Sharp Inequalities for the Numerical Radii 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 2×2 block operator matrix $$\left[ {\begin{array}{*{20}{c}} A&B \\ C&D \end{array}} \right]$$. Among extensions of some results of Kittaneh et al., it is shown that if $$T = \left[ {\begin{array}{*{20}{c}} A&0 \\ 0&D \end{array}} \right]$$, and f and g are non-negative continuous functions on [0,∞) such that f(t)g(t) = t (t ≥ 0), then for all non-negative nondecreasing convex functions h on [0,∞), we obtain that $$\begin{array}{*{20}{c}} {h({w^r}(T))} \\ { \leqslant \max \left( {\left\| {\frac{1}{p}h({f^{pr}}(|A|)) + \frac{1}{q}h({g^{qr}}(|A*|))} \right\|,\left\| {\frac{1}{p}h({f^{pr}}(|D|)) + \frac{1}{q}h({g^{qr}}(|D*|))} \right\|} \right)} \end{array}$$ where p, q > 1 with $$\tfrac{1}{p} + \tfrac{1}{q} = 1$$, and r min(p, q) ≥ 2.