2012/09/04 by Fumio Hiai, Dénes Petz, Denes Petz · 6 citations
Mathematics · Physics and Astronomy · #Class (philosophy) #Convexity #Function (biology) #Holomorphic and Operator Theory #Mathematical Inequalities and Applications #Monotone polygon #Operator (biology) #Skew #Statistical Mechanics and Entropy #Type (biology) #math-ph #math.FA #math.MP #msc:54C70 #msc:81P45
paper · pdf · doi:10.1063/1.4810781
published in Journal of Mathematical Physics 54(6) (American Institute of Physics) · 27 pages
arxiv created 2012/09/04 · openalex publication_date 2013/06/01 · arxiv updated 2015/06/11 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
Given a positive function f on (0, ∞) and a non-zero real parameter θ, we consider a function \documentclass[12pt]minimal\begindocumentIfθ (A,B,X)=Tr X^*(f(LARB-1)RB)-θ (X)\enddocumentIfθ(A,B,X)= Tr X*(f(LARB−1)RB)−θ(X) in three matrices A, B > 0 and X. This generalizes the notion of monotone metrics on positive definite matrices, and in the literature θ = ±1 has been typical. We investigate how operator monotony of f is sufficient and/or necessary for joint convexity/concavity of \documentclass[12pt]minimal\begindocumentIfθ (A,B,X)\enddocumentIfθ(A,B,X). Similar discussions are given for quasi-entropies and quantum skew informations.