Complete characterization of symmetric Kubo-Ando operator means satisfying Molnár's weak associativity

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 Molnár (2019) [24], where he gives a characterization of this subclass (which we call the Molnár class of means) in terms of the properties of their representing operator monotone functions. Molnár'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 Molnár mean by an explicit exponential-integral representation. From this we are able to understand the order structure of the Molnár class and construct several infinite families of explicit examples of Molnár means that are not the geometric mean. Our analysis also shows how to modify Molnár's original characterization so that the geometric mean is the only one satisfying the requisite set of properties.