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The Gohberg Anniversary Collection

Volume II: Topics in Analysis and Operator Theory

  • Editors
  • H. Dym
  • S. Goldberg
  • M. A. Kaashoek
  • P. Lancaster

Part of the Operator Theory: Advances and Applications book series (OT, volume 41)

Table of contents

  1. Front Matter
    Pages I-IX
  2. Joseph A. Ball, J. William Helton
    Pages 25-41
  3. Hari Bercovici, Dan Voiculescu
    Pages 57-60
  4. Branko Ćurgus, Aad Dijksma, Heinz Langer, Henk de Snoo
    Pages 125-152
  5. Avraham Feintuch
    Pages 241-254
  6. Ciprian Foias, Allen Tannenbaum
    Pages 255-277
  7. J. William Helton
    Pages 311-328
  8. Nikolai K. Nikolskii, Vasily I. Vasyunin
    Pages 405-434
  9. Donald Sarason
    Pages 485-496
  10. E. M. Semenov, I. Ya. Shneiberg
    Pages 497-509
  11. Keith F. Taylor
    Pages 511-518
  12. Harold Widom
    Pages 519-543
  13. Back Matter
    Pages 545-547

About this book

Introduction

In this article we shall use two special classes of reproducing kernel Hilbert spaces (which originate in the work of de Branges [dB) and de Branges-Rovnyak [dBRl), respectively) to solve matrix versions of a number of classical interpolation problems. Enroute we shall reinterpret de Branges' characterization of the first of these spaces, when it is finite dimensional, in terms of matrix equations of the Liapunov and Stein type and shall subsequently draw some general conclusions on rational m x m matrix valued functions which are "J unitary" a.e. on either the circle or the line. We shall also make some connections with the notation of displacement rank which has been introduced and extensively studied by Kailath and a number of his colleagues as well as the one used by Heinig and Rost [HR). The first of the two classes of spaces alluded to above is distinguished by a reproducing kernel of the special form K (>.) = J - U(>')JU(w)* (Ll) w Pw(>') , in which J is a constant m x m signature matrix and U is an m x m J inner matrix valued function over ~+, where ~+ is equal to either the open unit disc ID or the open upper half plane (1)+ and Pw(>') is defined in the table below.

Keywords

C*-algebra Hilbert space Operator theory calculus differential equation evolution minimum

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