Some operator and trace function convexity theorems

Abstract

We consider trace functions (A,B)↦Tr[(A^(q/2)B^pA^(q/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 A^(q/2)B^pA^(q/2) and convexity/concavity of the closely related trace functional Tr[A^(q/2)B^pA^(q/2)C^r]. 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.

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