Schur parameters, factorization, and dilation problems
Author(s)
Bibliographic Information
Schur parameters, factorization, and dilation problems
(Operator theory : advances and applications, v. 82)
Birkhäuser Verlag, c1996
- : sz
- : us
Available at 32 libraries
  Aomori
  Iwate
  Miyagi
  Akita
  Yamagata
  Fukushima
  Ibaraki
  Tochigi
  Gunma
  Saitama
  Chiba
  Tokyo
  Kanagawa
  Niigata
  Toyama
  Ishikawa
  Fukui
  Yamanashi
  Nagano
  Gifu
  Shizuoka
  Aichi
  Mie
  Shiga
  Kyoto
  Osaka
  Hyogo
  Nara
  Wakayama
  Tottori
  Shimane
  Okayama
  Hiroshima
  Yamaguchi
  Tokushima
  Kagawa
  Ehime
  Kochi
  Fukuoka
  Saga
  Nagasaki
  Kumamoto
  Oita
  Miyazaki
  Kagoshima
  Okinawa
  Korea
  China
  Thailand
  United Kingdom
  Germany
  Switzerland
  France
  Belgium
  Netherlands
  Sweden
  Norway
  United States of America
Note
Includes bibliographical references and index
Description and Table of Contents
Description
The subject of this book is about the ubiquity of the Schur parameters, whose introduction goes back to a paper of I. Schur in 1917 concerning an interpolation problem of C. Caratheodory. What followed there appears to be a truly fascinating story which, however, should be told by a professional historian. Here we provide the reader with a simplified version, mostly related to the contents of the book. In the twenties, thf~ theory of orthogonal polynomials on the unit circle was developed by G. Szego and the formulae relating these polynomials involved num- bers (usually called Szego parameters) similar to the Schur parameters. Mean- while, R. Nevanlinna and G. Pick studied the theory of another interpolation problem, known since then as the Nevanlinna-Pick problem, and an algorithm similar to Schur's one was obtained by Nevanlinna. In 1957, Z. Nehari solved OO an L problem which contained both Caratheodory-Schur and Nevannlina-Pick problems as particular cases. Apparently unrelated work of H. Weyl, J. von Neu- mann and K. Friedericks concerning selfadjoint extensions of symmetric operators was connected to interpolation by M. A. Naimark and M.
G Krein using some gen- eral dilation theoretic ideas. Classical moment problems, like the trigonometric moment and Hamburger moment problems, were also related to these topics and a comprehensive account of what can be called the classical period has appeared in the monograph of M. G. Krein and A. A. Nudelman, [KN].
Table of Contents
1 Schur Parameters and Positive Block Matrices.- 1.1 Preliminaries.- 1.2 Renorming Hilbert Spaces and Elementary Rotations.- 1.3 Kolmogorov Decompositions. I.- 1.4 Row and Column Contractions.- 1.5 The Structure of Positive Definite Kernels.- 1.6 Kolmogorov Decompositions. II.- 1.7 Notes.- 2 Models for Triangular Contractions.- 2.1 Preliminaries.- 2.2 The Structure of Triangular Contractions.- 2.3 Realization of Triangular Contractions.- 2.4 Unitary Couplings and Operator Ranges.- 2.5 Modeling Families of Contractions.- 2.6 Notes.- 3 Moment Problems and Interpolation.- 3.1 A Survey on Completion Problems.- 3.2 Extensions of Partial Isometries.- 3.3 Krein’s Formula.- 3.4 Moment Problems.- 3.5 The Commutant Lifting Method.- 3.6 Notes.- 4 Displacement Structures.- 4.1 Structured Matrices.- 4.2 Generalized Schur Algorithm.- 4.3 Discrete Transmission-Line Models.- 4.4 Displacement Structure and Completion Problems.- 4.5 Other Applications.- 4.6 Notes.- 5 Factorization of Positive Definite Kernels.- 5.1 Spectral Factors.- 5.2 Examples.- 5.3 Schur’s Algorithm, Szegö’s Theory and Spectral Factors.- 5.4 Maximum Entropy.- 5.5 Notes.- 6 Nonstationary Processes.- 6.1 Modeling Nonstationary Processes.- 6.2 Kolmogorov-Wiener Prediction.- 6.3 Other Prediction Problems.- 6.4 Szegö’s Limit Theorems.- 6.5 Notes.- 7 Graphs and Completion Problems.- 7.1 Preliminaries.- 7.2 Completing Positive Partial Matrices. I.- 7.3 Completing Positive Partial Matrices. II.- 7.4 Completing Contractive Partial Matrices.- 7.5 Notes.- 8 Determinantal Formulae and Optimization.- 8.1 Determinantal Formulae.- 8.2 Maximum Determinant Formulae.- 8.3 Maximum Determinant for Nonchordal Graphs.- 8.4 Inheritance Principles.- 8.5 Notes.- References.
by "Nielsen BookData"