BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY

In a first course in complex analysis, students learn a theorem that states that if an analytic function is zero on a non-discrete set inside a region in the complex plane, then the function must be identically zero. In particular, the values that an analytic function takes in the neighborhood of a single point completely determine the function in the whole region. This, of course, is very useful for proving many other theorems about analytic functions. However, it also presents a challenge when one is trying to construct examples with certain required properties. Unlike in a real analysis setting, one cannot just cut the region up into smaller pieces, construct examples locally, and hope to be able to glue everything back together. Over time many ingenious ways have been developed to deal with this problem. It is a large branch of modern complex analysis that tries to devise means to construct certain classes of analytic functions from real variable type parameters. If the region is the open unit disc D and one is interested in bounded analytic functions, then a fully developed theory is available. In fact, this theory extends to cover the Hardy spaces H for 0 < p < ∞. On the other hand, for the larger Bergman spaces A of the unit disc many new phenomena occur, new theorems and proofs had to be developed, and some basic questions are still not completely settled. Nevertheless, the past ten years have seen a remarkable number of breakthroughs in this area: perhaps most notably the geometric characterization of sequences of interpolation and sampling; a near closing of the gap between necessary and sufficient conditions for zero sequences of A-functions; characterizations of bounded Hankel operators, of compact Hankel operators, and of compact Toeplitz operators; the discovery of contractive zero divisors and an A-inner-outer factorization; the relationship between Bergman-inner functions and the biharmonic Green function; and other results concerning the invariant subspace structure of A. In the book under review the authors present certain aspects of these new developments. The main focus is on questions concerning the function theory and the invariant subspace structure of the spaces A. Hankel and Toeplitz operators are not discussed. For 0 < p < ∞ the Bergman space A is defined to be the set of all analytic functions f on the open unit disc D such that ||f ||pAp = ∫ D |f(z)| dA(z) π <∞. Here dA has been used to denote two-dimensional Lebesgue measure. For p ≥ 1 A is a Banach space, while for 0 < p < 1 A is a complete space with translation invariant metric given by d(f, g) = ||f − g||pAp . The reader should note that in this review we shall discuss only unweighted Bergman spaces. In the book, the authors consider the standard weighted Bergman spaces Aα, α > −1 and the growth spaces A−α, whenever this is feasible. Furthermore, to get maximum benefit out of reading either the book or this review, the reader should have at least a superficial familiarity with the H-function theory; see for example [D], [G], [K].