The fast track to Löwner’s theorem

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 Löwner’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.