Sharper bounds for the numerical radius of \({n}\times {n}\) operator matrices

Let \(A=\begin{bmatrix} A_{ij} \end{bmatrix}\) $A=\left[\begin{array}{c}A_{\mathit{ij}}^{}\end{array}\right]$ be an \(n\times n\) $n\times n$ operator matrix, where each \(A_{ij}\) $A_{\mathit{ij}}^{}$ is a bounded linear operator on a complex Hilbert space. Among other numerical radius bounds, we show that \(w(A)\le w({\hat{A}}),\) $w(A)\le w(\widehat{A}),$ where \({\hat{A}}=\begin{bmatrix} {\hat{a}}_{ij} \end{bmatrix}\) $\widehat{A}=\left[\begin{array}{c}\widehat{a}_{\mathit{ij}}^{}\end{array}\right]$ is an \(n\times n\) $n\times n$ complex matrix, with \(\begin{aligned} {\hat{a}}_{ij}= {\left\{ \begin{array}{ll} w(A_{ii}) &{}\text {when }i=j, \\ \left\| | A_{ij}|+ | A_{ji}^*| \right\| ^{1/2} \left\| | A_{ji}|+ | A_{ij}^*| \right\| ^{1/2} &{}\text {when }i<j, \\ 0 &{}\hbox {when}\ i>j . \end{array}\right. } \end{aligned}\) $$\begin{array}{c}\hfill \widehat{a}_{\mathit{ij}}^{}=\left\{\begin{array}{cc}w(A_{\mathit{ii}}^{})\hfill & \text{when}i=j,\hfill \\ ∥|,A_{\mathit{ij}}^{},|+|,A_{\mathit{ji}}^{\ast },|∥_{}^{1/2}∥|,A_{\mathit{ji}}^{},|+|,A_{\mathit{ij}}^{\ast },|∥_{}^{1/2}\hfill & \text{when}i<j,\hfill \\ 0\hfill & \text{when}i>j.\hfill \end{array}\right)\end{array}$$ This is a considerable improvement of the existing bound \(w(A)\le w({\tilde{A}}),\) $w(A)\le w(\tilde{A}),$ where \({\tilde{A}}=\begin{bmatrix} {\tilde{a}}_{ij} \end{bmatrix}\) $\tilde{A}=\left[\begin{array}{c}\tilde{a}_{\mathit{ij}}^{}\end{array}\right]$ is an \(n\times n\) $n\times n$ complex matrix, with \(\begin{aligned} {\tilde{a}}_{ij}= {\left\{ \begin{array}{ll} w(A_{ii}) &{}\hbox {when}\ i=j, \\ \Vert A_{ij}\Vert &{}\hbox {when}\ i\ne j. \end{array}\right. } \end{aligned}\) $$\begin{array}{c}\hfill \tilde{a}_{\mathit{ij}}^{}=\left\{\begin{array}{cc}w(A_{\mathit{ii}}^{})\hfill & \text{when}i=j,\hfill \\ \Vert A_{\mathit{ij}}^{}\Vert \hfill & \text{when}i\ne j.\hfill \end{array}\right)\end{array}$$ Further, applying the bounds, we develop the numerical radius bounds for the product of two operators and the commutator of operators. Also, we develop an upper bound for the spectral radius of the sum of the product of n pairs of operators, which improves the existing bound.