Extensions of Lieb's concavity theorem

Abstract

The operator function (A,B)\to\tr f(A,B)(K^*)K, defined on pairs of bounded self-adjoint operators in the domain of a function f of two real variables, is convex for every Hilbert Schmidt operator K, if and only if f is operator convex. As a special case we obtain a new proof of Lieb's concavity theorem for the function (A,B)\to\tr A^pK^*B^{q}K, where p and q are non-negative numbers with sum p+q\le 1. In addition, we prove concavity of the operator function (A,B)\to \tr(A(A+μ_1)^{-1}K^* B(B+μ_2)^{-1}K) on its natural domain D_2(μ_1,μ_2), cf. Definition 4.1