## 2 Mathematical Background  
Some preliminaries and notations are now introduced for the purposes of the study. Let \(\mathcal{H}\) be a complex Hilbert space with an inner product \(\langle\cdot,\cdot\rangle\) and \(\mathcal{B}(\mathcal{H})\) be the \(\mathcal{C}^{*}\)-algebra of all bounded linear operators from \(\mathcal{H}\) into itself. When \(\mathcal{H}=\mathbb{C}^{n}\), we identify \(\mathcal{B}(\mathcal{H})\) with the algebra \(\mathcal{M}_{n}\) of \(n\)-by-\(n\) complex matrices. For a bounded linear operator \(T\) on a Hilbert space \(\mathcal{H}\), the numerical range \(W(T)\) is the image of the unit sphere of \(\mathcal{H}\) under the quadratic form \(x\to\langle Tx,x\rangle\) associated with the operator. More precisely, we can set  
\[W(T)=\{\langle Tx,x\rangle:x\in\mathcal{H},\|x\|=1\}.\]  
In addition, the numerical radius is defined to be  
\[w(T)=\sup\{|\lambda|:\lambda\in W(T)\}=\sup_{\|x\|=1}|\langle Tx,x\rangle|.\]  
The spectral radius of an operator \(T\) is defined to be  
\[r(T)=\sup\{|\lambda|:\lambda\in\text{sp}(T)\}.\]  
We recall that the usual operator norm of an operator \(T\) is defined to be  
\[\|T\|=\sup\{\|Tx\|:x\in H,\|x\|=1\}.\]  
Several numerical radius type inequalities improving and refining the inequality  
\[\frac{1}{2}\|T\|\leq w(T)\leq\|T\|\qquad\qquad(T\in\mathcal{B}(\mathcal{H}))\]  
have been recently obtained by many other authors. See, for example, refs. [5; 6; 7; 8; 9; 10]. Four important facts concerning the numerical radius inequalities of \(n\times n\) operator matrices are obtained by different authors, which are grouped together as follows:  
\[\text{Let }\mathbf{S}=\big{[}S_{ij}\big{]}\in\mathcal{B}(\bigoplus_{i=1}^{n} \mathcal{H}_{i})\text{ such that }S_{ij}\in\mathcal{B}(\mathcal{H}_{j},\mathcal{H}_{i}).\text{ Then, Refs [5; 9; 10]}\]  
\[w(\mathbf{S})\leq\left\{ \begin{array}{l}w\Big{(}\big{[}t_{kj}^{(1)}\big{]} \Big{)},\\ \\ w\Big{(}\big{[}t_{kj}^{(2)}\big{]}\Big{)},\\ \\ w\Big{(}\big{[}t_{kj}^{(3)}\big{]}\Big{)},\\ \\ w\Big{(}\big{[}t_{kj}^{(4)}\big{]}\Big{)},\end{array} \right.\]where  
\[t_{kj}^{(1)}=w\Big{(}\big{[}\big{\|}S_{kj}\big{\|}\big{]}\Big{)};\qquad t_{kj}^{(2 )}=\left\{ \begin{array}{ll}\frac{1}{2}\Big{(}\big{\|}S_{kj}\big{\|}+\big{\|}S_ {kj}^{2}\big{\|}^{1/2}\Big{)},&k=j\\ \big{\|}S_{kj}\big{\|},&k\
eq j\end{array} \right. ;\]  
\[t_{kj}^{(3)}=\left\{ \begin{array}{ll}w\Big{(}S_{kj}\Big{)},&k=j\\ \big{\|}S_{kj}\Big{\|},&k\
eq j\end{array} \right. ;\qquad t_{kj}^{(4)}=\left\{ \begin{array}{ll}w\Big{(}S_{kj}\Big{)},&k=j\\ \\ w\Big{(}\begin{array}{cc}0&S_{kj}\\ S_{jk}&0\end{array}\Big{)},&k\
eq j\end{array} \right..\]  
Clearly, the third and fourth bounds above are gentle refinements of the first and second bounds; therefore, both \(w\Big{(}\big{[}t_{kj}^{(3)}\big{]}\Big{)}\) and \(w\Big{(}\big{[}t_{kj}^{(4)}\big{]}\Big{)}\) give better upper estimates for the numerical radius of \(\mathbf{S}=\big{[}S_{ij}\big{]}\). Let \(T_{n}\) be the tridiagonal Toeplitz matrix denoted by \(T_{n}=\mathrm{tridiag}(b,a,c)\), i.e.,  
\[T_{n}:=\left[ \begin{array}{cccc}a&c&0&\cdots&0\\ b&a&c&\ddots&\vdots\\ 0&b&\ddots&\ddots&0\\ \vdots&\ddots&\ddots&\ddots&c\\ 0&\cdots&0&b&a\end{array} \right]_{n\times n},\qquad\qquad n\geq 2.\]  
It is well known that the eigenvalues of \(T_{n}\) are given by [11] as  
\[\lambda_{k}=a+2\sqrt{|bc|}\cos\left(\frac{k\pi}{n+1}\right),\qquad k=1,2, \cdots,n\]  
and they have the polar form  
\[\lambda_{k}=a+2\sqrt{|bc|\mathrm{e}^{i(\theta+\phi)/2}\cos\left(\frac{k\pi}{ n+1}\right)},\qquad k=1,2,\cdots,n,\]  
where \(\theta=\arg(b)\) and \(\phi=\arg(c)\). In the case of \(bc\
eq 0\), \(T_{n}\) has \(n\) simple eigenvalues, and all of them lie in the closed segment  
\[S_{n,\lambda}=\bigg{\{}a+t\mathrm{e}^{i(\theta+\phi)/2}:t\in\mathbb{R},\,|t| \leq 2\sqrt{|bc|}\cos\left(\frac{\pi}{n+1}\right)\bigg{\}}\subset\mathbb{C}.\]  
The eigenvalues are located symmetrically with respect to \(a\).