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 0-387-94067-7 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