On the matrix Cauchy-Schwarz inequality

TL;DR

This paper develops new matrix inequalities related to the Cauchy-Schwarz inequality, focusing on Lieb functions and the mixed Cauchy-Schwarz inequality, leading to bounds on matrix norms and numerical radius.

Contribution

It introduces novel matrix inequalities of the Cauchy-Schwarz type, including bounds involving Lieb functions and the real part of matrices, expanding the theoretical framework.

Findings

Derived a new inequality for matrix norms involving Lieb functions.

Established bounds for the numerical radius using mixed Cauchy-Schwarz inequalities.

Extended the understanding of matrix inequalities related to the Cauchy-Schwarz framework.

Abstract

The main goal of this work is to present new matrix inequalities of the Cauchy-Schwarz type. In particular, we investigate the so-called Lieb functions, whose definition came as an umbrella of Cauchy-Schwarz-like inequalities, then we consider the mixed Cauchy-Schwarz inequality. This latter inequality has been influential in obtaining several other matrix inequalities, including numerical radius and norm results. Among many other results, we show that \[\left\| T \right\|\le \frac{1}{4}\left( \left\| \left| T \right|+\left| {{T}^{*}} \right|+2\mathfrak RT \right\|+\left\| \left| T \right|+\left| {{T}^{*}} \right|-2\mathfrak RT \right\| \right),\] where $\mathfrak RT$ is the real part of $T$.