Sharp inequalities for the numerical radius of block operator matrices

Abstract

In this paper, we present several sharp upper bounds for the numerical radii of the diagonal and off-diagonal parts of the $2\times2$ block operator matrix $\begin{bmatrix}A&B\\ C&D\end{bmatrix}$. Among extensions of some results of Kittaneh et al., it is shown that if $T=\begin{bmatrix}A&0\\ 0&D\end{bmatrix}$, and $f$ and $g$ are non-negative continuous functions on $[0,\infty)$ such that $f(t)g(t)=t\,\,(t\geq 0)$, then for all nonnegative nondecreasing convex functions $h$ on $[0,\infty)$ , we obtain that \begin{align*}h\left(w^r(T)\right)\leq \max\left(\left\|\frac{1}{p}h\left(f^{pr}(\left|A\right|)\right)+ \frac{1}{q}h\left(g^{qr}(\left|A^*\right|)\right)\right\|, \left\|\frac{1}{p}h\left(f^{pr}(\left|D\right|)\right)+ \frac{1}{q}h\left(g^{qr}(\left|D^*\right|)\right)\right\|\right), \end{align*} where $p, q>1$ with $\frac{1}{p}+\frac{1}{q}=1$ and $r\min(p,q)\geq 2$.