Nevanlinna representations in several variables

Abstract

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 polyhalf-plane, there are representation formulae in terms of densely-defined self-adjoint operators on a Hilbert space. We find four different representation formulae and we show that every function in the Loewner class has one of the four representations, corresponding precisely to four different growth conditions at infinity.

A subclass of boundary measures and the convex combination problem for Herglotz–Nevanlinna functions in several variables

“…This class of functions has, so far, proven a bit less prominent in both mathematics and applications, but relates nonetheless to multidimensional passive systems [21] and homogenization of multicomponent media [10,16,17]. However, the class of Herglotz-Nevanlinna functions in several variables has seen a renewed interest in the last few years, especially from a pure mathematical perspective, with results concerning both integral representations of this class of functions [14,15], as well as operator representations [2,3].…”

Cauchy transforms arising from homomorphic conditional expectations parametrize noncommutative Pick functions

“…(Here we have concretely written R(x) = V * xV and used a resolvent of the form (L − X) −1 (L + X) instead of (1 − UX) −1 (1 + UX) to agree with [2,1]. However, L is still a unitary.…”

Geometric Properties of Measures Related to Holomorphic Functions Having Positive Imaginary or Real Part

“…We also want to mention that a subclass of these functions, namely the Herglotz-Agler functions, are characterized via operator representations and so-called µresolvents, cf. [3,4,6]. However, these results do not imply the integral representation mentioned above and cannot be used for our current purpose.…”