On Applications of Reproducing Kernel Spaces to the Schur Algorithm and Rational J Unitary Factorization

  • Daniel Alpay
  • Harry Dym
Chapter
Part of the Operator Theory: Advances and Applications book series (OT, volume 18)

Abstract

The main theme of the first half of this paper rests upon the fact that there is a reproducing kernel Hilbert space of vector valued functions B (X) associated with each suitably restricted matrix valued analytic function X. The deep structural properties of certain classes of these spaces, and the theory of isometric and contractive inclusion of pairs of such spaces, which originates with de Branges, partially in collaboration with Rovnyak, is utilized to develop an algorithm for constructing a nested sequence B(X) ⊃ B(X 1) ⊃... of such spaces, each of which is included isometrically in its predecessor. This leads to a new and pleasing viewpoint of the Schur algorithm and various matrix generalizations thereof. The same methods are used to reinterpret the factorization of rational J inner matrices and a number of related issues, from the point of view of isometric inclusion of certain associated sequences of reproducing kernel Hilbert spaces.

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Copyright information

© Springer Basel AG 1986

Authors and Affiliations

  • Daniel Alpay
    • 1
  • Harry Dym
    • 1
  1. 1.Department of Theoretical MathematicsThe Weizmann Institute of ScienceRehovotIsrael

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