Further Numerical Radius Inequalities of Operators

Abstract

We investigate advanced numerical radius inequalities for \(2 \times 2\) block operator matrices acting on complex separable Hilbert spaces. Our primary contribution establishes a refined upper bound for the numerical radius of block matrices, expressed in terms of the numerical radii of diagonal blocks and the operator norms of off-diagonal components. We demonstrate that for operators \(A, B, C, D \in \mathcal{B}(H)\), the inequality \[\begin{align*}
w \begin{bmatrix} A & B \\ C & D \end{bmatrix} &\leq \frac{1}{2}\Big\|f^2(|A|) + g^2(|A^*|)\Big\| + \frac{1}{2} \Big\|f^2(|D|) + g^2(|D^*|)\Big\| \\
&\quad + \frac{\|B\| + \|C\|}{2}
\end{align*}\]provides an effective estimation tool. Additionally, we present a comprehensive comparative analysis with existing bounds, establishing conditions under which our result offers superior performance. The theoretical framework is supplemented with concrete examples and applications to polynomial zero estimation.