Concavity of certain matrix trace and norm functions. II

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.