94k:47049 47B38 30C99 46E20 46J15 
Shapiro, Joel H.(1-MIS)
Composition operators and classical function theory. (English) 
Universitext: Tracts in Mathematics.
Springer-Verlag, New York, 1993. xvi+223 pp. $34.00. ISBN

Composition operators are operators of the form f-->f o phi
acting on some space of functions on a set D, where phi is a
self-map of D. 

Written with infectious enthusiasm, this delightful book is an excellent
introductory text to the theory of composition operators on spaces of
analytic functions. It is not a monograph or a complete reference text on
the subject, but is "simply, an invitation to join in the fun", as stated in
the preface. As such it could not have been better. The most prominent
feature of the book is the shining revelation of the interplay between
classical function theory and operator theory. Classical concepts such as
subordination, angular derivatives, value distribution, iteration and
functional equations are given new meaning in the operator-theory
setting. They are linked to the most basic questions one can ask about
composition operators such as boundedness, compactness, cyclicity. The
inherent geometric flavor is prevalent throughout the book. The
development is on the Hardy space H^2, for phi an analytic
self-map of the unit disc, but these phenomena persist on many other
spaces of analytic functions. The classical concepts are developed fully
and this makes the book self-contained to a high degree and accessible to
anyone with a basic background in function theory and in operator
theory. Because of the intended audience, the treatment aims not to
obtain the most general results or to give all of what is known about
composition operators (for example, very little is said about spectra) but
to reveal the infrastructure on which the subject is built. 

Contents: 0. Linear fractional prologue: this chapter contains the basic
properties and classification of linear fractional transformations of the
unit disc. In the rest of the book, linear fractional transformations serve
not only as coordinate changing maps but also as a rich class of
manageable examples of composition operators. 1. Littlewood's theorem:
the Hardy space H^2 is introduced and the boundedness of
composition operators on H^2 is obtained as a consequence of
Littlewood's subordination principle. 2. Compactness; Introduction: the
motivation is set out and the first examples of compact and noncompact
composition operators are given. 3. Compactness and univalence: the
"right" proof of Littlewood's theorem for univalent phi in combination
with Schwarz's lemma gives insight for the compactness question. This
insight is confirmed by proving a characterization of compact composition
operators when the inducing map is univalent. The criterion brings into
the picture the Julia-Caratheodory theorem on the angular derivative. 4.
The angular derivative: the Julia-Caratheodory theorem is proved fully
with an emphasis on its geometric content. 5. Angular derivatives and
iteration: the Denjoy-Wolff theorem on the iterations of phi is
developed and its links with the material of Chapter 4 are exhibited. 6.
Compactness and eigenfunctions: eigenfunctions of composition operators
are solutions of the Schroder equation. Konigs's solution is presented, and
it is shown that compactness implies serious restrictions on the growth of
the eigenfunctions. 7. Linear fractional cyclicity: the cyclicity classification
for composition operators induced by linear fractional maps is given. 8.
Cyclicity and models: univalent maps phi are represented by simpler
ones on more complicated domains by means of linear fractional models.
The cyclicity results of Chapter 7 are then transferred to more general
univalent maps by means of models. 9. Compactness from models: the
results of Chapter 6 are turned around to show that, for univalently
induced composition operators, appropriate restrictions on the growth of
eigenfunctions implies compactness. 10. Compactness; the general case:
the general compactness criterion for composition operators in terms of
the Nevanlinna counting function is proved. 

Each chapter concludes with several exercises and a compilation of
informative notes. The reference section contains some 150 entries. This
book, together with the survey articles of E. A. Nordgren [in Hilbert space
operators (Long Beach, CA, 1977), 37--63, Lecture Notes in Math., 693,
Springer, Berlin, 1978; MR 80d:47046] and C. C. Cowen [in Operator
theory: operator algebras and applications, Part 1 (Durham, NH, 1988),
131--145, Proc. Sympos. Pure Math., 51, Part 1, Amer. Math. Soc.,
Providence, RI, 1990; MR 91m:47043], is an invaluable source for anyone
who wants to learn about composition operators. 

                    Reviewed by Aristomenis Siskakis 

Cited in: 97h:47023 95d:47036 

© Copyright American Mathematical Society 1994, 1997