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Boundary behavior of analytic functions of two variables via generalized models

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



We describe a generalization of the notion of a Hilbert space model of a function in the Schur class of the bidisc. This generalization is well adapted to the investigation of boundary behavior at a mild singularity of the function on the 2-torus. We prove the existence of a generalized model with certain properties corresponding to such a singularity and use this result to solve two function-theoretic problems. The first of these is to characterise the directional derivatives of a function in the Schur class at a singular point on the torus for which the Carath\'eodory condition holds. The second is to obtain a representation theorem for functions in the two-variable Pick class analogous to the refined Nevanlinna representation of functions in the one-variable Pick class.

Publication metadata

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

Publication type: Article

Publication status: Published

Journal: Indagationes Mathematicae

Year: 2012

Volume: 23

Issue: 4

Pages: 995-1027

Print publication date: 20/07/2012

Date deposited: 11/07/2012

ISSN (print): 0019-3577

ISSN (electronic): 1872-6100

Publisher: Elsevier BV


DOI: 10.1016/j.indag.2012.07.003


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Funder referenceFunder name
DMS 1068830National Science Foundation Grant on Extending Hilbert Space Operators
EP/J004545/1UK Engineering and Physical Sciences Research Council grant