## 2 Mathematical Background
Thus, the spectral radius of \(T_{n}\) is given by  
\[r(T_{n})=\max\bigg{\{}\bigg{|}a+2\sqrt{|bc|}\mathrm{e}^{i(\theta+\phi)/2}\cos \left(\frac{\pi}{n+1}\right)\bigg{|},\bigg{|}a+2\sqrt{|bc|}\mathrm{e}^{i( \theta+\phi)/2}\cos\left(\frac{n\pi}{n+1}\right)\bigg{|}\bigg{\}}. \tag{1}\]  
Furthermore, if \(bc\
eq 0\), then the eigenvectors \(x_{k}=\big{[}x_{1,n},x_{2,n},\cdots,x_{k,n}\big{]}^{T}\) associated with the eigenvalue \(\lambda_{k}\) of \(T_{n}\) are given in the form \(x_{k,j}=\big{(}\frac{c}{b}\big{)}^{k/2}\sin\left(\frac{k\pi j\pi}{n+1}\right),k,j=1,2,\cdots,n\). For a comprehensive study of Toeplitz matrices, the reader may refer to the interesting book [12]. The following result is of great interest in the results presented next [11]. **Lemma 1**. : _The tridiagonal Toeplitz matrix \(T_{n}=\mathrm{tridiag}(b,a,c)\) is normal (i.e., \(T_{n}^{*}T_{n}=T_{n}T_{n}^{*}\)) if and only if \(|b|=|c|\)._  
**Lemma 2**. : _Let \(\mathcal{H}_{1}\) and \(\mathcal{H}_{2}\) be Hilbert spaces and \(\mathbf{T}=\left[ \begin{array}{cc}A&B\\ C&D\end{array} \right]\) be an operator matrix with \(A\in\mathcal{B}(\mathcal{H}_{1})\), \(B\in\mathcal{B}(\mathcal{H}_{2},\mathcal{H}_{1})\), \(C\in\mathcal{B}(\mathcal{H}_{1},\mathcal{H}_{2})\) and \(D\in\mathcal{B}(\mathcal{H}_{2})\). Then,_  
\[\omega(\mathbf{T})\leq\frac{1}{2}\bigg{(}\omega(A)+\omega(D)+\sqrt{\left( \omega(A)-\omega(D)\right)^{2}+\left(\|B\|+\|C\|\right)^{2}}\bigg{)}.\]  
**Lemma 3**. : _If \(S:=\left[s_{kj}\right]\in\mathcal{M}_{n}(\mathbb{C})\), then,_  
\[\omega(S)\leq\omega\Big{(}\Big{[}\big{|}s_{kj}\big{|}\Big{]}\Big{)}=\frac{1}{2 }r\Big{(}\Big{[}\big{|}s_{kj}\big{|}+\Big{[}\big{|}s_{kj}\big{|}\Big{]}\Big{]} \Big{)}.\]  
Let \(A\in\mathcal{B}(\mathcal{H})\), then,  
\[|\langle Ax,y\rangle|^{2}\leq\Big{\langle}|A|^{2\alpha}x,x\Big{\rangle}\Big{ \langle}|A^{*}|^{2(1-\alpha)}y,y\Big{\rangle},\qquad 0\leq\alpha\leq 1,\]  
for any vectors \(x,y\in\mathcal{H}\), where \(|A|=\left(A^{*}A\right)^{1/2}\). This inequality is well known as the mixed Schwarz inequality, which was introduced in [13] and generalized later in [14]. The following result presents the Cartesian decomposition of the mixed Schwarz inequality [8]. **Lemma 4**. : _Let \(A\in\mathcal{B}(\mathcal{H})\) be with the Cartesian decomposition \(A=P+iQ\). If \(f\) and \(g\) are non-negative continuous functions on \([0,\infty)\) satisfying \(f(t)g(t)=t\)\((t\geq 0)\), then,_  
\[|\langle Ax,y\rangle|\leq\|f(|P|)x\|\|g(|P|)y\|+\|f(|Q|)x\|\|g(|Q|)y\|\]  
_for all \(x,y\in\mathcal{H}\)._  
**3. Numerical Radius Inequalities of \(m\times m\) Matrix Operator**  
Let us now turn our attention to numerical radius inequalities of the \(m\times m\) matrix operator, starting with the following theorem:  
**Theorem 1**. : _Let \(\mathbf{A}=\left[A_{kj}\right]\in\oplus_{m}\mathcal{M}_{n}(\mathbb{C})\) be an \(m\times m\) operator, such that \(P_{kj}+iQ_{kj}\) is the corresponding Cartesian decomposition of \(A_{kj}\). If \(f\) and \(g\) are non-negative continuous functions on \([0,\infty)\) satisfying \(f(t)g(t)=t\)\((t\geq 0)\), then,_  
\[w(\mathbf{A})\leq w^{1/2}\Big{(}\Big{[}c_{kj}\Big{]}\Big{)}, \tag{2}\]  
_where_  
\[c_{kj}=m\cdot\left\{ \begin{array}{ll}w\big{(}P_{kk}^{2}+Q_{kk}^{2}\big{)},&j= k\\ \\ \frac{1}{4}\Big{\|}f^{2}\Big{(}\Big{|}P_{kj}\Big{|}\Big{)}+g^{2}\Big{(}\Big{|} P_{kj}\Big{|}\Big{)}+f^{2}\Big{(}\Big{|}Q_{kj}\Big{|}\Big{)}+g^{2}\Big{(}\Big{|}Q_{kj} \Big{|}\Big{)}\Big{\|}^{2},&j\
eq k.\end{array} \right.\]  
Proof. : Let \(x=\left[x_{1},x_{2},\cdots,x_{m}\right]^{T}\in\bigoplus_{\ell=1}^{m}\mathcal{ M}_{n}(\mathbb{C})\) with \(\|x\|=1\); then, we have\[\frac{1}{m^{2}}|\langle\mathbf{A}x,x\rangle|^{2}=\frac{1}{m^{2}} \bigg{|}\sum_{k,j=1}^{m}\Big{\langle}A_{kj}x_{j},x_{k}\Big{\rangle}\bigg{|}^{2} \leq\frac{1}{m}\sum_{k,j=1}^{m}\Big{|}\Big{\langle}A_{kj}x_{j},x_{k}\Big{\rangle} \Big{|}^{2}\quad\text{(by [PERSON]'s inequality)}\] \[=\frac{1}{m}\sum_{k=1}^{m}|\langle A_{kk}x_{k},x_{k}\rangle|^{2} +\frac{1}{m}\sum_{ \begin{subarray}{c}j=1\\ j\
eq k\end{subarray}