On a comprehensive review of a proof of Löwner's theorem
Curt Healey
TL;DR
This work provides a self-contained, rigorous walkthrough of a proof of Löwner's theorem, bridging Kubo–Ando theory with operator monotone functions. It leverages Choquet theory to represent operator monotone functions via measures on extreme points, and combines a chain-rule/divided-differences framework with regularisation to establish smooth approximants and monotonicity properties. The analysis identifies extreme points of the normalized operator monotone function set as $f(t)=\frac{t}{\lambda+(1-\lambda)t}$ and uses Choquet's theorem to derive the integral representation $f(t)=\int_{[0,1]} \frac{t}{\lambda+(1-\lambda)t}\, d\mu(\lambda)$. This yields the classical Löwner representation and clarifies the connection between positive operator monotone functions and positive operator means via Kubo–Ando theory, with implications for operator algebra and functional calculus.
Abstract
Recent studies in Kubo-Ando theory make frequent use of the relationship between Kubo-Ando connections and positive operator monotone functions. This relationship is deeply connected to Löwner's theorem and our aim is to provide a comprehensive review of one of the proofs of Löwner's theorem. Our motivation arises from the fact that the foundational components upon which the theorem rests are found within a variety of sources, rendering it difficult to obtain a complete understanding of the proof without engaging in substantial external consultation. By consolidating these elements into a single, continuous account, the proof becomes substantially more accessible and may be assimilated with greater clarity and efficiency.