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Complex Analysis and Operator Theory

, Volume 3, Issue 1, pp 57–98 | Cite as

Boundary Relations, Unitary Colligations, and Functional Models

  • Jussi BehrndtEmail author
  • Seppo Hassi
  • Henk de Snoo
Article

Abstract.

Recently a new notion, the so-called boundary relation, has been introduced involving an analytic object, the so-called Weyl family. Weyl families and boundary relations establish a link between the class of Nevanlinna families and unitary relations acting from one Kreĭn space, a basic (state) space, to another Kreĭn space, a parameter space where the Nevanlinna family or Weyl family is acting. Nevanlinna families are a natural generalization of the class of operator-valued Nevanlinna functions and they are closely connected with the class of operator-valued Schur functions. This paper establishes the connection between boundary relations and their Weyl families on the one hand, and unitary colligations and their transfer functions on the other hand. From this connection there are various advances which will benefit the investigations on both sides, including operator theoretic as well as analytic aspects. As one of the main consequences a functional model for Nevanlinna families is obtained from a variant of the functional model of L. de Branges and J. Rovnyak for Schur functions. Here the model space is a reproducing kernel Hilbert space in which multiplication by the independent variable defines a closed simple symmetric operator. This operator gives rise to a boundary relation such that the given Nevanlinna family is realized as the corresponding Weyl family.

Mathematics Subject Classification (2000).

Primary 46E22, 47B32 Secondary 47A06, 47B25 

Keywords.

Boundary relation boundary triplet Weyl family Weyl function Nevanlinna family Nevanlinna function unitary colligation transfer function Schur function reproducing kernel Hilbert space functional model 

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Copyright information

© Birkhäuser Verlag Basel/Switzerland 2008

Authors and Affiliations

  1. 1.Technische Universität BerlinInstitut für Mathematik MA 6-4BerlinGermany
  2. 2.Department of Mathematics and StatisticsUniversity of VaasaVaasaFinland
  3. 3.Department of Mathematics and Computing ScienceUniversity of GroningenGroningenNederland

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