## The Theory of \mathcal{H}(b) Spaces, Volumes 1 and 2

by Emmanuel Fricain and Javad Mashreghi

Cambridge University Press, 2016

ISBN 978-1-107-02777-0, 978-1-107-02778-7

To quote from the publisher’s description of this massive 2-volume treatise, “An \mathcal{H}(b) space is defined as a collection of analytic functions which are in the image of an operator. The theory of \mathcal{H}(b) spaces bridges two classical subjects: complex analysis and operator theory, which makes it both appealing and demanding.

“The first volume of this comprehensive treatment is devoted to the preliminary subjects required to understand the foundation of \mathcal{H}(b) spaces, such as Hardy spaces, Fourier analysis, integral representation theorems, Carleson measures, Toeplitz and Hankel operators, various types of shift operators, and Clark measures.

“The second volume focuses on the central theory. Both books are accessible to graduate students as well as researchers: each volume contains numerous exercises and hints, and figures are included throughout to illustrate the theory. Together, these two volumes provide everything the reader needs to understand and appreciate this beautiful branch of mathematics.”

These two volumes have a total of 1300 pages. In addition to the exercises mentioned above, each of the 31 chapters ends with a few pages of notes. The lists of references contain 572, resp. 194 entries.

A long and very detailed review of these two volumes, written by Brett D. Wicks, recently appeared in the Bulletin of the AMS (Volume 56, No. 3, July 2019, Pages 535-542), or online at https://www.ams.org/journals/bull/2019-56-03/

I urge the interested reader to consult this review. To quote from the final section: “[The volumes] are of a monographic nature and are designed for a person who wants to learn the theory of these spaces and understand the state of the art in the area. All major results are included. In some situations the original proofs are provided, while in other cases they provide the ‘better’ proofs that have become available since. The books are designed to be accessible to both experts and newcomers to the area. Comments at the end of each section are very helpful, and the numerous exercises were clearly chosen to help master some of the techniques and tools used. [The two volumes] are probably suitable for a year-long sequence in a topics course on complex analysis and operator theory. They are also appropriate for helping a novice learn the techniques and undertake research in the area.”

The reviewer ends by writing, “In sum, these are excellent books that are bound to become standard references for the theory of \mathcal{H}(b) spaces.”