# Operator Theory and Arithmetic in \(H^{∞}\)

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*Hari Bercovici*

Jordan's classification theorem for linear transformations on a
finite-dimensional vector space is a natural highlight of the deep relationship
between linear algebra and the arithmetical properties of polynomial rings.
Because the methods and results of finite-dimensional linear algebra seldom
extend to or have analogs in infinite-dimensional operator theory, it is
therefore remarkable to have a class of operators which has a classification
theorem analogous to Jordan's classical result and has properties closely
related to the arithmetic of the ring \(H^{\infty}\) of bounded analytic
functions in the unit disk. \(C_0\) is such a class and is the central object of
study in this book.

A contraction operator belongs to \(C_0\) if and only if the associated
functional calculus on \(H^{\infty}\) has a nontrivial kernel.
\(C_0\) was
discovered by Béla Sz.-Nagy and Ciprian Foiaş in their work on canonical
models for contraction operators on Hilbert space. Besides their intrinsic
interest and direct applications, operators of class \(C_0\) are very helpful in
constructing examples and counterexamples in other branches of operator theory.
In addition, \(C_0\) arises in certain problems of control and realization
theory.

In this survey work, the author provides a unified and concise presentation of
a subject that was covered in many articles. The book describes the
classification theory of \(C_0\) and relates this class to other subjects such as
general dilation theory, stochastic realization, representations of convolution
algebras, and Fredholm theory.

This book should be of interest to operator theorists as well as theoretical
engineers interested in the applications of operator theory. In an effort to
make the book as self-contained as possible, the author gives an introduction
to the theory of dilations and functional models for contraction operators.
Prerequisites for this book are a course in functional analysis and an
acquaintance with the theory of Hardy spaces in the unit disk. In addition,
knowledge of the trace class of operators is necessary in the chapter on weak
contractions.

#### Table of Contents

# Table of Contents

## Operator Theory and Arithmetic in $H^{}$

- Contents ix10 free
- Introduction xi12 free
- Chapter I. An Introduction to Dilation Theory 114 free
- Chapter II. The Class C[sub(0)] 1528
- Chapter III. Classification Theory 3548
- Chapter IV. Applications of Jordan Models 7386
- Chapter V. Characteristic Functions and the Class C[sub(0)] 109122
- Chapter VI. Weak Contractions 139152
- Chapter VII. Fredholm Theory 181194
- Chapter VIII. Miscellaneous Applications 231244
- Appendix: Notes and Comments 251264
- References 257270
- List of Notation 267280
- Subject Index 271284