Mitja Nedic: An integral representation for Herglotz-Nevanlinna functions in two variables

Abstract: Herglotz-Nevanlinna functions are holomorphic functions mapping the poly-upper half-plane to the closed upper half-plane. In the case of one variable we have classical results due to Herglotz, Nevanlinna, Pick, Cauer and others regarding integral representations of such functions. In the case of several variables the most work has been done by Vladimirov in the 1970s, but his results are incomplete in the sense that they do not provide an “if and only if” characterization.

In this talk we will present an integral representation for Herglotz-Nevanlinna functions in two variables which provides a complete characterization of this class in terms of a real number, two non-negative numbers and a positive measure satisfying certain conditions. We will also discuss further properties of the representing measures which highlight the difference between one and two variables.