Sharp inequalities for the numerical radii of block operator matrices (Q2288342)

Let \(A,B,C,D\) be bounded linear Hilbert space operators. The authors obtain sharp upper bounds for the numerical radii \(w(T)\) and \(w(S)\) of the block operator matrices \(T = \left(\begin{smallmatrix} A & O \\ O & D \end{smallmatrix}\right)\) and \(S = \left(\begin{smallmatrix} 0 & B \\ C & 0 \end{smallmatrix}\right)\). For example, it is shown that, if \(f(t),g(t)\) are continuous non-negative functions on \([0,\infty)\) with \(f(t)g(t)=t\), then, for any non-negative non-decreasing convex function \(h(t)\) on \([0,\infty)\), the following inequality holds: \[h(w(T))\le \frac{1}{2} \max \left(||h(f^{2} (|A|))+h(g^{2} (|A|))||,\, ||h(f^{2} (|A|))+h(g^{2} (|A|))||\right).\] A similar inequality is proved for the numerical radius \(w(S)\). The paper also contains other variations of these inequalities.