Holomorphic Operators Between Krein Spaces and the Number of Squares of Associated Kernels View Full Text


Ontology type: schema:Chapter     


Chapter Info

DATE

1992

AUTHORS

D. Alpay , A. Dijksma , J. van der Ploeg , H. S. V. de Snoo

ABSTRACT

Suppose that Θ(z) is a bounded linear mapping from the Kreĭ space to the Kreĭn space G, which is defined and holomorphic in a small neighborhood of z = 0. Then often Θ admits realizations as the characteristic function of an isometric, a coisometric and of a unitary colligation in which for each case the state space is a Kreĭn space. If the colligations satisfy minimality conditions (i.e., are controllable, observable or closely connected, respectively) then the positive and negative indices of the state space can be expressed in terms of the number of positive and negative squares of certain kernels associated with Θ, depending on the kind of colligation. In this note we study the relations between the numbers of positive and negative squares of these kernels. Using the Potapov-Ginzburg transform we give a reduction to the case where the spaces and G are Hilbert spaces. For this case these relations has been considered in detail in [DLS1]. More... »

PAGES

11-29

References to SciGraph publications

  • 1991. De Branges-Rovnyak Operator Models and Systems Theory: A Survey in TOPICS IN MATRIX AND OPERATOR THEORY
  • 1985-03. Hilbert spaces of analytic functions, inverse scattering and operator models.II in INTEGRAL EQUATIONS AND OPERATOR THEORY
  • 1974. Monotone Matrix Functions and Analytic Continuation in NONE
  • 1986. On Applications of Reproducing Kernel Spaces to the Schur Algorithm and Rational J Unitary Factorization in I. SCHUR METHODS IN OPERATOR THEORY AND SIGNAL PROCESSING
  • 1989. Characteristic Functions of Unitary Colligations and of Bounded Operators in Krein Spaces in THE GOHBERG ANNIVERSARY COLLECTION
  • 1990-09. Unitary colligations of operators in Krein spaces in INTEGRAL EQUATIONS AND OPERATOR THEORY
  • 1984-09. Hilbert spaces of analytic functions, inverse scattering and operator models.I in INTEGRAL EQUATIONS AND OPERATOR THEORY
  • 1984-01. Extensions of J-isometric and J-symmetric operators in FUNCTIONAL ANALYSIS AND ITS APPLICATIONS
  • 1990. Extension Theorems for Contraction Operators on Kreĭn Spaces in EXTENSION AND INTERPOLATION OF LINEAR OPERATORS AND MATRIX FUNCTIONS
  • Book

    TITLE

    Operator Theory and Complex Analysis

    ISBN

    978-3-0348-9699-3
    978-3-0348-8606-2

    Identifiers

    URI

    http://scigraph.springernature.com/pub.10.1007/978-3-0348-8606-2_2

    DOI

    http://dx.doi.org/10.1007/978-3-0348-8606-2_2

    DIMENSIONS

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