Refinement of numerical radius inequalities of complex Hilbert space operators

Abstract

We develop upper and lower bounds for the numerical radius of $2\times 2$ off-diagonal operator matrices, which generalize and improve on some existing ones. We also show that if A is a bounded linear operator on a complex Hilbert space, then for all $r\ge 1$,

$$\begin{aligned} w^{2r}(A) \le \frac{1}{4} \big \Vert |A|^{2r}+|A^*|^{2r} \big \Vert + \frac{1}{2} \min \left\{ \big \Vert \Re \big (|A|^r\, |A^*|^r \big ) \big \Vert , w^r(A^2) \right\} \end{aligned}$$

where w(A), $\Vert A\Vert$ and $\Re (A)$, respectively, stand for the numerical radius, the operator norm and the real part of A. This (for $r=1$) improves on some existing well-known numerical radius inequalities.