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*}

Keywords: Numerical radius; operator norm; inequality.

MSC: 47A12, 47A30

DOI: 10.57016/MV-mrhd2011

Pages:  50--57     

Volume  75 ,  Issue  1 ,  2023