This text appears as number 237 in the Graduate Texts in Mathematics series published by Springer. It is based on a graduate course taught at the University of Toronto and it is intended as a possible textbook for graduate courses, as well as independent study and reference for research.
The book is very well written and contains very elegant proofs. It assumes background knowledge from basic courses in complex analysis and functional analysis and discusses the Hardy-Hilbert spaces (a natural setting for operator theory) and operators on them (the Shift Operator, Toeplitz Operators, Hankel Operators, and Composition Operators). As the authors mention in the Preface, this book was written with the following quote from Hardy in mind: “Beauty is the first test: there is no permanent place in the world for ugly mathematics.”
Each chapter in the book has plenty of exercises, as well as sections with “notes and remarks” with references to the many papers and books related to the subject. The references at the end of the text are very helpful for the reader interested in future research.
In short: this is a wonderful book, a pleasure to read and use as a text, or add to any mathematician’s “collection” of references to a beautiful and rich subject.
Some possible typos have been posted as a list of errata at http://www.math.toronto.edu/rosent .
Mihaela Poplicher is an associate professor of mathematics at the University of Cincinnati. Her research interests include functional analysis, harmonic analysis, and complex analysis. She is also interested in the teaching of mathematics. Her email address is Mihaela.Poplicher@uc.edu.
Introduction.- The Unilateral Shift and Factorization of Functions.- Toeplitz Operators.- Hankel Operators.- Composition Operators.- Further reading.- References.- Notation Index.- Author Index.- Subject Index.