Sharpening Some Classical Numerical Radius Inequalities

TL;DR

This paper introduces new bounds for the numerical radius of Hilbert space operators, especially for hyponormal operators, refining existing inequalities through novel upper and lower bounds involving operator convex functions.

Contribution

It provides new upper and lower bounds for the numerical radius of operators, generalizing and refining previous inequalities specifically for hyponormal operators.

Findings

Established new bounds for numerical radii of operators.

Generalized inequalities for hyponormal operators.

Refined existing bounds with sharper estimates.

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.