##
**Composition operators and classical function theory.**
*(English)*
Zbl 0791.30033

Universitext: Tracts in Mathematics. New York: Springer-Verlag. xiii, 223 p. (1993).

The subject of the book is the theory of composition operators on the Hardy space \(H^ 2\) of the unit disc, with the main emphasis being compact composition operators. The book is not an exhaustive treatise on the subject matter; rather it is an introduction emphasizing the interplay of operator theory and classical analytic function theory.

To describe the setting, \(H^ 2\) denotes the Hardy space of holomorphic functions \(f\) on the unit disc \(U\) in \(\mathbb{C}\) for which \(\| f \|^ 2_ 2=\sup_{0<r<1} {1 \over 2 \pi} \int^{2 \pi}_ 0 | f(re^{it}) |^ 2dt<\infty\). For each holomorphic function \(\varphi:U \to U\), the composition operator \(C_ \varphi\) is defined by \[ C_ \varphi f=f \circ \varphi. \] Each such operator is bounded on \(H^ 2\) with \[ \| C_ \varphi \| \leq \sqrt{{1+| \varphi (0) | \over 1-| \varphi(0) |}}. \] This result is known as Littlewood’s theorem, the proof of which is the topic of chapter 1.

The main thrust of the work is on the relationship of function theoretic properties of \(\varphi\) and the behavior of the operator \(C_ \varphi\). Much of the text is concerned with characterizing those self-maps \(\varphi\) of \(U\) for which \(C_ \varphi\) is compact on \(H^ 2\). Although all the results are stated for \(H^ 2\), there is no loss of generality since \(C_ \varphi\) is compact on \(H^ p\), \(0<p<\infty\), if and only if it is compact on \(H^ 2\). In chapter 3 it is proved that if \(\varphi\) is a univalent self-map of \(U\), then \(C_ \varphi\) is compact on \(H^ 2\) if and only if \(\lim_{| z | \to 1-} {1-| \varphi (z) | \over 1-| z |} =+ \infty\).

In the subsequent chapter the geometric interpretation of the above is provided by the Julia-Carathéodory theorem which is used to prove the following: a) If \(C_ \varphi\) is compact on \(H^ 2\), then \(\varphi\) has an angular derivative at no point of \(\partial U\); b) If \(\varphi\) is univalent and has no angular derivative at any point of \(\partial U\), then \(C_ \varphi\) is compact on \(H^ 2\).

In chapter 6 the author uses the Denjoy-Wolff theorem to determine the spectrum of compact composition operators and then connects this with Königs’s classical work on holomorphic solutions of Schröders functional equation \(f \circ \varphi=\lambda f\). Chapters 7 and 8 deal with a discussion of linear fractional models and their use as a tool for investigating other properties of composition operators. This is illustrated in the study of cyclicity in chapter 7 and compactness in chapter 9. In the final chapter the connection is made between the solution of the compactness problem for general composition operators and the Nevanlinna counting function.

The book is easily accessible to anyone with a basic background in complex function theory and functional analysis. The author’s exposition presents a lively and interesting discussion of the topics and provides significant insight into the problems and results. The numerous exercises at the end of each chapter further help to expand upon the subject matter. Although the book does not have the usual list of unsolved problems or new directions for research, the reading of the monograph is sufficient to motivate problems for further investigation.

To describe the setting, \(H^ 2\) denotes the Hardy space of holomorphic functions \(f\) on the unit disc \(U\) in \(\mathbb{C}\) for which \(\| f \|^ 2_ 2=\sup_{0<r<1} {1 \over 2 \pi} \int^{2 \pi}_ 0 | f(re^{it}) |^ 2dt<\infty\). For each holomorphic function \(\varphi:U \to U\), the composition operator \(C_ \varphi\) is defined by \[ C_ \varphi f=f \circ \varphi. \] Each such operator is bounded on \(H^ 2\) with \[ \| C_ \varphi \| \leq \sqrt{{1+| \varphi (0) | \over 1-| \varphi(0) |}}. \] This result is known as Littlewood’s theorem, the proof of which is the topic of chapter 1.

The main thrust of the work is on the relationship of function theoretic properties of \(\varphi\) and the behavior of the operator \(C_ \varphi\). Much of the text is concerned with characterizing those self-maps \(\varphi\) of \(U\) for which \(C_ \varphi\) is compact on \(H^ 2\). Although all the results are stated for \(H^ 2\), there is no loss of generality since \(C_ \varphi\) is compact on \(H^ p\), \(0<p<\infty\), if and only if it is compact on \(H^ 2\). In chapter 3 it is proved that if \(\varphi\) is a univalent self-map of \(U\), then \(C_ \varphi\) is compact on \(H^ 2\) if and only if \(\lim_{| z | \to 1-} {1-| \varphi (z) | \over 1-| z |} =+ \infty\).

In the subsequent chapter the geometric interpretation of the above is provided by the Julia-Carathéodory theorem which is used to prove the following: a) If \(C_ \varphi\) is compact on \(H^ 2\), then \(\varphi\) has an angular derivative at no point of \(\partial U\); b) If \(\varphi\) is univalent and has no angular derivative at any point of \(\partial U\), then \(C_ \varphi\) is compact on \(H^ 2\).

In chapter 6 the author uses the Denjoy-Wolff theorem to determine the spectrum of compact composition operators and then connects this with Königs’s classical work on holomorphic solutions of Schröders functional equation \(f \circ \varphi=\lambda f\). Chapters 7 and 8 deal with a discussion of linear fractional models and their use as a tool for investigating other properties of composition operators. This is illustrated in the study of cyclicity in chapter 7 and compactness in chapter 9. In the final chapter the connection is made between the solution of the compactness problem for general composition operators and the Nevanlinna counting function.

The book is easily accessible to anyone with a basic background in complex function theory and functional analysis. The author’s exposition presents a lively and interesting discussion of the topics and provides significant insight into the problems and results. The numerous exercises at the end of each chapter further help to expand upon the subject matter. Although the book does not have the usual list of unsolved problems or new directions for research, the reading of the monograph is sufficient to motivate problems for further investigation.

Reviewer: M.Stoll (Columbia)

### MSC:

30D55 | \(H^p\)-classes (MSC2000) |

30-02 | Research exposition (monographs, survey articles) pertaining to functions of a complex variable |

47B38 | Linear operators on function spaces (general) |