1112.0098 (Frank Hansen)

The fast track to Löwner's theorem    [PDF]

Frank Hansen

The theory of operator monotone functions began with L\"owner's seminal work in 1934 in which a detailed analysis of matrix monotone functions of a fixed order is given. L\"owner furthermore proved that a function defined in any type of interval is operator monotone if and only if it allows an analytic continuation to a Pick function. There are several integral formulas for Pick functions, and it is these concrete representations rather than L\"owner's theorem itself that are used in most applications. The operator monotone functions defined in the positive half-axis are of particular importance, and we give a version of the theory where these formulas can be established directly without using the theory of Pick functions. We found a canonical relation between positive and arbitrary operator monotone functions defined in the positive half-axis, and this result effectively reduces the theory to the case of positive functions. Key words and phrases: operator monotone function, integral representation, L\"owner's theorem.

View original: http://arxiv.org/abs/1112.0098