A new norm on the space of reproducing kernel Hilbert space operators and Berezin number inequalities

Abstract: In this note, we introduce a novel norm, termed the $t-$Berezin norm, on the algebra of all bounded linear operators defined on a reproducing kernel Hilbert space $\mathcal{H}$ as $$|A|{t-ber} = \sup{ \lambda, \mu \in \Omega} \left{ t|\langle A \hat{k}\lambda, \hat{k}\mu \rangle| + (1-t) |\langle A^* \hat{k}\lambda, \hat{k}\mu \rangle| \right}, \quad t\in [0,1],$$ where $A \in \mathcal{B}(\mathcal{H})$ is a bounded linear operator. This norm characterizes those invertible operators which are also unitary. Using this newly defined norm, we establish various upper bounds for the Berezin number, thereby refining the existing results. Additionally, we derive several sharp bounds for the Berezin number of an operator via the Orlicz function.