Extensions of Lieb's concavity theorem

Details Extensions of Lieb's concavity theorem Extensions of Lieb's concavity theorem

2005-11-28

arxiv.org/abs/math-ph/0511090v5

Journal of Statistical Physics, 124 (2006) 87-101

Physics

Mathematical Physics

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Scientific paper

10.1007/s10955-006-9155-2

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+\mu_1)^{-1}K^* B(B+\mu_2)^{-1}K) on its natural domain D_2(\mu_1,\mu_2), cf. Definition 4.1

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