Abstract
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
in its natural domain D 2(μ1,μ2), cf. Definition 3.
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Hansen, F. Extensions of Lieb’s Concavity Theorem. J Stat Phys 124, 87–101 (2006). https://doi.org/10.1007/s10955-006-9155-2
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DOI: https://doi.org/10.1007/s10955-006-9155-2