Extensions of Lieb's concavity theorem

Frank Hansen

Research output: Contribution to journalArticlepeer-review

25 Citations (Scopus)


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 212), cf. Definition 3.

Original languageEnglish
Pages (from-to)87-101
Number of pages15
JournalJournal of Statistical Physics
Issue number1
Publication statusPublished - 2006 Jul


  • Generalized Hessian
  • Lieb's concavity theorem
  • Operator convex function

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics


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