Sharper bounds for the numerical radius of $n \times n$ operator matrices II

TL;DR

This paper introduces sharper bounds for the numerical radius of operator matrices, refining existing bounds and applying these results to individual operators, products, and commutators to improve estimation accuracy.

Contribution

The paper provides new bounds for the numerical radius of operator matrices, refining previous bounds and extending these results to single operators, their products, and commutators.

Findings

New bounds for the numerical radius of operator matrices.

Refined bounds for the numerical radius of single operators.

Bounds for products and commutators of operators.

Abstract

Let $A=[A_{ij}]$ be an $n\times n$ operator matrix where each $A_{ij}$ is a bounded linear operator on a complex Hilbert space $\mathcal{H}$. With other numerical radius bounds via contraction operators, we show that $w(A) \leq w(\tilde{A}),$ where $\tilde{A}=[a_{ij}]$ is an $n\times n$ complex matrix with \begin{eqnarray*} a_{ij}=\begin{cases} w(A_{ii}) \quad \text{if } i=j\\ \underset{0\leq t \leq 1}{\min} \left\| |A_{ij}|^{2t} + |A_{ji}^*|^{2t} \right\|^{1/2} \left\| |A_{ij}^*|^{2(1-t)}+ |A_{ji}|^{2(1-t)} \right\|^{1/2} \quad \text{if } i< j 0 \quad \text{if } i> j. \end{cases} \end{eqnarray*} This bound refines the well known bound $w(A) \leq w(\hat{A}),$ where $\hat{A}=[\hat{a}_{ij}]$ is an $n\times n$ matrix with $\hat{a}_{ij}= w(A_{ii}) $ \text{if } $i=j$ and $\hat{a}_{ij}= \|A_{ij}\| $ \text{if } $i\neq j$ [Linear Algebra Appl. 468 (2015), 18--26]. We deduce that if $A$, $B$ are bounded linear operators on $\mathcal{H},$ then \begin{eqnarray*} w\left(\begin{bmatrix} 0&A\\ B&0 \end{bmatrix}\right) \leq \frac12 \left\| |A|^{2t} + |B^*|^{2t} \right\|^{1/2} \left\| |A^*|^{2(1-t)}+ |B|^{2(1-t)} \right\|^{1/2} \quad \text{for all } t\in [0,1]. \end{eqnarray*} Further by applying the numerical radius bounds of operator matrices, we deduce some numerical radius bounds for a single operator, the product of two operators, the commutator of operators. We show that if $A$ is a bounded linear operator on $\mathcal{H},$ then $w(A) \leq \frac12 \|A\|^t \left\| |A|^{1-t}+|A^*|^{1-t} \right\| \quad \text{for all } t\in [0,1],$ which refines as well as generalizes the existing ones.