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Nevanlinna representations in several variables

Lookup NU author(s): Professor Jim Agler, Professor Nicholas Young



This work is licensed under a Creative Commons Attribution 4.0 International License (CC BY 4.0).


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 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.

Publication metadata

Author(s): Agler J, Tully-Doyle R, Young NJ

Publication type: Article

Publication status: Published

Journal: Journal of Functional Analysis

Year: 2016

Volume: 270

Issue: 8

Pages: 3000-3046

Print publication date: 15/04/2016

Online publication date: 23/02/2016

Acceptance date: 02/02/2016

Date deposited: 08/03/2016

ISSN (print): 0022-1236

ISSN (electronic): 1096-0783

Publisher: Elsevier


DOI: 10.1016/j.jfa.2016.02.004


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Funder referenceFunder name
1361720National Science Foundation
EP/N03242X/1UK Engineering and Physical Sciences Research Council
DMS 1068830National Science Foundation
EP/J004545/1UK Engineering and Physical Sciences Research Council