Numerical radius inequalities for certain \(2 \times 2\) operator matrices (Q649043)

The authors use the properties of the numerical radius of bounded linear operators on a Hilbert space \(H\), in particular its weak unitary invariance, to provide some auxiliary inequalities such as \(w\left( \left[\begin{matrix} X & Y \\ Y & X \end{matrix} \right]\right)=\max\{w(X+Y),w(X-Y)\}\). They then give upper and lower bounds for the numerical radius of the off-diagonal part \(\left[\begin{matrix} 0 & X \\ Y & 0 \end{matrix} \right]\) of \(2 \times 2\) operator matrix \(\left[\begin{matrix} Z & X \\ Y & W\end{matrix} \right]\). They utilize their results as well as the known result \(w\left( \left[\begin{matrix} X & 0 \\ 0 & Y \end{matrix} \right]\right)=\max\{w(X),w(Y)\}\) to prove that if \(X, Y, Z, W \in B(H)\), then \[ w\left( \left[\begin{matrix} X & Y \\ Z & W \end{matrix} \right] \right) \geq \max \left\{w(X),w(W),\frac{w(Y+Z)}{2},\frac{w(Y-Z)}{2}\right\} \] and \[ w\left( \left[\begin{matrix} X & Y \\ Z & W \end{matrix} \right] \right) \leq \max \left\{ w(X), w(W)\right\}+\frac{w(Y+Z)+w(Y-Z)}{2}. \] Several improvements of some norm inequalities such as \(\|X\| \leq 2w(X)\) are also established.