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