On Cauchy-Schwarz type inequalities and applications to numerical radius inequalities

TL;DR

This paper refines classical inequalities like Cauchy-Schwarz and Kato's inequality within inner product spaces, leading to improved bounds on the numerical radius of operators, thus advancing theoretical understanding and potential applications in operator theory.

Contribution

It introduces new refinements of fundamental inequalities and applies them to derive sharper bounds on the numerical radius of operators, extending recent results in the literature.

Findings

Refined Cauchy-Schwarz inequality in inner product spaces.

Established a more general refinement of Kato's inequality.

Derived improved bounds for the numerical radius of operators.

Abstract

In this work, a refinement of the Cauchy--Schwarz inequality in inner product space is proved. A more general refinement of the Kato's inequality or the so called mixed Schwarz inequality is established. Refinements of some famous numerical radius inequalities are also pointed out. As shown in this work, these refinements generalize and refine some recent and old results obtained in literature. Among others, it is proved that if $T\in\mathscr{B}\left(\mathscr{H}\right)$, then \begin{align*} \omega^{2}\left(T\right) &\le \frac{1}{12} \left\| \left| T \right|+\left| {T^* } \right|\right\|^2 + \frac{1}{3} \omega\left(T\right)\left\| \left| T \right|+\left| {T^* } \right|\right\| \\ &\le\frac{1}{6} \left\| \left| T \right|^2+ \left| {T^* } \right|^2 \right\| + \frac{1}{3} \omega\left(T\right)\left\| \left| T \right|+\left| {T^* } \right|\right\|, \end{align*} which refines the recent inequality obtained by Kittaneh and Moradi in \cite{KM}.