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arXiv · 1409.0564

Some Operator and Trace Function Convexity Theorems

Eric A. Carlen·Rupert L. Frank·Elliott H. Lieb
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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 R\'enyi entropy conjectures are proved for some cases.

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Eric A. Carlen, Rupert L. Frank, Elliott H. Lieb. 2014-09-01. Some Operator and Trace Function Convexity Theorems. https://arxiv.org/abs/1409.0564

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