Some operator and trace function convexity theorems

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

doi:10.1016/j.laa.2015.11.006

Some operator and trace function convexity theorems

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

Mathematics

Research output: Contribution to journal › Article › peer-review

19 Scopus citations

Abstract

We consider trace functions (A,B) & Tr[ (A^q/2B^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/2B^pA^q/2 and convexity/concavity of the closely related trace functional Tr[A^q/2B^pA^q/2C^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)
Pages (from-to)	174-185
Number of pages	12
Journal	Linear Algebra and Its Applications
Volume	490
DOIs	https://doi.org/10.1016/j.laa.2015.11.006
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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10.1016/j.laa.2015.11.006

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title = "Some operator and trace function convexity theorems",

abstract = "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 R{\'e}nyi entropy conjectures are proved for some cases.",

keywords = "Operator concavity, Operator convexity, R{\'e}nyi entropy, Trace inequality",

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doi = "10.1016/j.laa.2015.11.006",

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T1 - Some operator and trace function convexity theorems

AU - Carlen, Eric A.

AU - Frank, Rupert L.

AU - Lieb, Elliott H.

PY - 2016/2/1

Y1 - 2016/2/1

N2 - 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.

AB - 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.

KW - Operator concavity

KW - Operator convexity

KW - Rényi entropy

KW - Trace inequality

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SP - 174

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JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

ER -

Some operator and trace function convexity theorems

Abstract

All Science Journal Classification (ASJC) codes

Keywords

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