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 Trf(Φ(Ap)1/2Ψ(Bq)Φ(Ap)1/2) and symmetric (anti-) norm functions of the form ‖f(Φ(Ap)σΨ(Bq))‖, where Φ and Ψ are positive linear maps, σ is an operator mean, and f(xγ) with a certain power γ is an operator monotone function on (0,∞). Moreover, the variational method of Carlen, Frank and Lieb is extended to general non-decreasing convex/concave functions on (0,∞) so that we prove joint concavity/convexity of more trace functions of Lieb type.