Refinement of numerical radius inequalities of complex Hilbert space operators

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.