The fast track to L\"owner's theorem

TL;DR

This paper presents a streamlined approach to L"owner's theorem, establishing integral representations for operator monotone functions directly without complex prior analyses, simplifying the theoretical framework.

Contribution

It introduces a direct method for integral representations of operator monotone functions, reducing reliance on L"owner's detailed matrix analysis and linking positive and general cases.

Findings

Direct integral representations for operator monotone functions

Canonical relationship between positive and general operator monotone functions

Simplified theoretical framework for L"owner's theorem

Abstract

The operator monotone functions defined in the positive half-line are of particular importance. We give a version of the theory in which integral representations for these functions can be established directly without invoking L\"owner's detailed analysis of matrix monotone functions of a fixed order or the theory of analytic functions. We found a canonical relationship between positive and arbitrary operator monotone functions defined in the positive half-line, and this result effectively reduces the theory to the case of positive functions. MSC2010 classification: 26A48; 26A51; 47A63. Key words and phrases: operator monotone function; integral representation; L\"owner's theorem.