On the monotonicity of weighted power means of matrices

TL;DR

This paper investigates the monotonicity of the Euclidean norm of the difference between positive operators and their weighted power means, establishing conditions under which the in-betweenness property holds or fails.

Contribution

It proves the monotonicity of the Euclidean norm of the difference for weighted power means when 1/2 ≤ p ≤ 1 and shows failure of this property for p > 2.

Findings

Monotonicity holds for 1/2 ≤ p ≤ 1.

In-betweenness property is satisfied in this range.

Counterexamples exist for p > 2.

Abstract

Let $\mu_p(A,B,t)=(tA^p+(1-t)B^p)^{1/p}$ denote the weighted power mean between positive operators $A$ and $B$. We show that the function $t\to \|A-\mu_p(A,B,t)\|_2$ is monotonically decreasing whenever $1/2 \leq p \leq 1$. Hence showing that the weighted power means satisfy Audenaert's "in-betweenness" property for positive operators for power satisfying $1/2 \leq p \leq 1$. 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.