Concavity of certain matrix trace and norm functions. II

TL;DR

This paper advances the understanding of the concavity and convexity properties of matrix trace and norm functions, refining existing methods and extending variational techniques to broader classes of functions.

Contribution

It refines Epstein's method and extends the variational approach to prove joint concavity/convexity of a wider class of matrix functions of Lieb type.

Findings

Proved joint concavity/convexity of new matrix trace functions.

Extended variational methods to general non-decreasing convex/concave functions.

Generalized results to operator monotone functions with specific powers.

Abstract

We refine Epstein's method to prove joint concavity/convexity of matrix trace functions of Lieb type $\mathrm{Tr}\,f(\Phi(A^p)^{1/2}\Psi(B^q)\Phi(A^p)^{1/2})$ and symmetric (anti-) norm functions of the form $\|f(\Phi(A^p)\,\sigma\,\Psi(B^q))\|$, where $\Phi$ and $\Psi$ are positive linear maps, $\sigma$ is an operator mean, and $f(x^\gamma)$ with a certain power $\gamma$ is an operator monotone function on $(0,\infty)$. Moreover, the variational method of Carlen, Frank and Lieb is extended to general non-decreasing convex/concave functions on $(0,\infty)$ so that we prove joint concavity/convexity of more trace functions of Lieb type.