Further Inequalities for the Numerical Radius of Hilbert Space Operators

Research paper by S. Tafazoli, H. R. Moradi, S. Furuichi, P. Harikrishnan

Indexed on: 01 Mar '12Published on: 12 Jul '19Published in: arXiv - Mathematics - Functional Analysis

Abstract

In this article, we present some new inequalities for numerical radius of Hilbert space operators via convex functions. Our results generalize and improve earlier results by El-Haddad and Kittaneh. Among several results, we show that if $A\in \mathbb{B}\left( \mathcal{H} \right)$ and $r\ge 2$, then ${{w}^{r}}\left( A \right)\le {{\left\| A \right\|}^{r}}-\underset{\left\| x \right\|=1}{\mathop{\inf }}\,{{\left\| {{\left| \left| A \right|-w\left( A \right) \right|}^{\frac{r}{2}}}x \right\|}^{2}}$ where $w\left( \cdot \right)$ and $\left\| \cdot \right\|$ denote the numerical radius and usual operator norm, respectively.