Improvements of Some Numerical radius inequalities

TL;DR

This paper refines and improves existing inequalities related to the numerical radius of Hilbert space operators, providing tighter bounds and new inequalities that enhance the understanding of operator behavior.

Contribution

The paper introduces several new refined upper bounds for the numerical radius inequalities, improving upon the classical Kittaneh inequality with additional terms involving infimums.

Findings

New upper bounds for the numerical radius are established.

Refinements improve the classical Kittaneh inequality.

Additional related results are discussed.

Abstract

In this work, we improve and refine some numerical radius inequalities. In particular, for all Hilbert space operators $T$, the celebrated Kittaneh inequality reads: \begin{align*} \frac{1}{4}\left\| T^*T + TT^*\right\|\le w^{2 }\left(T \right) \le \frac{1}{2}\left\| T^*T + TT^*\right\|. \end{align*} In this work we provide some important refinements for the upper bound of the Kittaned inequality. Indeed, we establish \begin{align*} w^{2 }\left(T \right) \le \frac{1}{2}\left\| T^*T + TT^*\right\| - \frac{1}{4} \mathop {\inf }\limits_{\left\| x \right\| = 1} \left( {\left\langle {\left| T \right|x,x} \right\rangle - \left\langle {\left| T^* \right|x,x} \right\rangle } \right)^2, \end{align*} which also refined and improved as \begin{align*} w^{2 }\left(T \right) \le \frac{1}{2}\left\| T^*T + TT^*\right\| - \frac{1}{2} \mathop {\inf }\limits_{\left\| x \right\| = 1} \left( {\left\langle {\left| T \right|x,x} \right\rangle - \left\langle {\left| T^* \right|x,x} \right\rangle } \right)^2, \end{align*} and \begin{align*} w^{2 }\left(T \right) \le \frac{1}{2} \left\|T^*T+TT^* \right\| -\frac{1}{2} \mathop {\inf }\limits_{\left\| x \right\| = 1} \left(\left\langle {\left| T \right|^{2 }x,x} \right\rangle^{\frac{1}{2}} - \left\langle {\left| T^* \right|^{2 } x,x} \right\rangle^{\frac{1}{2}}\right)^2, \end{align*} with third improvement \begin{align*} w^2 \left( {T } \right) \le \frac{1}{4 }\left\| {\left| T \right| + \left| {T^* } \right| } \right\|^{2} - \frac{1}{{4 }}\mathop {\inf }\limits_{\left\| x \right\| = 1} \left( {\left\langle {\left| T \right| x,x} \right\rangle - \left\langle {\left| {T^* } \right| x,x} \right\rangle } \right)^2. \end{align*} Other general related results are also considered.