Improvement of numerical radius inequalities

Abstract: We develop upper and lower bounds for the numerical radius of $2\times 2$ off-diagonal operator matrices, which generalize and improve on the existing ones. We also show that if $A$ is a bounded linear operator on a complex Hilbert space and $|A|$ stands for the positive square root of $A$, i.e., $|A|=(A^*A)^{1/2}$, then for all $r\geq 1$, $w^{2r}(A) \leq \frac{1}{4} \big | |A|^{2r}+|A^*|^{2r} \big | + \frac{1}{2} \min\left{ \big |\Re\big(|A|^r\, |A^*|^r \big) \big |, w^r(A^2) \right} $ where $w(A)$, $|A|$ and $\Re(A)$, respectively, stand for the numerical radius, the operator norm and the real part of $A$. This (for $r=1$) improves on existing well-known numerical radius inequalities.