On Araki-Type Trace Inequalities

TL;DR

This paper establishes new trace inequalities involving positive semi-definite matrices and functions, extending Araki-type inequalities with conditions on functions and parameters, contributing to matrix analysis and operator inequalities.

Contribution

The paper proves novel trace inequalities for positive semi-definite matrices involving monotone functions and specific parameter ranges, extending existing Araki-type inequalities.

Findings

Proved a trace inequality for positive monotone functions and matrices.

Established conditions under which the reverse inequality holds.

Extended the class of Araki-type inequalities in matrix analysis.

Abstract

In this paper, we prove a trace inequality $\text{Tr}[ f(A) A^s B^s ] \leq \text{Tr}[ f(A) (A^{1/2} B A^{1/2} )^s ]$ for any positive and monotone increasing function $f$, $s\in[0,1]$, and positive semi-definite matrices $A$ and $B$. On the other hand, for $s\in[0,1]$ such that the map $x\mapsto x^s g(x)$ is positive and decreasing, then $ \text{Tr}[ g(A) (A^{1/2} B A^{1/2} )^s ] \leq \text{Tr}[ g(A) A^s B^s ]$.