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

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.