Numerical radius inequalities for \(2\times 2\) operator matrices (Q2907264)

The authors present some upper bounds for the numerical radius of a general \(2\times 2 \) operator matrix \(\left[\begin{matrix} A & B \\ C & D \end{matrix} \right]\) and give a generalization of the known double inequality \(\frac{1}{2}\|A\| \leq w(A)\leq \|A\|\,\,(A\in B(H))\). They also establish some lower bounds for \(\left[\begin{matrix} A & B \\ 0 & 0 \end{matrix} \right]\) and obtain some numerical radius inequalities for sums and products of operators and thus extend some inequalities of \textit{S. S. Dragomir} [in: C. Bandle (ed.) et al., Inequalities and applications. Proceedings of the conference on inequalities and applications, Noszvaj, Hungary, September 9--15, 2007. Basel: Birkhäuser. International Series of Numerical Mathematics 157, 135--146 (2009; Zbl 1266.26036)].