### Abstract

We consider trace functions (A,B) & Tr[ (A^{q/2}B^{p}A^{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^{p}A^{q/2} and convexity/concavity of the closely related trace functional Tr[A^{q/2}B^{p}A^{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.

Original language | English (US) |
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Pages (from-to) | 174-185 |

Number of pages | 12 |

Journal | Linear Algebra and Its Applications |

Volume | 490 |

DOIs | |

State | Published - Feb 1 2016 |

### All Science Journal Classification (ASJC) codes

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

### Keywords

- Operator concavity
- Operator convexity
- Rényi entropy
- Trace inequality

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## Cite this

Carlen, E. A., Frank, R. L., & Lieb, E. H. (2016). Some operator and trace function convexity theorems.

*Linear Algebra and Its Applications*,*490*, 174-185. https://doi.org/10.1016/j.laa.2015.11.006