A FEW WORDS ABOUT THE BOOK

vn

book contains 1428 propositions conventionally called theorems, lemmas, corollar-

ies, and exercises. For Exercises and Further Results, the proofs are called "hints",

and while they are shorter they still contain all the principal ingredients to under-

stand the proof. All exercises were tested by a team of volunteer readers whose

names are listed below (there were no casualties...). Some (rare) facts included in

exercises and not proved are marked by asterisk *.

Sections Notes and Remarks usually contain surveys of the rest of the theory

presented in the main body of the corresponding chapter.

Reference A.a.(3.7 means subsection a./?.7 of Part A; a.(3 means section (3 of

chapter a of the Part where you are; a.f3.^(2) means point (3) of subsection a./3/y,

etc. Sign • indicates the end of a proof or a reasoning.

T H E READER AND THE AUTHOR

As it is clear from the subtitle of the book, I expect that some readers are

novices, graduate or undergraduate students, possessing the needed knowledge in-

dicated above. It is also supposed that some readers are experts. Well, I shall be

rewarded if there is at least one. In this case, and also anticipating the inevitable

reproaches as to why I selected such and such subjects and not others, I permit

myself to quote (in my translation from Russian) a great philologist, an expert of

texts as such.

"The answers books give us are to questions that are not exactly the same as

the author set before himself, but to those that we are able to raise ourselves... The

books encircle us like mirrors, in which we see only our own reflexion; the reason

why it is, perhaps, not everywhere the same is because all these mirrors are curved,

each in its own way."

M.L. Gasparov

"Philology as morality"

WHA T IS MISSED

Of course, I do not intend to list here the rest of mathematics but just to men-

tion explicitly some border subjects that could have been included but were not.

These are (without any ordering) extremal problems of complex analysis (start-

ing from results of S.Ya. Khavinson and H. Shapiro in the 1960's); the problems of

harmonic analysis-synthesis (from L. Schwartz and B. Malgrange of the 1950's); in-

variant subspaces, from the existence problem in a Hilbert space, up to (more impor-

tant) classification problems for concrete operators (including descriptions of closed

ideals in algebras of holomorphic functions); singular integral models for hypo-

and semi-normal operators; scattering; univalent functions via quasi-orthogonal

decompositions; realization theory; operator valued constrained interpolation, and

some other themes. I have no better way to excuse these omissions than to fol-

low E. Beckenbach and R. Bellman who quoted the following verses (for a similar

purpose):

Oh, the little more, and how much it is!

And the little less, and what worlds away!

R. Browning (Saul.st.39)