Improved Inequalities for the Extension of Euclidean Numerical Radius

Abstract

This paper aims to discuss inequalities involving extension of Euclidean numerical radius. We obtain a refinement of the inequality shown by Sattari et al. We give an improvement of the inequality presented by Kittaneh for the numerical radius. In fact we show that  if $T\in \mathcal B({\mathcal H})$, then
   \[\omega ^2(T)\leq \frac{1}{2}\lVert T^*T+ TT^*\rVert
      -\underset {\left\| x \right\|=1}\inf\phi(x),\]
     where $\phi(x)=
     \bigg\langle
    \bigg( \bigg\lvert
     \lvert T\rvert - \langle \lvert T\rvert x, x\rangle \bigg\rvert^2+ \bigg\lvert \lvert T^*\rvert -\langle \lvert T^*\rvert x, x \rangle\bigg\rvert^2\bigg)x, x\bigg\rangle.$