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 \[f\left( ω\left( A \right) \right)\le \frac{1}{2}\left\| f\left( \frac{1}{1+\frac{ξ_{\left| A \right|}^{2}}{8}}\left| A \right| \right)+f\left( \frac{1}{1+\frac{ξ_{\left| A \right|}^{2}}{8}}\left| {{A}^{*}} \right| \right) \right\|,\] where ${\xi }_{|A|}=\underset{|x|=1}{\inf }\left\{\frac{⟨\left(|A|-|A^{\ast }|\right)x,x⟩}{⟨\left(|A|+|A^{\ast }|\right)x,x⟩}\right\}$. Our results refine and generalize earlier inequalities for hyponormal operator.

to appear in Oper. Matrices