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

TL;DR

This work addresses the problem of fully characterizing the Molnár class of symmetric Kubo–Ando means that satisfy a weak associativity condition. It establishes a bijection between the Molnár class $\mathcal{M}$ and a class $\mathcal{W}$ of real measurable odd $p$-periodic functions bounded by $|\Psi|\le \tfrac{1}{2}$, via the exponential–integral representation $f(z)=\sqrt{z}\,e^{S(\log z)}$ with $S(w)=\tfrac{1}{2}\int_{\mathbb{R}} \frac{\Psi(\lambda)\sinh(w/2)}{\cosh(\lambda/2)\cosh((w-\lambda)/2)} d\lambda$, where $p=2\log c$ for a scaling $c>1$. This yields explicit nongeometric Molnár means through Fourier-type constructions $\Psi(\lambda)=\pm\tfrac{1}{2}\sin\left(\frac{\pi\lambda}{\log c}\right)$, producing $f_n(x)=\sqrt{x}\exp\left\{\frac{\pi\sin^{2}\left(\frac{\pi n\log x}{2\log c}\right)}{\sinh\left(\frac{\pi^{2}n}{\log c}\right)}\right\}$, and extremal elements $f_{\min}$ and $f_{\max}$ expressed via Jacobi elliptic functions with a parameter $m$ determined by $p=2\log c$. The authors show a sharp order structure in the Molnár class and prove that the geometric mean is recovered only when two incommensurate scaling constants are present, thereby providing a complete negative answer to Molnár’s question. The results deliver a comprehensive framework for constructing Molnár means, explicitly characterizing the entire class, and clarifying how a geometric mean can be uniquely singled out under strengthened scaling conditions.

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. Molnár, ``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 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 Molnar 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.

Figures (3)

• Figure 1: A Molnár mean $\sigma=\sigma_f$ corresponds to a representing function $f\in \mathcal{M}_c$ for some $c\in (1,\infty)$. As represented in the transition from (a) to (b) in the figure, by using the invertible transform $w=\log z$ (with inverse $z=e^w$) from $\mathbb{C}\setminus (-\infty,0]$ onto the strip $D=\{w\in \mathbb{C}:|\operatorname{Im}w|<\pi\}=\mathbb{R}\times (-\pi,\pi)$, the analytic functions $f\in \mathcal{M}_c$ are mapped bijectively onto the analytic functions $S\in \mathcal{W}_p$ via $W_p(f)=\log f(e^{w})-w/2$, where $p=2\log c$. Then using the exponential representation of $OM_{+}\setminus\{0\}$ functions from 77KN we obtain the explicit representation of the class ${\mathcal{W}}_{p}$ and, perforce, of ${\mathcal{M}}_{c}$.
• Figure 2: Graphs of the minimal and the maximal elements of ${\mathcal{M}}_{c}$ relative to $f_{\#}(x)=\sqrt{x}$ for $c=e^{10}$, corresponding to $p=20$. All other functions in ${\mathcal{M}}_c$, such as $f_1$, given by (\ref{['SeqNonGeomMolnarMeans']}) with $n=1$, are sandwiched between $f_{\min}$ and $f_{\max}$.
• Figure 3: $f_{\nabla}$ and $f_{!}$ are the limits of $f_{\max}$ and $f_{\min}$, respectively, as $c\to\infty$. The plots of $f_{\max}/f_{\#}$ and $f_{\min}/f_{\#}$ for $p=10,15,20,25$ are shown.

Theorems & Definitions (28)

• Theorem 2: L. Molnár
• Theorem 3: L. Molnár
• Definition 4: Molnár mean
• Theorem 6
• Lemma 7
• Definition 8: Molnár class of functions
• Lemma 9
• proof
• Lemma 10
• proof