Inequalities involving 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 $\mathcal{H}.$ Among many inequalities obtained here, it is shown that if $A$ is a positive bounded linear operator on $\mathcal{H}$, then $\Vert A{\Vert }_{ber}=\text{ber}(A)$, where $\Vert A{\Vert }_{ber}$ and $\text{ber}(A)$ 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.

13 pages