Schur Parameters, Factorization and Dilation Problems

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.