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Canonical Agler decompositions and transfer function realizations

Authors: Kelly Bickel and Greg Knese
Journal: Trans. Amer. Math. Soc. 368 (2016), 6293-6324
MSC (2010): Primary 47B32; Secondary 47A57, 47A40, 93C35, 46E22
Published electronically: January 13, 2016
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Abstract: A seminal result of Agler proves that the natural de Branges-Rovnyak kernel function associated to a bounded analytic function on the bidisk can be decomposed into two shift-invariant pieces. Agler's decomposition is non-constructive--a problem remedied by work of Ball-Sadosky-Vinnikov, which uses scattering systems to produce Agler decompositions through concrete Hilbert space geometry. This method, while constructive, so far has not revealed the rich structure shown to be present for special classes of functions--inner and rational inner functions. In this paper, we show that most of the important structure present in these special cases extends to general bounded analytic functions. We give characterizations of all Agler decompositions, we prove the existence of coisometric transfer function realizations with natural state spaces, and we characterize when Schur functions on the bidisk possess analytic extensions past the boundary in terms of associated Hilbert spaces.

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Additional Information

Kelly Bickel
Affiliation: Department of Mathematics, Bucknell University, Lewisburg, Pennsylvania 17837

Greg Knese
Affiliation: Department of Mathematics, Washington University in St. Louis, St. Louis, Missouri 63130

Keywords: Agler decomposition, scattering systems, transfer function, colligation, Lax-Phillips, input/state/output system, bidisk, polydisk, Schur class, de Branges-Rovnyak spaces, analytic extension, reproducing kernel Hilbert space, operator ranges
Received by editor(s): December 1, 2013
Received by editor(s) in revised form: August 14, 2014
Published electronically: January 13, 2016
Additional Notes: The first author was supported by the American Association of University Women and National Science Foundation Grants DMS 0955432 and DMS 1448846.
The second author was supported by National Science Foundation Grant DMS 1363239
Article copyright: © Copyright 2016 American Mathematical Society