Some operator and trace function convexity theorems

Eric A. Carlen, Rupert L. Frank, Elliott H. Lieb

Research output: Contribution to journalArticlepeer-review

18 Scopus citations


We consider trace functions (A,B) & Tr[ (Aq/2BpAq/2)s] where A and B are positive n×n matrices and ask when these functions are convex or concave. We also consider operator convexity/concavity of Aq/2BpAq/2 and convexity/concavity of the closely related trace functional Tr[Aq/2BpAq/2Cr]. The concavity questions are completely resolved, thereby settling cases left open by Hiai; the convexity questions are settled in many cases. As a consequence, the Audenaert-Datta Rényi entropy conjectures are proved for some cases.

Original languageEnglish (US)
Pages (from-to)174-185
Number of pages12
JournalLinear Algebra and Its Applications
StatePublished - Feb 1 2016

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory
  • Numerical Analysis
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics


  • Operator concavity
  • Operator convexity
  • Rényi entropy
  • Trace inequality


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