The fast track to L\"owner'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 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.