A Few Words about the Book

WHA T THIS BOOK IS ABOUT

The book represents a mixture of harmonic and complex analysis with operator

theory. The interplay between these disciplines is one of the most significant features

of the second half of Twentieth century mathematics. It gave rise to several jewels

of analysis, such as the theory of singular integral operators, Toeplitz operators,

mathematical scattering theory, Sz.-Nagy-Foia§ model theory, the L. de Branges

proof of the Bieberbach conjecture, as well as solving the principal interpolation

problems in complex analysis and discovering the structural properties of function

spaces (from Besov to Bergman).

The principal ingredients of the book are clear from the Contents and Subject

Index, and indeed a simple list of key words tells more than long explanations.

Without reproducing these lists nor the introductions to the four parts A, B, C,

and D of the book, I would like give an abridged list of my favorite subjects, ordered

by their appearance in the book:

Hardy classes

The Hilbert transformation

Weighted polynomial approximation

Cyclicity phenomena

Maximal and Littlewood-Paley functions

The Marcinkiewicz weak type interpolation

Wiener filtering theory

Riemann ( function

Hankel operators: spectral theory, Feller's theory, moment problems

Reproducing kernel Hilbert spaces

Schatten-von Neumann operator ideals

Toeplitz operators

The operator corona problem

Spectral theory of normal operators

Sz.-Nagy-Foia§ function model

Von Neumann inequalities

Carleson and generalized free interpolations

Theory of spectral multiplicities

Elements of semigroup theory

Classical control theory of dynamical systems

Bases of exponentials on intervals of the real line

Elements of the H°° control theory