Nevanlinna representations in several variables

We generalize to several variables the classical theorem of Nevanlinna that characterizes the Cauchy transforms of positive measures on the real line. We show that for the Loewner class, a large class of analytic functions that have non-negative imaginary part on the upper polyhalfplane, there are representation formulae in terms of densely-defined self-adjoint operators on a Hilbert space. We identify four types of such representations, and we obtain function-theoretic conditions that are necessary and sufficient for a given function to possess a representation of each of the four types.