin radius of bounded linear operators, enhancing previously known estimates. Furthermore, we study the convexity of the Berezin range of a class of composition operators and weighted shift operators on both the Hardy space and the Bergman space.