Concavity of certain matrix trace and norm functions

2012/10/28 by Fumio Hiai, Hiai, Fumio
Computer Science · Mathematics · #15A60 #47A30 #47A60 #Analytic and geometric function theory #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Mathematical Inequalities and Applications #Mathematical Physics (math-ph) #Operator Algebras (math.OA) #Optimization and Variational Analysis

paper · pdf · doi:10.48550/arxiv.1210.7524

openalex publication_date 2012/10/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We refine Epstein's method to prove joint concavity/convexity of matrix trace functions of the extended Lieb type TrΦ(A^p)^1/2Ψ(B^q)Φ(A^p)^1/2s, where Φ and Ψ are positive linear maps. By the same method combined with majorization technique, similar properties are proved for symmetric (anti-) norm functions of the form ||Φ(A^p)σΨ(B^q)^s|| involving an operator mean σ. Carlen and Lieb's variational method is also used to improve the convexity property of norm functions ||Φ(A^p)^s||.

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