Banach Spaces of Analytic Functions

the book is a cornerstone of any serious inquiry in Hardy spaces and the invariant subspace problem; it is also hightly readable and well written. people interested in a second course on complex functions, harmonic analysis and functional analysis (banach and hilbert spaces) should have a look at it; it deserves it and the reader will be richly rewarded...

The theory of Hardy spaces is vast, along with its applications. This book overviews what was known about them in the early 1960s. In spite of its age, it can still be read profitably by anyone interested in harmonic analysis and Hardy spaces. Chapter 1 gives a quick review of the mathematical background needed for reading the rest of the book, mostly dealing with measure theory, and Banach and Hilbert spaces. In chapter 2, the author gives a detailed treatment of Fourier series over the closed interval from -pi to pi. The chapter is designed to answer two questions, namely whether a function is determined by its Fourier series, and given a particular Fourier series, how one can recapture the function. These questions must be addressed in the appropriate norm on the Banach space of Lp spaces of Lebesgue integrable functions. There are many methods of recapturing the function, and the author discusses a few such methods, one being the Cesaro means. The authors proves that for a function in Lp, the Cesaro means of the Fourier series of the function converge to it in the Lp norm (when p is greater than or equal to 1 but less than infinity). When p is infinity, the author shows this is true in the weak-star topology. The author then shows how the Cesaro means can be used to characterize the different types of Fourier series. Analytic and harmonic functions in the unit disk are defined and studied in chapter 3. The first question the author addresses is to what extent these functions are determined by their boundary values. The author shows how to represent these functions on the closed unit disk using the Cauchy and Poisson integral formulas, thus answering this question. The second question he addresses is the behavior of these functions on the boundary, i.e. the Dirichlet problem. His methods for harmonic functions are analagous to those for Lp under the guise of Cesaro means, i.e. Cesaro summability becomes Abel summability. The author shows this connection more rigorously by proving Fatou's theorem. Hp spaces are defined in this chapter, and the author illustrates one of the major differences between the harmonic and analytic functions. The author begins the study of H1 spaces in chapter 4, initially via the Helson-Lowdenslager approach. He first proves Fejer's theorem for functions which are continuous on the closed unit disk and analytic at each interior point: the real parts of these functions are uniformly dense in the space of real-valued continuous functions on the unit circle. Szego's theorem, which gives a measure of the "distance" from the constant function 1 to the subspace of these functions that vanish at the origin, is proved, as well as the Riesz theorem, which shows that analytic measures on the unit circle are absolutely continuous with respect to Lebesgue measure. He then applies these results to H1 functions, showing that such functions cannot vanish on a set of positive Lebesgue measure on the circle without being identical