### 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 (A,B) → Tr [ A/A+μ _{1}K*B/B+μ_{2}K] in its natural domain D _{2}(μ_{1},μ_{2}), cf. Definition 3.

Original language | English |
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Pages (from-to) | 87-101 |

Number of pages | 15 |

Journal | Journal of Statistical Physics |

Volume | 124 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2006 Jul 1 |

### Keywords

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

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

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## Cite this

Hansen, F. (2006). Extensions of Lieb's concavity theorem.

*Journal of Statistical Physics*,*124*(1), 87-101. https://doi.org/10.1007/s10955-006-9155-2