A FEW WORDS ABOUT THE BOOK
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."
"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
Oh, the little more, and how much it is!
And the little less, and what worlds away!
R. Browning (Saul.st.39)