REFINING NUMERICAL RADIUS INEQUALITIES OF HILBERT SPACE OPERATORS

Abstract

Several upper estimates for the numerical radius of Hilbert space operators are given. Among many other inequalities, it is shown that $$\begin{array}{r}{\omega }^2\left(A\right)\le \frac{1}{4}\Vert {\left|A\right|}^2+{\left|A^{\ast }\right|}^2\Vert +\frac{1}{2}\omega \left(A^2\right)-\frac{1}{2}\underset{\Vert x\Vert =1}{\underset{x\in \mathcal{H}}{inf}}{(\sqrt{⟨{\left|A\right|}^{2}x,x⟩}-\sqrt{⟨{\left|A^{\ast }\right|}^{2}x,x⟩})}^2.\end{array}$$