The operator function (A,B)→ Trf(A,B)(K *)K, defined in 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. We obtain, as a special case, a new proof of Lieb's concavity theorem for the function (A,B)→ TrA p K* B q K, where p and q are non-negative numbers with sum p+q ≤ 1. In addition, we prove concavity of the operator function (A,B) → Tr [ A/A+μ 1K*B/B+μ2K] in its natural domain D 2(μ1,μ2), cf. Definition 3.
ASJC Scopus subject areas
- Statistical and Nonlinear Physics
- Mathematical Physics