Complete characterization of symmetric Kubo-Ando operator means satisfying Moln\'ar's weak associativity

TL;DR

This paper fully characterizes a subclass of symmetric Kubo-Ando means satisfying weak associativity, revealing multiple means beyond the geometric mean and clarifying their structure through operator monotone functions and measurable odd periodic functions.

Contribution

It provides a complete characterization of the Molnár class of means, constructs explicit examples beyond the geometric mean, and refines the conditions to isolate the geometric mean.

Findings

Constructed an order-preserving bijection with measurable odd periodic functions.

Generated several infinite families of Molnár means not equal to the geometric mean.

Modified Molnár's characterization to uniquely identify the geometric mean.

Abstract

We provide a complete characterization of a subclass of weakly associative means of positive operators in the class of symmetric Kubo-Ando means. This class, which includes the geometric mean, was first introduced and studied in L. Moln\'ar, ``Characterizations of certain means of positive operators," Linear Algebra Appl. 567 (2019) 143-166, where he gives a characterization of this subclass (which we call the Moln\'ar class of means) in terms of the properties of their representing operator monotone functions. Moln\'ar's paper leaves open the problem of determining if the geometric mean is the only such mean in that subclass. Here we give a negative answer to this question by constructing an order-preserving bijection between this class and a class of real measurable odd periodic functions bounded in absolute value by $1/2$. Each member of the latter class defines a Molnar mean by an explicit exponential-integral representation. From this we are able to understand the order structure of the Moln\'ar class and construct several infinite families of explicit examples of Moln\'ar means that are not the geometric mean. Our analysis also shows how to modify Moln\'ar's original characterization so that the geometric mean is the only one satisfying the requisite set of properties.