Boundary Relations, Unitary Colligations, and Functional Models

Jussi Behrndt*, Seppo Hassi, Henk de Snoo

*Corresponding author for this work

    Research output: Contribution to journalArticleAcademicpeer-review

    19 Citations (Scopus)


    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 Krein in space, a basic (state) space, to another Krein in 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.

    Original languageEnglish
    Pages (from-to)57-98
    Number of pages42
    JournalComplex analysis and operator theory
    Issue number1
    Publication statusPublished - Mar-2009


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


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