##
**Composition operators on function spaces.**
*(English)*
Zbl 0788.47021

North-Holland Mathematics Studies. 179. Amsterdam: North-Holland. x, 315 p. (1993).

Let \((X,{\mathcal X},m)\) be a \(\sigma\)-definite measure space, let \(S\) be a non-empty set, and let \(\{F_ s,\;s\in S\}\) be a class of vector spaces over the field of real numbers \(\mathbb{R}\) or over the field of complex numbers \(\mathbb{C}\). The Cartesian product of sets \(F_ s\), \(s\in S\), is denoted by \(\prod F_ s\). If \(T: X\to X\) is a transformation on \(X\) and \(M(X)\) is a set of measurable functions on \(X\) into \(\prod F_ s\), then the composition operator \(C_ T\) is defined by \(C_ T(f)(x)= f\circ T(x)= f(T(x))\). If \(\rho\) is a transformation on \(M(X)\) such that \(\rho f\circ T\in \prod F_ s\) for \(f\) in \(\prod F_ s\), then the ‘weighted composition operator’, \(W_{\rho,T}\), is defined by \(W_{\rho,T}(f)(x)=\rho f\circ T(x)= \rho f(T(x))\).

In this book, the authors have endeavoured to present an account of a variety of known results relating to composition and weighted composition operators. Chapter I contains terminology relating to a number of function spaces. Chapter 2 introduces some of the essential basic properties of composition operators, with statements of results involving normality, compactness and boundedness of composition operators in \(L^ p(X)\). In particular, the introduced of the Radon-Nikodym derivative in a change of variable identifies some weighted composition operators as isometries.

The results of Chapter 3 relate to composition operators in \(H^ p\)- spaces, and include considerations of \(H^ p(D^ n)\), where \[ D^ n=\{(x_ 1,x_ 2,\dots,x_ n): x_ i\in \mathbb{C}, \;0\leq | x_ j|\leq 1,\;j=1,2,\dots,n\} \] and \(H^ p(P^ +)\) where \(P^ +=\{w= u+ iv\in C, v>0\}\). The considerations of Chapter 4 include invertibility and compactness of weighted composition operators in locally convex function spaces. The results of Chapter 5 deal with the connection between isometries and composition operators, and include the Banach- Lamperti theorem on the representation of isometries in \(L^ p(X)\), \(1\leq p<\infty\), \(p\neq 2\). Considerations of Chapter 5 also include some connections of composition operators with Ergodic theory, and statements involving semigroups of operators and dynamical systems.

The book elaborates upon most of the topics treated but does not include some recent developments including some new considerations on isometries in \(L^ p\)-spaces, semigroups of operators generated by ‘differential operators’ and properties of ‘isometry invariant operators’ [cf. papers by the reviewer, Bull. Math. 5, 1-30 (1982; Zbl 0504.47038), Bull. Math. No. 8, 1-48 (1983; Zbl 0514.47026)].

Although there are no obviously misleading statements in most parts of the main text, a keen reader may notice a slight confusion in the definition of composition operators in the opening sections of Chapter 1. The confusion would seem to involve the use of the same character \(X\) for two different sets.

In this book, the authors have endeavoured to present an account of a variety of known results relating to composition and weighted composition operators. Chapter I contains terminology relating to a number of function spaces. Chapter 2 introduces some of the essential basic properties of composition operators, with statements of results involving normality, compactness and boundedness of composition operators in \(L^ p(X)\). In particular, the introduced of the Radon-Nikodym derivative in a change of variable identifies some weighted composition operators as isometries.

The results of Chapter 3 relate to composition operators in \(H^ p\)- spaces, and include considerations of \(H^ p(D^ n)\), where \[ D^ n=\{(x_ 1,x_ 2,\dots,x_ n): x_ i\in \mathbb{C}, \;0\leq | x_ j|\leq 1,\;j=1,2,\dots,n\} \] and \(H^ p(P^ +)\) where \(P^ +=\{w= u+ iv\in C, v>0\}\). The considerations of Chapter 4 include invertibility and compactness of weighted composition operators in locally convex function spaces. The results of Chapter 5 deal with the connection between isometries and composition operators, and include the Banach- Lamperti theorem on the representation of isometries in \(L^ p(X)\), \(1\leq p<\infty\), \(p\neq 2\). Considerations of Chapter 5 also include some connections of composition operators with Ergodic theory, and statements involving semigroups of operators and dynamical systems.

The book elaborates upon most of the topics treated but does not include some recent developments including some new considerations on isometries in \(L^ p\)-spaces, semigroups of operators generated by ‘differential operators’ and properties of ‘isometry invariant operators’ [cf. papers by the reviewer, Bull. Math. 5, 1-30 (1982; Zbl 0504.47038), Bull. Math. No. 8, 1-48 (1983; Zbl 0514.47026)].

Although there are no obviously misleading statements in most parts of the main text, a keen reader may notice a slight confusion in the definition of composition operators in the opening sections of Chapter 1. The confusion would seem to involve the use of the same character \(X\) for two different sets.

Reviewer: G.O.Okikiolu (London)

### MSC:

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

46J15 | Banach algebras of differentiable or analytic functions, \(H^p\)-spaces |

47-02 | Research exposition (monographs, survey articles) pertaining to operator theory |

47A35 | Ergodic theory of linear operators |

47D06 | One-parameter semigroups and linear evolution equations |

46E30 | Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) |

46E15 | Banach spaces of continuous, differentiable or analytic functions |