Extensions of Lieb's Concavity Theorem

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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)→ TrApK*BqK, where p and q are non-negative numbers with sum p+q ≤ 1. In addition, we prove concavity of the operator function $$ in its natural domain D2(μ1,μ2), cf. Definition 3.

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