Operator Theory and Arithmetic in H (Mathematical Surveys and Monographs)

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 Foias 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.