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

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 $w(A)\le w(\tilde{A}),$ 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 $w(A)\le w(\widehat{A}),$ where $\widehat{A}=[{\widehat{a}}_{ij}]$ is an $n\times n$ matrix with ${\widehat{a}}_{ij}=w(A_{ii})$ $\text{if }$ $i=j$ and ${\widehat{a}}_{ij}=\mathrm{\parallel }{A}_{ij}\mathrm{\parallel }$ $\text{if }$ $i\mathrm{\ne }j$ [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 $w(A)\le \frac{1}{2}\mathrm{\parallel }A{\mathrm{\parallel }}^t\parallel \mathrm{\mid }A{\mathrm{\mid }}^{1-t}+\mathrm{\mid }{A}^{\ast }{\mathrm{\mid }}^{1-t}\parallel \text{for all }t\in [0,1],$ which refines as well as generalizes the existing ones.