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

Abstract

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 $A_{ij}\in \mathcal{B}({\mathcal{H}}_j,{\mathcal{H}}_i).$ 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 $$a_{ij}=\{\begin{array}{llcc}w(A_{ii})& \text{if i=j},{\sqrt{{\left(\Vert A_{ij}\Vert +\Vert A_{ji}\Vert \right)}^2-\left(\Vert A_{ij}\Vert \Vert A_{ji}\Vert -w(A_{ji}{A}_{ij})\right)}}^{}& \text{if i<j},0& \text{if i>j.}\end{array}$$ This bound improves $w(\mathbf{A})\le w\left({\left[\begin{array}{c}{a}_{ij}^{\prime }\end{array}\right]}_{i,j=1}^n\right),$ where $a_{ij}^{\prime }=w(A_{ii})$ if $i=j$ and $a_{ij}^{\prime }=\Vert A_{ij}\Vert$ if $i\ne j$. 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 $w(A\otimes B)\le w(A)\Vert B\Vert$, 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.

arXiv comment14 pages