The main theme of the book is the structure of modules over function algebras of holomorphic functions in several complex variables. The book concentrates mainly on topics developed by both authors. The book starts with a nice introduction summarizing main results described in the book. Chapter 2 describes the technique of characteristic space theory for analytic Hilbert modules, which was developed by K. Guo. Chapter 3 shows that analytic Hilbert modules in several variables have a much more rigid structure than in one variable. The equivalence problem for Hardy submodules in the cases of the polydisk or the unit ball is treated in Chapter 4. The next chapter describes the structure of the Fock space, or more generally, reproducing function spaces on Cn. Chapter 6 contains a discussion of modules over the Arveson space of square integrable functions on the unit ball in Cn. In the last chapter, the authors describe the extension theory of Hilbert modules over function algebras. The book is written in a clear and systematic way and it describes an interesting part of the recent development in the field. Each chapter ends with useful bibliographic comments.

Reviewer:

vs