Over the last ten years, the theory of Bergman spaces has undergone a
remarkable metamorphosis. In a series of major advances, central problems
once considered intractable were solved, and a rich theory has emerged.
Although progress continues, the time seems ripe for a full and unified
account of the subject, weaving old and new results together.
The modern subject of Bergman spaces is a blend of complex function
theory with functional analysis and operator theory. It comes in contact with
harmonic analysis, approximation theory, hyperbolic geometry, potential
theory, and partial differential equations. Our aim has been to develop
background material and make the subject accessible to a broad segment of
the mathematical community. We hope the book will prove useful not only
as a reference for research workers, but as a text for graduate students.
In fact, the book evolved from a rough set of notes prepared for grad-
uate students in a two-week course that one of us gave in 1996 at the
Norwegian University of Science and Technology in Trondheim, in conjunc-
tion with a conference on Bergman spaces supported by the Research Coun-
cil of Norway. Since that time we have used successive versions of the man-
uscript in graduate courses we taught at the University of Michigan (1998),
Washington University in St. Louis (1999), and San Francisco State Univer-
sity (2001). The last course was supported by the NSF CIRE (Collaborative
to Integrate Research and Education) program. The students in all of these
courses were enthusiastic, and their perceptive remarks on the manuscript
often led to substantial improvements.
In striving for clear presentations of material, we have had the benefit
of expert advice from many friends and colleagues. We are most grateful to
Kristian Seip for guiding us to a self-contained account of his deep results
on interpolation and sampling. Harold Shapiro showed us an elegant way to
develop the biharmonic Green function and helped with other construc-
tions. Dmitry Khavinson fielded a steady barrage of technical questions
and offered many useful suggestions on the manuscript. Sheldon Axler
made a careful reading of several chapters and gave valuable criticism.
Mathematical help of various sorts came also from Marcin Bownik, Brent
Carswell, Eric Hayashi, Hakan Hedenmalm, Anton Kim, John McCarthy,
Maria Nowak, Stefan Richter, Richard Rochberg, Joel Shapiro, Michael
Stessin, Carl Sundberg, James Tung, Dror Varolin, Dragan Vukotic, Rachel
Weir, and Kehe Zhu. Special thanks go to Joel Shapiro for permission to