A FEW WORDS ABOUT THE BOOK
STYLE
I have tried to follow the logic of the above subjects as I understand it. As
a consequence, this book is neither a function theory monograph, nor an opera-
tor theory manual. It is a treatise on operator-based function theory, or, if you
prefer, function-based operator theory. As in my previous book "Treatise on the
shift operator" (Springer, 1986) I have in mind a picture close to mathematical
reality, where the most interesting and important facts take part of several disci-
plines simultaneously. This is why the way in which things proceed in this book is
sometimes different from the appoved didactic style of presentation, when, first of
all, background materials should be developed (even if you will need it 300 pages
later...), then you go to the next preparatory level, and so on.
Here, new concepts and auxiliary materials appear when they are needed to
continue the main theme. This theme is developed as theory of functions on the
circle group and of operators acting on them, starting with the basic shift oper-
ator, then passing to stationary filtering, and Hankel and Toeplitz operators as
compressions of the multiplication operators. Next, we arrive at the model theory
for Hilbert space operators as (advanced) compressions of the same shift operator,
and, finally, all this machinery is brought together to control dynamical systems.
Therefore, taken as a style to telling mathematics, this is more a passion or a tale
of mental intrigue than a rationally arranged catalog of facts.
It is also worth mentioning that this book has its origins in four courses I
gave in 1992-1996 to graduate students in the University of Bordeaux, France.
Although the courses were considerably extended when preparing this book, the
text, perhaps, preserves the flavor of interaction with the audience: sometimes I
repeat some notions or ideas already stated some tens (or hundreds...) of pages
earlier to remind the reader of something what he may have forgotten from the last
course.
BACKGROUND
As it is clear from the preceding lines, the book can be read by anyone having
a standard analysis background: Lebesgue measure,
Lp
spaces, elements of Fourier
series and Fourier transforms (the Plancherel theorem), elementary holomorphic
functions, Stone-Weierstrass theorem, Hilbert and Banach spaces, reflexivity, the
Hahn-Banach theorem, compactness, and so on.
FORMAL STRUCTURE
Parts A and B form the first volume of the book, and parts C and D form the
second. Formally speaking, parts A, B, C and D are (reasonably) independent of
each other in the sense that, for example, I may employ in part B some results of
parts A, C, or D, but in the same way that I use (rarely) results from some exterior
basic monographs.
The Parts are divided into chapters; there are 25 in the book. All chapters
but one contain two special sections: Exercises and Further Results, and Notes and
Remarks. These are important and inseparable parts of the book. To illustrate, the
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