Some Berezin number inequalities for operator matrices

The Berezin symbol $\tilde{A}$ of an operator $A$ acting on the reproducing kernel Hilbert space $\mathcal{H}=\mathcal{H}(\mathrm{\Omega })$ over some (non-empty) set is defined by $\widetilde{A}(\lambda)=\langle A\hat{k}{\lambda},\hat{k}{\lambda}\rangle,,,(\lambda\in\Omega)$,where$\hat{k}{\lambda}=\frac{{k}{\lambda}}{|{k}{\lambda}|}$ is the normalized reproducing kernel of $\mathcal{H}$. The Berezin number of operator $A$ is defined by $$\mathbf{b}\mathbf{e}\mathbf{r}$(A) = \underset{\lambda \in \Omega}{\sup} \big|\tilde{A}(\lambda)\big|=\underset{\lambda \in \Omega }{\sup} \big|\langle A\hat{k}{\lambda}, \hat{k}_{\lambda}\rangle\big|$\mathrm{.}Moreover$$\mathbf{b}\mathbf{e}\mathbf{r}$(A)\leqslant w(A)$ (numerical radius). In this paper, we present some Berezin number inequalities. Among other inequalities, it is shown that if $$\mathbf{T}$=\left[\begin{array}{cc} A&B C&D \end{array}\right]\in {\mathbb B}({\mathscr H(\Omega_1)}\oplus{\mathscr H(\Omega_2)})$, then \begin{align*} $\mathbf{b}\mathbf{e}\mathbf{r}$($\mathbf{T}$) \leqslant\frac{1}{2}\left( $\mathbf{b}\mathbf{e}\mathbf{r}$(A)+ $\mathbf{b}\mathbf{e}\mathbf{r}$(D)\right)+\frac{1}{2}\sqrt{\left( $\mathbf{b}\mathbf{e}\mathbf{r}$(A)- $\mathbf{b}\mathbf{e}\mathbf{r}$(D)\right)^2+(|B|+|C|)^2}. \end{align*}