Banach Spaces of Analytic Functions

by Kenneth Hoffman

Overview

A classic of pure mathematics, this advanced graduate-level text explores the intersection of functional analysis and analytic function theory. Close in spirit to abstract harmonic analysis, it is confined to Banach spaces of analytic functions in the unit disc.
The author devotes the first four chapters to proofs of classical theorems on boundary values and boundary integral representations of analytic functions in the unit disc, including generalizations to Dirichlet algebras. The fifth chapter contains the factorization theory of Hp functions, a discussion of some partial extensions of the factorization, and a brief description of the classical approach to the theorems of the first five chapters. The remainder of the book addresses the structure of various Banach spaces and Banach algebras of analytic functions in the unit disc.
Enhanced with 100 challenging exercises, a bibliography, and an index, this text belongs in the libraries of students, professional mathematicians, as well as anyone interested in a rigorous, high-level treatment of this topic.

Product Details

ISBN-13:	9780486149967
Publisher:	Dover Publications
Publication date:	06/10/2014
Sold by:	Barnes & Noble
Format:	eBook
Pages:	224
File size:	15 MB
Note:	This product may take a few minutes to download.

Table of Contents

Preliminaries     1
Measure and Integration     1
Banach Spaces     5
Hilbert Space and Fourier Series     8
Fourier Series     15
Cesaro Means     16
Characterization of Types of Fourier Series     22
Notes     24
Exercises     25
Analytic and Harmonic Functions in the Unit Disc     27
The Cauchy and Poisson Kernels     28
Boundary Values     32
Fatou's Theorem     34
H[superscript p] Spaces     39
Notes     39
Exercises     40
The Space H[superscript 1]     42
The Helson-Lowdenslager Approach     42
Szego's Theorem     48
Completion of the Discussion of H[superscript 1]     50
Dirichlet Algebras     54
Notes     57
Exercises     58
Factorization for H[superscript p] Functions     61
Inner and Outer Functions     61
Blaschke Products and Singular Functions     63
The Factorization Theorem     67
Absolute Convergence of Taylor Series     70
Remarks on the Classical Approach     72
Functionsof Bounded Characteristic     73
Notes     74
Exercises     74
Analytic Functions with Continuous Boundary Values     77
Conjugate Harmonic Functions     78
Theorems of Fatou and Rudin     80
The Closed Ideals of A     82
Commutative Banach Algebras     89
Wermer's Maximality Theorem     93
Notes     95
Exercises     95
The Shift Operator     98
The Shift Operator on H[superscript 2]     98
More about Dirichlet Algebras     101
Invariant Subspaces for H[superscript 2] of the Half-plane     103
Isometries     108
The Shift Operator on L[superscript 2]     111
The Vector-valued Case     114
Representations of H[superscript infinity]     116
Notes     119
Exercises     119
H[superscript p] Spaces in a Half-plane     121
H[superscript p] of the Half-plane     121
Boundary Values for H[superscript p] Functions     124
The Paley-Wiener Theorem     131
Factorization for H[superscript p] Functions in a Half-plane     132
Notes     133
Exercises      133
H[superscript p] as a Banach Space     136
Extreme Points     136
Isometries     142
Projections from L[superscript p] to H[superscript p]     149
Notes     155
Exercise     156
H[superscript infinity] as a Banach Algebra     158
Maximal Ideals in H[superscript infinity]     159
Topological Structure of M(H[superscript infinity])     162
Discs in Fibers     166
L[superscript infinity] as a Banach Algebra     169
The Silov Boundary     172
Inner Functions and the Silov Boundary     175
Representing Measures and Annihilating Measures     180
Algebras on the Fibers     187
Maximality     193
Interpolation     194
Notes     206
Exercises     207
Bibliography     209
Index     214