Further Properties of Accretive Matrices

TL;DR

This paper investigates the properties of accretive matrices, providing new inequalities, bounds, and order-preserving results to enhance understanding of their algebraic structure and relation to positive definite matrices.

Contribution

It introduces novel inequalities, bounds, and order-preserving properties for accretive matrices, expanding the theoretical framework beyond existing literature.

Findings

Order-preserving results established

New bounds for the absolute value of accretive matrices

Choi-Davis-type and mean-convex inequalities derived

Abstract

To better understand the algebra $\mathcal{M}_n$ of all $n\times n$ complex matrices, we explore the class of accretive matrices. This class has received renowned attention in recent years due to its role in complementing those results known for positive definite matrices. More precisely, we have several results that allow a better understanding of accretive matrices. Among many results, we present order-preserving results, Choi-Davis-type inequalities, mean-convex inequalities, sub-multiplicative results for the real part, and new bounds of the absolute value of accretive matrices. These results will be compared with the existing literature. In the end, we quickly pass through related entropy results for accretive matrices.