Title: On a comprehensive review of a proof of Löwner's theorem Show Chinese title

Title: 关于Löwner定理证明的综合回顾

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 L\"owner's theorem and our aim is to provide a comprehensive review of one of the proofs of L\"owner'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. Show Chinese abstract

Abstract: 最近在Kubo-Ando理论中的研究频繁使用Kubo-Ando连接与正算子单调函数之间的关系。 这种关系与Löwner定理密切相关，我们的目标是提供对Löwner定理其中一个证明的全面综述。 我们的动机源于这样一个事实，即该定理所依赖的基础组成部分出现在各种来源中，使得在不进行大量外部咨询的情况下很难完全理解该证明。 通过将这些要素整合成一个连续的叙述，证明将变得更容易理解，并可以更清晰和高效地被掌握。