Some convexity and monotonicity results of trace functionals

TL;DR

This paper establishes the convexity and monotonicity properties of certain trace functionals involving positive definite matrices, extending previous results and resolving a conjecture in matrix analysis.

Contribution

It proves the optimal convexity conditions for trace functionals and extends existing results, including resolving a key conjecture in the field.

Findings

Proved convexity of trace functionals for optimal parameters

Extended results in previous literature on trace inequalities

Resolved a conjecture related to trace functional convexity

Abstract

In this paper, we prove the convexity of trace functionals $$(A,B,C)\mapsto \text{Tr}|B^{p}AC^{q}|^{s},$$ for parameters $(p,q,s)$ that are best possible, where $B$ and $C$ are any $n$-by-$n$ positive definite matrices, and $A$ is any $n$-by-$n$ matrix. We also obtain the monotonicity versions of trace functionals of this type. As applications, we extend some results in \cite{HP12quasi,CFL16some} and resolve a conjecture in \cite{RZ14} in the matrix setting. Other conjectures in \cite{RZ14} will also be discussed. We also show that some related trace functionals are not concave in general. Such concavity results were expected to hold in different problems.