Strengthening of spectral radius, numerical radius, and Berezin radius inequalities

Suppose $\mathcal{H}_1, \mathcal{H}_2, \ldots, \mathcal{H}_n$ are arbitrary complex Hilbert spaces, and ${\bf A}=[A_{ij}]$ is an $n\times n$ operator matrix with $A_{ij}\in \mathcal{B}(\mathcal{H}_j, \mathcal{H}_i).$ We show that $w({\bf A}) \leq w\left(\begin{bmatrix} a_{ij} \end{bmatrix}_{i,j=1}^n \right),$ where $w(\cdot)$ denotes the numerical radius and the entries $$ a_{ij}=\begin{cases} w(A_{ii}) & \textit{if $i=j$}, \sqrt{ \left( \|A_{ij}\|+\|A_{ji}\| \right)^2- \left(\|A_{ij}\| \|A_{ji}\|-w(A_{ji}A_{ij}) \right)}^{} & \textit{if $i<j$}, 0 & \textit{if $i>j$.} \end{cases}$$ This bound improves $w({\bf A}) \leq w\left(\begin{bmatrix} a'_{ij} \end{bmatrix}_{i,j=1}^n \right),$ where $a'_{ij}=w(A_{ii})$ if $i=j$ and $a'_{ij}=\|A_{ij}\|$ if $i\neq j$. We deduce an upper bound for the Kronecker products $A\otimes B$, where $A\in \mathcal{M}_n(\mathbb{C})$ and $B\in \mathcal{B}(\mathcal{H}_1)$, which refines Holbrook's classical bound $w(A\otimes B)\leq w(A)\|B\|$, when all entries of $A$ are non-negative. Further, we obtain the Berezin radius inequalities for $n\times n$ operator matrices where the entries are reproducing kernel Hilbert space operators. We provide an example, which illustrates these inequalities for some concrete operators on the Hardy--Hilbert space. Applying the numerical radius bounds, we show that if $A_i \in \mathcal{B}(\mathcal{H}_i, \mathcal{H}_1) $ and $B_i\in \mathcal{B}(\mathcal{H}_1, \mathcal{H}_i)$ for $i=1,2,$ then \begin{eqnarray*} r(A_1B_1+A_2B_2) \leq \frac{ 1 }{2 } \left(w(B_1A_1)+w(B_2A_2) \right) + \frac{ 1 }{2 } \sqrt{ \left(w(B_1A_1)-w(B_2A_2)\right)^2 + 3\|B_1A_2\|\|B_2A_1\| + \eta}, \end{eqnarray*} where $\eta=w(B_2A_1 B_1A_2)$, and $r(\cdot)$ denotes the spectral radius. We also achieve a bound for the roots of an algebraic equation.