Matrix power means and new characterizations of operator monotone functions

2021/06/10 by Trung Hoa Dinh, Công Trình Lê, Cong Trinh Le +6
Decision Sciences · Mathematics · #15B48 #46E05 #47A56 #47A63 #47A64 #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Mathematical Inequalities and Applications #Multi-Criteria Decision Making #Operator Algebras (math.OA) #math.FA #math.OA #msc:15B48 #msc:46E05 #msc:47A56 #msc:47A63 #msc:47A64

paper · pdf · doi:10.48550/arxiv.2106.05914

13 pages, some typos are fixed

openalex publication_date 2021/06/10 · arxiv created 2021/07/13 · arxiv updated 2021/07/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For positive definite matrices A and B, the Kubo-Ando matrix power mean is defined as P_μ(p, A, B) = A^1/2(\frac1+(A^-1/2BA^-1/2)^p2 )^1/p A^1/2 (p ≥ 0). In this paper, for 0≤ p ≤ 1 ≤ q, we show that if one of the following inequalities f(P_μ(p, A, B)) ≤ f(P_μ(1, A, B)) ≤ f(P_μ(q, A, B))holds for any positive definite matrices A and B, then the function f is operator monotone on (0, ∞). We also study the inverse problem for non-Kubo-Ando matrix power means with the powers 1/2 and 2. As a consequence, we establish new charaterizations of operator monotone functions with the non-Kubo-Ando matrix power means.

Related

• On matrix inequalities between the power means: counterexamples
• On the monotonicity of weighted power means of matrices
• Some inequalities of matrix power and Karcher means for positive linear maps
• New characterizations of operator monotone functions
• Matrix limit theorems of Kato type related to positive linear maps and operator means