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., $|A|=(A^{\ast }A{)}^{1/2}$, then for all $r\ge 1$, $w^{2r}(A)\le \frac{1}{4}\Vert |A{|}^{2r}+|A^{\ast }{|}^{2r}\Vert +\frac{1}{2}\min \left\{\Vert \Re (|A{|}^r|A^{\ast }{|}^r)\Vert ,w^r(A^2)\right\}$ 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 existing well-known numerical radius inequalities.

11 pages