Nevanlinna representations in several variables

2012/03/10 by Jim Agler, Agler, Jim, R. Tully-Doyle +3 · 1 citation
Mathematics · #30E05 #30E20 #32A30 #47A10 #47B25 #Complex Variables (math.CV) #FOS: Mathematics #math.CV #msc:30E05 #msc:30E20 #msc:32A30 #msc:47A10 #msc:47B25

paper · pdf · doi:10.48550/arxiv.1203.2261

37 pages. In this version we have added some references and expanded the introduction

arxiv created 2012/06/24 · arxiv updated 2012/06/26

Abstract

We generalize two integral representation formulae of Nevanlinna to functions of several variables. We show that for 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 introduce three types of structured resolvent of a self-adjoint operator and identify four different types of representation in terms of these resolvents. We relate the types of representation that a function admits to its growth at infinity.

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