Sharpening Some Classical Numerical Radius Inequalities

Abstract

New upper and lower bounds for the numerical radii of Hilbert space operators are given. Among our results, we prove that if $A\in \mathcal{B} \left( \mathcal{H}\right) $ is a hyponormal operator, then for all non-negative non-decreasing operator convex $f$ on $ [0,\infty ),$ we have \[f\left( \omega \left( A \right) \right)\le \frac{1}{2}\left\| f\left( \frac{1}{1+\frac{\xi_{\left| A \right|}^{2}}{8}}\left| A \right| \right)+f\left( \frac{1}{1+\frac{\xi_{\left| A \right|}^{2}}{8}}\left| {{A}^{*}} \right| \right) \right\|,\] where ${{\xi }_{\left| A\right| }}=\underset{\left| x\right| =1}{\mathop{\inf }}\,\left\{ \frac{\left\langle \left( \left| A\right| -\left| {{A}^{\ast }}\right| \right) x,x\right\rangle }{ \left\langle \left( \left| A\right| +\left| {A^{\ast }} \right| \right) x,x\right\rangle }\right\} $. Our results refine and generalize earlier inequalities for hyponormal operator.