The book first rigorously develops the theory of reproducing kernel Hilbert spaces. The authors then discuss the Pick problem of finding the function of smallest $H^\infty$ norm that has specified values at a finite number of points in the disk. Their viewpoint is to consider $H^\infty$ as the multiplier algebra of the Hardy space and to use Hilbert space techniques to solve the problem. This approach generalizes to a wide collection of spaces. The authors then consider the interpolation problem in the space of bounded analytic functions on the bidisk and give a complete description of the solution. They then consider very general interpolation problems. The book includes developments of all the theory that is needed, including operator model theory, the Arveson extension theorem, and the hereditary functional calculus.
Based on a 1999 graduate course held at Washington University, this volume is intended to be accessible to graduate students interested in operator theory or holomorphic spaces. Readers should have basic knowledge of functional analysis (Lebesgue integration, the closed graph theorem, Hahn-Banach theorem, Banach-Alaoglu theorem; some operator theory on Hilbert spaces such as knowing what the strong operator and weak-star topologies are, knowing what a unitary operator is) and complex analysis (knowing Schwarz's lemma and what the Poisson kernel is). Includes both routine and challenging exercises; an appendix presents results. Annotation c. Book News, Inc., Portland, OR (booknews.com)
