Berezin number and Berezin norm inequalities for operator matrices

Abstract: We establish new upper bounds for Berezin number and Berezin norm of operator matrices, which are refinements of the existing bounds. Among other bounds, we prove that if $A=[A_{ij}]$ is an $n\times n$ operator matrix with $A_{ij}\in\mathbb{B}(\mathcal{H})$ for $i,j=1,2\dots n$, then $|A|{ber} \leq \left|\left[|A{ij}|{ber}\right]\right|$ and $\textbf{ber}(A) \leq w([a{ij}]),$ where $a_{ii}=\textbf{ber}(A_{ii}),$ $a_{ij}=\big||A_{ij}|+|A^*{ji}|\big|^{\frac{1}{2}}{ber} \big||A_{ji}|+|A^*{ij}|\big|^{\frac{1}{2}}{ber}$ if $i<j$ and $a_{ij}=0$ if $i>j$. Further, we give some examples for the Berezin number and Berezin norm estimation of operator matrices on the Hardy-Hilbert space.