Some Operator and Trace Function Convexity Theorems

Abstract

We consider convex trace functions ${\Phi }_{p,q,s}=Trace[(A^{q/2}{B}^p{A}^{q/2}{)}^s]$ where $A$ and $B$ are positive $n\times n$ matrices and ask when these functions are convex or concave. We also consider operator convexity/concavity of $A^{q/2}{B}^p{A}^{q/2}$ and convexity/concavity of the closely related trace functional $Trace[A^{q/2}{B}^p{A}^{q/2}{C}^r]$. For concavity, these questions are completely settled, thereby settling cases left open by Hiai, while the convexity questions are settled in many cases. As a consequence, the Audenaert-Datta Rényi entropy conjectures are proved for some cases.

arXiv comment11 pages latex2e. Some new results added and some proofs simplified using the triple convexity theorem. Minor error in a proof detected and corrected 10/10/14. Some typos and a comment added at the end of the proof of Theorem 3.2 02/05/15. The revision corrects a few typos