Jointly convex mappings related to Lieb’s theorem and Minkowski type operator inequalities

Abstract

Employing the notion of operator log-convexity, we study joint concavity/convexity of multivariable operator functions related to Lieb’s theorem: $$(A,B)\mapsto F(A,B)=\psi \left[ \Phi (f(A))\ \sigma \ \Psi (g(B))\right] $$ , where $$\Phi $$ and $$\Psi $$ are positive linear mappings and $$\sigma $$ is an operator mean. As applications, we prove jointly concavity/convexity of matrix trace functions $$\mathrm {Tr}\left\{ F(A,B)\right\} $$ . Moreover, considering positive multi-linear mappings in F(A, B), our study of the joint concavity/convexity of $$(A_1,\ldots ,A_k)\mapsto \psi \left[ \Phi (f(A_1),\ldots ,f(A_k))\right] $$ provides some generalizations and complement to results of Ando and Lieb concerning the concavity/convexity of mappings involving tensor product. In addition, we present Minkowski type operator inequalities for a unital positive linear mapping, which is an operator version of Minkowski type matrix trace inequalities under a more general setting than Carlen and Lieb, Bekjan, and Hiai.