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{align*}{{\omega }^{2}}\left( A \right)\le \frac{1}{4}\left\| {{\left| A \right|}^{2}}+{{\left| {{A}^{*}} \right|}^{2}} \right\| +\frac{1}{2}\omega \left( {{A}^{2}} \right)-\frac{1}{2}\underset{\left\| x \right\|=1} {\mathop{\underset{x\in \mathscr H}{\mathop{\inf }}\,}}\,{{\left( \sqrt{\left\langle {{\left| A \right|}^{2}}x,x \right\rangle } -\sqrt{\left\langle {{\left| {{A}^{*}} \right|}^{2}}x,x \right\rangle } \right)}^{2}}.\end{align*}