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
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