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Numerical radius inequalities for certain $$2 \times 2$$ operator matrices. (English) Zbl 1238.47004
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.

##### MSC:
 47A12 Numerical range, numerical radius 47A30 Norms (inequalities, more than one norm, etc.) of linear operators 47A63 Linear operator inequalities
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##### References:
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