On the monotonicity of weighted power means of matrices

2017/01/24 by Raluca Dumitru, Dumitru, Raluca, Jose Franco +1
Mathematics · #FOS: Mathematics #Functional Analysis (math.FA) #math.FA

paper · pdf · doi:10.48550/arxiv.1701.07023

Proposition 4 is not true. Therefore, the article needs major revisions

arxiv created 2017/01/30 · arxiv updated 2017/01/31

Abstract

Let μ_p(A,B,t)=(tA^p+(1-t)B^p)^1/p denote the weighted power mean between positive operators A and B. We show that the function t→ ‖A-μ_p(A,B,t)‖₂ is monotonically decreasing whenever 1/2 ≤ p ≤ 1. Hence showing that the weighted power means satisfy Audenaert's "in-betweenness" property for positive operators for power satisfying 1/2 ≤ p ≤ 1. We also show that when p>2 there exist operators for which the weighted power mean does not satisfy this "in-betweenness" property with respect to the Euclidean metric.

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