Berezin number and Berezin norm inequalities for operator matrices

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.