Inequalities involving Berezin norm and Berezin number

TL;DR

This paper establishes new inequalities relating Berezin norm and Berezin number for operators on reproducing kernel Hilbert spaces, highlighting differences from numerical radius properties and providing specific equalities for positive operators.

Contribution

It introduces novel inequalities involving Berezin norm and Berezin number, including an equality for positive operators, and emphasizes the distinct behavior from numerical radius for selfadjoint operators.

Findings

For positive operators, Berezin norm equals Berezin number.

The equality between Berezin norm and Berezin number does not hold for selfadjoint operators.

New inequalities provide bounds and relations between Berezin norm and Berezin number.

Abstract

We obtain new inequalities involving Berezin norm and Berezin number of bounded linear operators defined on a reproducing kernel Hilbert space $\mathscr{H}.$ Among many inequalities obtained here, it is shown that if $A$ is a positive bounded linear operator on $\mathscr{H}$, then $\|A\|_{ber}=\textbf{ber}(A)$, where $\|A\|_{ber}$ and $\textbf{ber}(A)$ are the Berezin norm and Berezin number of $A$, respectively. In contrast to the numerical radius, this equality does not hold for selfadjoint operators, which highlights the necessity of studying Berezin number inequalities independently.