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

Suppose ${\mathcal{H}}_1,{\mathcal{H}}_2,\dots ,{\mathcal{H}}_n$ are arbitrary complex Hilbert spaces, and $\mathbf{A}=[A_{ij}]$ is an $n\times n$ operator matrix with $A_{ij}\in \mathcal{B}({\mathcal{H}}_j,{\mathcal{H}}_i)\mathrm{.}$ We show that $w(\mathbf{A})\le w\left({\left[\begin{array}{c}{a}_{ij}\end{array}\right]}_{i,j=1}^n\right),$ where $w(\cdot )$ denotes the numerical radius and the entries $$a_{ij}=\{\begin{array}{llll}w(A_{ii})& \mathit{\text{if}}i=j,{\sqrt{{(\mathrm{\parallel }{A}_{ij}\mathrm{\parallel }+\mathrm{\parallel }{A}_{ji}\mathrm{\parallel })}^2-(\mathrm{\parallel }{A}_{ij}\mathrm{\parallel }\mathrm{\parallel }{A}_{ji}\mathrm{\parallel }-w(A_{ji}{A}_{ij}))}}^{}& \mathit{\text{if}}i<j,0& \mathit{\text{if}}i>j\mathit{\text{.}}\end{array}$$ This bound improves $w(\mathbf{A})\le w\left({\left[\begin{array}{c}{a}_{ij}^{\mathrm{\prime }}\end{array}\right]}_{i,j=1}^n\right),$ where $a_{ij}^{\mathrm{\prime }}=w(A_{ii})$ if $i=j$ and $a_{ij}^{\mathrm{\prime }}=\mathrm{\parallel }{A}_{ij}\mathrm{\parallel }$ if $i\mathrm{\ne }j$. 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 $w(A\otimes B)\le w(A)\mathrm{\parallel }B\mathrm{\parallel }$, 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| + η}, \end{eqnarray*} where $\eta =w(B_2{A}_1{B}_1{A}_2)$, and $r(\cdot )$ denotes the spectral radius. We also achieve a bound for the roots of an algebraic equation.