Strengthening of spectral radius, numerical radius, and Berezin radius inequalities
Pintu Bhunia
TL;DR
The paper develops refined numerical radius and Berezin radius bounds for $n\times n$ operator matrices and RKHS operators, enabling sharper estimates for the spectral radius of sums, products, and commutators. The main technique combines a key inner-product inequality for pairs of bounded operators with a matrix-entry bound, yielding $w({\bf A}) \le w([a_{ij}])$ and $\textbf{ber}(\mathbf{A}) \le w([a_{ij}])$ with explicitly structured entries. These bounds then feed into applications including improved Kronecker-product bounds, spectral-radius estimates, and sharper bounds for the zeros of algebraic equations via the Frobenius companion matrix. The results refine and generalize several known inequalities in the literature and are illustrated with Hardy space examples, highlighting potential impact in operator theory and numerical analysis.
Abstract
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\| + η}, \end{eqnarray*} where $η=w(B_2A_1 B_1A_2)$, and $r(\cdot)$ denotes the spectral radius. We also achieve a bound for the roots of an algebraic equation.