Function Theory and Lp Spaces (University Lecture Series) 1470455935, 9781470455934

Table of contents :
Preface
Overview
Notation
Chapter 1. The Basics of ℓ^{𝑝}
1.1. Definition of ℓ^{𝑝}
1.2. Classical Inequalities
1.3. Completeness
1.4. The Case 0

Citation preview

UNIVERSITY LECTURE SERIES VOLUME 75

Function Theory p and  Spaces Raymond Cheng Javad Mashreghi William T. Ross

10.1090/ulect/075

Function Theory p and  Spaces

UNIVERSITY LECTURE SERIES VOLUME 75

Function Theory p and  Spaces Raymond Cheng Javad Mashreghi William T. Ross

EDITORIAL COMMITTEE Robert Guralnick Emily Riehl

William P. Minicozzi II (Chair) Tatiana Toro

2010 Mathematics Subject Classification. Primary 46B10, 46B25, 30H50, 30J05, 30B10, 30B30.

For additional information and updates on this book, visit www.ams.org/bookpages/ulect-75

Library of Congress Cataloging-in-Publication Data Names: Cheng, Raymond, 1961- author. | Mashreghi, Javad, author. | Ross, William T., 1964author. Title: Function theory and Lp spaces / Raymond Cheng, Javad Mashreghi, William T. Ross. Description: Providence, Rhode Island : American Mathematical Society, [2020] | Series: University lecture series, 1047-3998 ; volume 75 | Includes bibliographical references and index. Identifiers: LCCN 2020008686 | ISBN 9781470455934 (paperback) | ISBN 9781470460099 (ebook) Subjects: LCSH: Functions. | Functions of complex variables. | Lp spaces. | Functional analysis. | AMS: Functional analysis {For manifolds modeled on topological linear spaces, see 57Nxx, 58Bxx} – Normed linear spaces and Banach spaces; Banach lattices {For function spaces, see 46Exx} – Duality and re | Functional analysis {For manifolds modeled on topological linear spaces, see 57Nxx, 58Bxx} – Normed linear spaces and Banach spaces; Banach lattices {For function spaces, see 46Exx} – Classical Bana | Functions of a complex variable {For analysis on manifolds, see 58-XX} – Spaces and algebras of analytic functions – Algebras of analytic functions. | Functions of a complex variable {For analysis on manifolds, see 58-XX} – Function theory on the disc – Inner functions. | Functions of a complex variable {For analysis on manifolds, see 58-XX} – Series expansions – Power series (including lacunary series). | Functions of a complex variable {For analysis on manifolds, see 58-XX} – Series expansions – Boundary behavior of power series, over-convergence. Classification: LCC QA331 .C4486 2020 | DDC 515/.73–dc23 LC record available at https://lccn.loc.gov/2020008686

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established to ensure permanence and durability. Visit the AMS home page at https://www.ams.org/ 10 9 8 7 6 5 4 3 2 1

25 24 23 22 21 20

Contents Preface

ix

Overview

xi

Notation

xv

Chapter 1. The Basics of p 1.1. Definition of p 1.2. Classical Inequalities 1.3. Completeness 1.4. The Case 0 < p < 1 1.5. The Embedding Operator 1.6. Schauder Basis 1.7. The Dual of p 1.8. Norming Functionals 1.9. Notes

1 1 2 5 6 9 10 10 14 15

Chapter 2. Frames 2.1. Frames in Hilbert Spaces 2.2. Frames in p 2.3. Parseval and Tight Frames 2.4. Riesz Bases 2.5. An Analogue of the Feichtinger Conjecture for p 2.6. Notes

17 17 18 21 24 27 28

Chapter 3. The Geometry of p 3.1. Convexity 3.2. Metric Projection 3.3. Birkhoff-James Orthogonality 3.4. Smoothness 3.5. Notes

29 29 33 34 36 40

Chapter 4. Weak Parallelogram Laws 4.1. The Parallelogram Law 4.2. Basic Properties 4.3. Geometric Consequences of Weak Parallelogram Laws 4.4. Duality of the Weak Parallelogram Laws 4.5. Weak Parallelogram Laws for p 4.6. Pythagorean Inequalities 4.7. Lack of Weak Parallelogram Laws 4.8. Metric Projections onto Nested Subspaces

41 41 43 47 48 50 55 59 60

v

vi

CONTENTS

4.9. Notes

61

Chapter 5. Hardy and Bergman Spaces 5.1. The Hardy Space 5.2. The Bergman Space 5.3. The Shift Operator 5.4. The Backward Shift Operator 5.5. Notes

63 63 68 69 70 72

Chapter 6. p as a Function Space 6.1. The Space pA 6.2. Some Inhabitants of pA 6.3. Relationship to H p Spaces 6.4. Evaluation Functionals and Duality 6.5. Boundary Behavior in pA 6.6. pA is an Algebra When 0 < p  1 6.7. Notes

73 73 74 81 82 83 84 87

Chapter 7. Some Operators on pA 7.1. The Shift Operator 7.2. The Difference Quotient Operator 7.3. Hadamard Multipliers 7.4. Isometries on pA 7.5. Composition Operators 7.6. Notes

89 89 91 92 93 95 99

Chapter 8. Extremal Functions 8.1. Zero Sets 8.2. Solving an Extremal Problem 8.3. Extremal Functions as Inner Functions 8.4. A Related Extremal Problem 8.5. One Point Extremal Function 8.6. Finite Point Extremal Functions 8.7. Extra Zeros 8.8. Bounds for Extra Zeros 8.9. Notes

101 101 102 105 109 111 113 115 120 122

Chapter 9. Zeros of pA Functions 9.1. The Blaschke Condition 9.2. Zero Sets and p-Inner Functions 9.3. Geometric Convergence to the Boundary 9.4. Slower Than Geometric Convergence to the Boundary 9.5. A Non-Blaschke Zero Set for p > 2 9.6. Blaschke, But Not a Zero Set 9.7. Zero Sets When 0 < p  1 9.8. A Note About Sampling in pA 9.9. Notes

123 123 126 129 132 134 138 144 145 148

Chapter 10. The Shift 10.1. Finite Co-Dimensional Invariant Subspaces

149 149

10.2. 10.3. 10.4. 10.5. Chapter 11.1. 11.2. 11.3. 11.4. 11.5.

CONTENTS

vii

A Quick Review of Fredholm Theory The Division Property Beurling’s Theorem Notes

151 152 154 156

11. The Backward Shift Pseudocontinuations Other Types of Continuations Finite Dimensional Invariant Subspaces Gap Series Theorems Notes

159 159 162 168 168 171

Chapter 12. Multipliers of pA 12.1. Convolutions 12.2. The Space of Multipliers 12.3. Mp as the Commutant 12.4. A Multiplier Norm Estimate 12.5. Connection to Fourier Multipliers 12.6. Boundary Properties of Multipliers 12.7. Isometric Multipliers 12.8. Smooth Multipliers 12.9. 1A Embeds Contractively in Mp 12.10. Quotients of Multipliers 12.11. Inner Multipliers 12.12. Notes

173 173 173 176 179 180 182 184 186 188 189 190 191

Chapter 13.1. 13.2. 13.3. 13.4. 13.5. 13.6.

193 193 195 198 199 200 203

13. The Wiener Algebra Some Inhabitants of the Wiener Algebra Wiener’s 1/f Theorem Composition Zero Sets Ideals Notes

Bibliography

205

Author Index

213

Subject Index

217

Preface This book covers the classical sequence space p , where p ∈ (0, ∞], along with the associated space pA of analytic functions on the open unit disk whose sequence of Taylor coefficients belongs to p . We are particularly interested in the case p ∈ [1, ∞], where p and pA are Banach spaces. There is a large literature on Banach spaces emphasizing various characterization and embedding theorems, basis theory, and other aspects. We refer the reader to some great books such as these classics [48, 105] or the more recent tomes [4, 24]. The main emphasis of this book, however, is on the function theory of the pA spaces, that is, the connections between the algebraic and functional analysis properties of the sequence space p , the analytical behavior of associated functions from pA , as well as some natural linear operators on these spaces. Despite their classical origins and simple definition, the pA spaces are not as well understood as other function spaces such as the Hardy, Bergman, and Dirichlet spaces, all of which have a well-developed and extensive literature along with their own volumes [13, 53, 54, 57, 68, 81, 104]. In particular, the Hardy space represents quite a triumph of a century of mathematical analysis. The zero sets of the Hardy spaces are fully understood; an inner-outer factorization reveals the structure of their member functions; Beurling’s Theorem describes the shift-invariant subspaces; various extremal problems and interpolation problems are solved; the Corona Theorem discusses the maximal ideals; and the multipliers (Hadamard and pointwise) and composition operators are also well known. All of these riches ultimately owe their presence to the boundary growth properties that define the Hardy space. In contrast to this, the p-summability condition on the Taylor coefficients of pA functions seems to purchase much less regularity in the functions themselves. Nevertheless, pA is an equally fascinating class of analytic functions with a developing, but scattered, literature [1, 2, 11, 27–30, 33, 34, 36, 37, 69, 74, 75, 79, 93, 99, 108–114, 129, 133, 156, 163–166] that also deserves its own book. The geometric tools developed within these pages, which include an extension of the Pythagorean Theorem, have made possible some inroads into important issues. These advances include a description of the zero sets of pA , along with some gains in our understanding of canonical factorization and the shift-invariant subspaces. Despite these advances, many fundamental questions concerning the structure of pA remain without complete answers. We will point out these open questions as opportunities for the reader to join the discussion. For the sequence space p , we will examine various properties of its norm, along with embeddings, completeness, and duality. We will also explore some geometric concepts including convexity, smoothness, Birkhoff-James orthogonality, the weak

ix

x

PREFACE

parallelogram laws, and the aforementioned Pythagorean inequalities. For the associated function space pA , we will discuss boundary growth and boundary behavior, extremal problems, zero sets, and duality, along with some well-known operators on pA such as the shift, backward shift, composition operators, and the pointwise and Hadamard multipliers. Much of the literature on this subject, especially the applications of certain geometric tools to describe the functional behavior of pA , has not yet been brought together under a single title. Except for some basic measure theory, functional analysis, and complex analysis, which we expect the reader to know, the material in this book is self contained and we give detailed proofs of nearly all of the results. We place great emphasis on our desire to educate and not merely catalogue the main ideas. As such, we not only go through the material in a deliberate and careful way, but we also furnish end notes for each chapter where we give proper references, historical background, and avenues for further exploration. R. Cheng, Norfolk

J. Mashreghi, Quebec City

W. Ross, Richmond

Overview The sequence space p is one of the first Banach spaces a graduate student encounters when studying functional analysis. In fact, this sequence space, along with its close cousins, were among the first Banach spaces to be systematically studied and often served as motivation for many questions. It is natural then to examine pA , the analytic functions whose sequence of Taylor coefficients lies in p . This combines the Banach space theory of the sequence space with the complex analysis of the corresponding function space. By classical theory, the sequence space p , p ∈ (1, ∞), is a reflexive Banach space with a readily identifiable dual space. When p ∈ (0, 1), it becomes an F space. The p = 1 case is of special interest since it corresponds to the well-studied Wiener algebra. With these p spaces one can even define a notion of orthogonality that is inspired by Hilbert space orthogonality with its associated concepts such as a projection and the unique closest vector to a convex set. With p also comes a version of the parallelogram law that one has for a Hilbert space and even a version of the classical Pythagorean Theorem. The first several chapters of the book comprise this material. We begin our discussion of p in Chapter 1 with a treatment of the classical inequalities of Young, H¨older, and Minkowski. These classical results show that p is a Banach space when p ∈ [1, ∞] and an F -space when p ∈ (0, 1). We then identify the dual of p and include a description of the norming functionals for p , since they will be needed later in our investigation of orthogonality in Banach spaces. Frames and Riesz bases in Hilbert spaces are well-developed concepts with many applications to image compression and data recovery. As it turns out, there are analogues of these notions for reasonable Banach spaces. One of these reasonable spaces is of course p , and in Chapter 2 we discuss frames, with a special emphasis on Parseval frames and Riesz bases in this setting. We will also cover a version of the now-settled Feichtinger conjecture for the p spaces. We take up the geometry of p in Chapter 3 with particular attention to convexity. Since Birkhoff-James orthogonality is an important tool in our book, and is not at the fingertips of every Banach space student, we also spend some time unpacking this topic. To make much of the orthogonality material fit together, we also treat the issue of the smoothness of p in the final section of this chapter. Birkhoff-James orthogonality segues quite naturally into versions of the classical parallelogram laws and the Pythagorean Theorem for certain Banach spaces, which we take up in Chapter 4. These Pythagorean Theorems, which comprise a family of inequalities, will be of great use to us when we move on to the function theory portion of the book. Chapter 5 is somewhat of an interlude as we pass from the properties of the sequence spaces p to those of the function spaces pA . Roughly speaking, pA lies xi

xii

OVERVIEW

somewhere between the classical Hardy spaces, where much is known through the classical results of Hardy, Littlewood, Riesz, Nevanlinna, and Beurling; and the Bergman spaces, where the function theory, though well studied by Aleman, Richter, Korenblum, Hedenmalm, Seip, and Sundberg, is more complicated and less understood. With the Hardy spaces, there is an enormous amount of structure (boundary regularity, well understood zero sets, factorization theorems) and since pA is contained in the Hardy spaces when p ∈ (1, 2], it enjoys some, but not all, of these properties. On the other hand, when p ∈ (2, ∞], pA is much larger and acts more like the complicated Bergman spaces in that there is little boundary regularity for these functions. Thus the classical theory no longer provides us with any of the structure that Hardy spaces enjoy. The function theory portion of the book begins in earnest with Chapter 6, where we cover some elementary properties of the pA spaces. In particular, we discuss the boundary growth and boundary behavior of these functions as well as examples of interesting functions that belong to these spaces. Making our case that pA acts like a Hardy space for some p and a Bergman space for other p, we cover the Hausdorff-Young inequalities which place the Hardy spaces in relationship to the pA spaces. When p ∈ (0, 1], pA is algebra of functions that are smooth up to the boundary, while when p = 1, 1A is the well-known Wiener algebra and will be studied further in Chapter 13. Though there is much function theory for pA still to cover, their zero sets for example, we devote our attention in Chapter 7 towards some of the basic linear transformations (operators) on pA . These are all well-known operators that play important roles in functional analysis and operator theory, and include the shift, the backward shift, Hadamard multipliers, isometries, and composition operators. This is only an introductory chapter for these operators since two of them in particular, the shift and the backward shift, will get their own chapters later on. The zero sets for pA are not completely understood, as compared to the Hardy spaces, where they are completely characterized by the Blaschke condition. An avenue into understanding these zero sets is through certain linear extremal problems which are taken up in Chapter 8. These problems have a rich history dating back to the beginnings of complex analysis and quite a lot can be said. In the pA case we explore extremal problems through Birkhoff-James orthogonality and develop a notion of an “inner function” for pA . These inner functions have some remarkable properties and guide us in understanding the zero sets for pA . We work out some extremal function computations for finite zero sets and show that unlike the Hardy and Bergman spaces, the extremal functions in pA , even for finite zero sets, can have extra zeros. Chapter 9 is focused entirely on the zero sets for pA . We begin our discussion with the classical Blaschke condition and survey how close and how far the zero sets for pA deviate from this condition. Our general treatment of the zeros of pA relies on developing a Banach space version of a Hilbert space technique of Shapiro and Shields which examines zero sets through solutions of extremal problems. Here is where we bring in the connection to the Birkhoff-James theory developed in the previous chapter. Towards the end of this chapter, we present two results which show that for certain p there are zero sets for pA which are not Blaschke sequences and for other p there are Blaschke sequences which are not zero sets for pA . Thus

OVERVIEW

xiii

the Blaschke condition, although related to the zeros of pA , is not a complete characterization, except of course when p = 2. We covered some of the basic properties of the shift operator in Chapter 7. In Chapter 10 we expand this discussion and cover the invariant subspaces and cyclic vectors for the shift. We describe the invariant subspaces of finite co-dimension using some ideas of Axler and Bourdon. When p = 2 there is a complete description of the invariant subspaces of 2A due to Beurling. However, for other p, there is no such result and, in fact the invariant subspaces can have quite wild behavior. Here we touch upon some results of Abakumov and Borichev, which show that the invariant subspaces of pA , p > 2, can have large indices similar to those of the Bergman space. Chapter 11 is devoted to the invariant subspaces and the cyclic vectors for the backward shift operator on pA . When p = 2 the notion of a pseudocontinuation comes into play, in that a function belonging to an invariant subspace for the backward shift has an associated meromorphic function on the exterior disk that shares the same boundary values on the unit circle. For other p, the concept of a pseudocontinuation needs to be replaced by another generalized analytic continuation involving formal multiplication of Laurent series. We also explore some lacunary series results of Shapiro and Abakumov, which extend the classical noncontinuability gap theorems of Hadamard and Fabry to the generalized analytic continuations surveyed in this chapter. There is quite a large literature, mostly Russian, concerning the pointwise multipliers of pA , which is what we cover in Chapter 12. In general, for most Banach spaces of analytic functions, the pointwise multipliers are better behaved in some ways than generic functions from the space. In the Hardy and Bergman spaces for instance, the multipliers must be bounded analytic functions, and in fact being a bounded analytic function is the exact criterion for being a multiplier for these two spaces. For the pA spaces however, though every multiplier must be bounded, this is not sufficient. Results in this area show that the multipliers of pA must have a certain amount of extra boundary regularity. There are also many results which determine whether certain classical inner functions, such as Blaschke products or singular inner functions, are multipliers for pA . Chapter 13 is a selected survey of the Wiener algebra 1A . This is a vast area with several texts on the subject, and we will not provide a complete account. Instead, we will outline some of the main results such as the Wiener 1/f theorem and the description of the maximal ideals of 1A . Other ideas such as a characterization of the zero sets, the ideals, and function composition results are complicated and unresolved. This book is meant not to be the definitive work on the subject but instead to provide the reader with the necessary fundamentals to begin their own work to complete the picture. As such, we look forward to reading about your discoveries.

Notation • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •

:= . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . equality by definition ≈ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . approximate equality C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . the complex numbers  = C ∪ {∞} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . the Riemann sphere C ,  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . the real and imaginary parts N = {1, 2, . . .} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . the natural numbers N0 = {0, 1, 2, . . .} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . the nonnegative integers D = {z ∈ C : |z| < 1} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . the open unit disk D = {z ∈ C : |z|  1} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . the closed unit disk De = {z ∈ C : |z| > 1} ∪ {∞} . . . . . . . . . . . . . . . . the extended exterior disk T = {z ∈ C : |z| = 1} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .the unit circle dim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . the dimension of a vector space  ·  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . the norm on a vector space B(X , Y ) . . . . . . . . . . . . . . . . . . . . . . . . the bounded operators from X to Y X ∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . the dual space of a Banach space X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . isometric isomorphism A∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . the adjoint of an operator A σ(A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . the spectrum of an operator A K (X ) . . . . . . . . . . . . . . . . . . . the compact operators on a Banach space X p . . . . . . . . . . . . . . . . . . . . . . . . . . . . the space of p-summable sequences (p. 1)  · p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . the norm or quasinorm on p (p. 1) ek . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . the kth standard basis vector (p. 1) Lp = Lp (X, B, μ) . . . . . . . . . . . . . . . . . . . . . . . . . . . .the Lebesgue spaces (p. 2) p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . the H¨older conjugate of p (p. 2) (·, ·) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . the inner product (p. 3) δjk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . the Kronecker delta (p. 10)

·, · . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . the bilinear pairing (p. 11) λx . . . . . . . . . . . . . . . . . . . . . . . . . . . the norming linear functional for x (p. 15) a· . . . . . . . . . . . . . . . . . . . . . the signed power of a complex number a (p. 15) X1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . the closed unit ball of X (p. 29) ∂X1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . the unit sphere of X (p. 29) PM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . metric projection onto M (p. 34) ⊥X , ⊥p . . . . . . . . . . . . . . . . . . . . . . . . . . . Birkhoff-James orthogonality (p. 34) r-LWP(C), r-UWP(C) . . . . . . . . . . . . . the weak parallelogram laws (p. 43) dm. . . . . . . . . . . . . . . . . . . . . . . . . .normalized Lesbesgue measure on T (p. 63) Hol(D) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . the analytic functions on D (p. 63) H p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .the Hardy space (p. 63)  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . the closed linear span (p. 66) dα . . . . . . . . . . . . . . . . . . . normalized planar Lebesgue measure on D (p. 68) xv

xvi

NOTATION

• • • • • • • • • • • • • • • • • • • • • • • • •

Ap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . the Bergman space (p. 68) S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . the shift operator (p. 69) [f ] . . . . . . . . . . . . . . . . . . the shift-invariant subspace generated by f (p. 70) S ∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . the backward shift operator (p. 70) [g]∗ . . . . . . . . . . . . . . . . . . . . the S ∗ -invariant subspace generated by g (p. 71) pA . . . . . . . . . . . . . . . . . . . . . . . . .the function space associated with p (p. 73) Qw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . the difference quotient operator (p. 91)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . the Hadamard product (p. 92) Mh . . . . . . . the Hadamard multiplication operator with symbol h (p. 92) Cϕ . . . . . . . . . . the composition operator induced by the symbol ϕ (p. 95) RW . . . . . . . . . . . . . . . . . . . . the functions in pA which vanish on W (p. 102) f . . . . . . . . . . . . the metric projection of f onto the subspace [Sf ] (p. 106) kw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . the reproducing kernel (p. 107) i(T ) . . . . . . . . . . . . . . . . . . the index of a semi-Fredholm operator T (p. 151) Z(M ) . . . . . . . . . . . . . . . . . . . . . . . . . the set of common zeros for M (p. 153) H 2 (De ) . . . . . . . . . . the Hardy space of the extended exterior disk (p. 160) dA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . planar Lebesgue measure on C (p. 165) a ∗ b . . . . . . . . . . . . . . . . . . . . . . . . . the convolution of two sequences (p. 173) Mp . . . . . . . . . . . . . . . . . . . . . . . . . . . the (pointwise) multipliers of pA (p. 173) Mϕ . . . . . . . . . . . the multiplication operator with symbol ϕ on pA (p. 174) ϕMp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . the multiplier norm of ϕ (p. 174) Σb (h) . . . . . . . . . . . the boundary spectrum of an inner function h (p. 191) A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . the disk algebra (p. 193) 1A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . the analytic Wiener algebra (p. 193) W . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . the Wiener algebra (p. 195)

10.1090/ulect/075/01

CHAPTER 1

The Basics of p The p spaces were among the first examples of Banach spaces to be systematically studied and have often served as the inspiration for many concepts in Banach space theory. In this chapter we will show that when p ∈ [1, ∞] they are indeed Banach spaces, that is, normed vector spaces that are complete under the induced metric. Toward this goal, important tools such as the H¨older and Minkowski inequalities are derived. Some attention is also given to the case when p ∈ (0, 1) where p , though no longer a Banach space, is a complete metric space with many interesting properties. We will then use these classical inequalities to characterize the dual of p . 1.1. Definition of p Definition 1.1.1. Let p ∈ (0, ∞) and define p to be the set of sequences a = {a0 , a1 , a2 , . . .} of complex numbers for which the quantity 1  p (1.1.2) ap := |ak |p k0 ∞

is finite. We define 

to be the set of bounded sequences and set a∞ := sup{|ak | : k  0}.

The choice of the nonnegative integers N0 = {0, 1, 2, . . .} to be the indexing set for p is traditional and is further motivated by connections to function theory. As we will see, beginning in Chapter 6, the sequence {a0 , a1 , a2 , . . .} will correspond to the power series a0 + a1 z + a2 z 2 + · · · . However, from time to time it will be necessary to utilize other indexing sets. For example, we could write p (Z) to indicate that the index set is the integers Z, that is,    p (Z) = {. . . , a−3 , a−2 , a−1 , a0 , a1 , a2 , a3 , . . .} : |ak |p < ∞ . k∈Z

We will also use the notation e0 := {1, 0, 0, 0, 0, . . .}, e1 := {0, 1, 0, 0, 0, . . .}, e2 := {0, 0, 1, 0, 0, . . .}, .. . for the standard basis vectors for p . Of course we still need to show that p is a vector space. This will be done in the next section. 1

1. THE BASICS OF p

2

Some of what we know about p is true in the more general setting of the Lebesgue spaces Lp = Lp (X, B, μ), where X is a set, B is a σ-algebra of subsets of X, and μ is a positive measure on X. Observe that p can be thought of as Lp (N0 , B, μ), where the σ-algebra B consists of all the subsets of N0 and μ is counting measure on N0 . Recall the norm on Lp is given by  1 p f p := |f |p dμ , p < ∞, X

and f ∞ := ess-sup{|f (x)| : x ∈ X}, where ess-sup denotes the essential supremum ess-sup{|f (x)| : x ∈ X} := inf{y  0 : μ(f −1 {(y, ∞)}) = 0}. 1.2. Classical Inequalities We shall now establish that when p ∈ [1, ∞], the set Lp (and hence p ), endowed with  · p , is a Banach space. In doing so, we will acquire a number of tools that will be useful again and again. Let us begin with Young’s Inequality where we adhere to the following definition. Definition 1.2.1. For a fixed p ∈ [1, ∞] define p to be the H¨ older conjugate index corresponding to p, that is, 1 1 +  = 1. p p Observe that p = p/(p − 1) ∈ [1, ∞]. Theorem 1.2.2 (Young’s Inequality). If a, b > 0 and p > 1, then 

ab 

(1.2.3)

ap bp + , p p



with equality if and only if ap = bp . Proof. Let x = p1 and observe that 1 − x = natural logarithm function we have

1 p .

By the strict concavity of the

log ab = log a + log b 

= x log(ap ) + (1 − x) log(bp ) 

 log [xap + (1 − x)bp ]   ap bp  +  . = log p p The exponential function is strictly monotone increasing, and so by exponentiating both sides of the above, we see that (1.2.3) holds along with the stated condition for equality.  Theorem 1.2.4 (H¨ older’s Inequality). Let (X, B, μ) be a measure space and let p ∈ [1, ∞]. Then for any complex valued measurable functions f and g on X, f g1  f p gp .

(1.2.5) 

When p and p are both finite, equality in (1.2.5) holds if and only if there exist  α, β  0, not both zero, such that α|f (x)|p = β|g(x)|p for almost all x ∈ X.

1.2. CLASSICAL INEQUALITIES

3

Proof. We may assume that 0 < f p < ∞ and 0 < gp < ∞, for otherwise (1.2.5) is trivially true. Furthermore, when p = 1 and p = ∞, the condition |f (x)g(x)|  |f (x)|g∞ holds for almost every x ∈ X. Integrating both sides of this inequality results in |f (x)g(x)| dμ  |f (x)|g∞ dμ, X

X

which yields f g1  f 1 g∞ . We may therefore assume that p ∈ (1, ∞), and perforce, p ∈ (1, ∞). Now define F (x) :=

f (x) , f p

G(x) :=

g(x) , gp

x ∈ X.

Notice that F p = 1 and Gp = 1. Young’s Inequality (Theorem 1.2.2) shows that  |G(x)|p |F (x)|p + |F (x)G(x)|  p p for almost all x ∈ X. Integrating both sides of the previous inequality, we obtain 

Gpp F pp f g1 1 1 = F G1  = +  = 1. + f p gp p p p p The inequality in (1.2.5) now follows. Now observe that equality holds if and only  if |F (x)|p = |G(x)|p for almost every x ∈ X, which is equivalent to the condition 

α|f (x)|p = β|g(x)|p , 

where α = gpp and β = f pp .



H¨older’s inequality is, in spirit, a generalization of the Cauchy-Schwarz Inequality, which states that the inner product of two vectors in a Hilbert space is absolutely bounded by the product of their lengths. Indeed, the case p = p = 2 is associated with the Hilbert space L2 (X, B, μ), for which the inner product is given by (f, g) :=

f (x)g(x) dμ. X

We can use H¨older’s inequality to derive Minkowski’s Inequality, which serves as the Triangle Inequality for Lp . Theorem 1.2.6 (Minkowski’s Inequality). Let (X, B, μ) be a measure space and let p ∈ [1, ∞]. If f, g ∈ Lp (X, B, μ), then f + g ∈ Lp (X, B, μ) and (1.2.7)

f + gp  f p + gp .

Moreover, when p ∈ (1, ∞), equality holds in (1.2.7) if and only if g = 0, or f = αg for some α  0. Proof. The Triangle Inequality for the complex numbers C ensures that |f (x) + g(x)|  |f (x)| + |g(x)| for every x ∈ X. Integrating both sides of the previous inequality shows that f + g1  f 1 + g1 ,

1. THE BASICS OF p

4

for any integrable functions f and g on X. This verifies the p = 1 case. Similarly, from the inequalities sup{|f (x) + g(x)| : x ∈ X}  sup{|f (x)| + |g(x)| : x ∈ X}  sup{|f (x)| : x ∈ X} + sup{|g(x)| : x ∈ X}, the p = ∞ case is confirmed. For p ∈ (1, ∞) the function t → tp is convex and monotone increasing on the interval [0, ∞). Thus for any f, g ∈ Lp and any x ∈ X we have

f (x) + g(x) p  |f (x)| + |g(x)| p |f (x)|p + |g(x)|p





 2 2 2 and thus |f (x) + g(x)|p  2p−1 (|f (x)|p + |g(x)|p ). From here it follows that f + g ∈ Lp . Now we need to verify (1.2.7). If f + gp = 0 there is nothing to prove. Therefore suppose otherwise, and apply the Triangle Inequality on C to get p f + gp = |f (x) + g(x)|p dμ X = |f (x) + g(x)| · |f (x) + g(x)|p−1 dμ X (|f (x)| + |g(x)|) · |f (x) + g(x)|p−1 dμ  X  |f (x)| · |f (x) + g(x)|p−1 dμ X |g(x)| · |f (x) + g(x)|p−1 dμ. + X

Note that p − 1 = pp . H¨older’s Inequality (Theorem 1.2.4), applied to each of the last two terms above, yields  1   1  p p (1.2.8) |f (x)|p dμ |f (x) + g(x)|(p−1)p dμ f + gpp  X



+

X

|g(x)| dμ p

1  p

X

|f (x) + g(x)|

(p−1)p

dμ

 1 p

X

= (f p + gp )f + gp−1 . p This affirms the claim for p ∈ (1, ∞). The condition for equality follows from the H¨older’s Inequality estimate in (1.2.8).  Let f ∈ Lp and let c ∈ C. Then f p = 0 precisely when f (x) = 0 for almost every x ∈ X. Thus we identify two Lp functions if they are equal almost everywhere. Minkowski’s Inequality shows that Lp is closed under addition, and that  · p satisfies the Triangle Inequality. Finally, from |cf (x)|p dμ = |c|p |f (x)|p dμ X

X

we deduce that Lp is closed under multiplication by scalars, and cf p = |c|f p . In conclusion, Lp is a vector space over C, and  · p determines a norm on Lp .

1.3. COMPLETENESS

5

1.3. Completeness Recall that a metric space (X , d) is complete if every Cauchy sequence in X converges to a point in X . Any normed vector space (V ,  · ) is a metric space under the natural metric given by d(a, b) = a − b,

a, b ∈ V .

When p ∈ [1, ∞], the normed vector space L is complete, and hence a Banach space. Let us prove completeness for p , the subject of this book. p

Theorem 1.3.1. For any p ∈ [1, ∞], the space p is complete. Proof. Fix p ∈ [1, ∞], and suppose that a(1) , a(2) , a(3) , . . . is a Cauchy sequence of vectors in p . Thus for each n  1, the vector a(n) is itself a sequence of complex numbers, given by (n)

(n)

(n)

(n)

a(n) = {a0 , a1 , a2 , a3 , . . .}. For any index k  0 we have (m)

|ak

(n)

− ak |  a(m) − a(n) p .

From here we see that for each fixed k, (1)

(2)

(3)

{ak , ak , ak , . . .} is a Cauchy sequence of complex numbers and therefore converges to some complex number ak (no superscript). The vector a := {a0 , a1 , a2 , . . .} serves as a candidate for the limit of the given Cauchy sequence of vectors. These next several steps verify this. By hypothesis, for any > 0 there exists an index N such that a(n) − a(m) p < ,

m, n  N.

In particular, for each k  0, this gives k  1/p (n) (m) |aj − aj |p < . j=0

By letting n tend to infinity we obtain k  1/p (m) |aj − aj |p  . j=0

Since this holds for all k, it follows that 1/p  (m) (1.3.2) a(m) − ap  |aj − aj |p  . j0

By Minkowski’s Inequality ap  a(m) − ap + a(m) p < ∞, and so a ∈ p . From (1.3.2) we conclude that the sequence {a(n) }n1 converges to a in p norm.

1. THE BASICS OF p

6

We have shown that the normed vector space p is complete.



When p = 2 the space 2 is not only a Banach space but also a complex Hilbert space with inner product  (a, b)2 := ak bk . k0

Notice the complex conjugate in the second slot. Also notice that p = 2. 1.4. The Case 0 < p < 1 Many of the inequalities of the previous section are a result of the convexity of the function t → tp when p  1. When p ∈ (0, 1), this function is concave, and some of the corresponding inequalities are true in the reverse sense. This tells us something about the spaces Lp = Lp (X, B, μ) when p ∈ (0, 1). Proposition 1.4.1. If u, v  0, and p ∈ (0, 1), then (1.4.2)

(u + v)p  up + v p ,

with equality if and only if uv = 0. Proof. If either u or v is zero, then equality holds in (1.4.2). Otherwise, from the fact that p ∈ (0, 1), we have  u p  v p u v < < and . u+v u+v u+v u+v Thus  u p  v p v u + < + . 1= u+v u+v u+v u+v Multiplying through by (u + v)p completes the proof.  Proposition 1.4.3. Let p ∈ (0, 1) and suppose that f, g ∈ Lp (μ). Then (1.4.4)

f + gp  2(1−p)/p (f p + gp ),

with equality if and only if f (x)g(x) = 0 μ-almost everywhere and f p = gp . Proof. From Proposition 1.4.1, we have for all x |f (x) + g(x)|p  (|f (x)| + |g(x)|)p  |f (x)|p + |g(x)|p . Integrating over the space shows that (1.4.5)

f + gpp  f pp + gpp .

And now since the function ϕ(t) = tp is strictly concave, we have (1.4.6)

(1.4.7)

f + gpp  f pp + gpp  f p + gp  p p =2 2  f  + g p p p 2 2 = 21−p (f p + gp )p .

Take pth roots to obtain (1.4.4). Equality holds in (1.4.4) if and only if there is equality in both (1.4.6) and (1.4.7). This implies that f (x)g(x) = 0 μ-almost everywhere and f p = gp , respectively. 

1.4. THE CASE 0 < p < 1

7

As a consequence of Proposition 1.4.3, we see that Lp for p ∈ (0, 1) is closed under addition. Since it is also closed under scalar multiplication, it is a vector space. It is not a normed space under  · p , however, since the Triangle Inequality can fail. In fact, the Triangle Inequality reverses for nonnegative functions. This will be important in proving Theorem 3.1.16 (part of Clarkson’s Inequalities) below. Proposition 1.4.8. If p ∈ (0, 1), and f and g are nonnegative functions in Lp (μ), then (1.4.9)

f + gp  f p + gp ,

with equality if and only if there are nonnegative constants α and β, not both zero, such that αf (x) = βg(x) μ-almost everywhere. Proof. If either f or g is the zero function, then the claim holds with equality. Otherwise, let f p t= . f p + gp Use the strict concavity of the function ϕ(t) = tp to obtain  f (x) g(x) p f (x)p g(x)p + (1 − t) (f (x) + g(x))p = t  t p + (1 − t) . t 1−t t (1 − t)p for every x. Equality holds precisely when g(x) f (x) = . t 1−t Integrating over the space results in f pp gpp f + gpp  t p + (1 − t) t (1 − t)p  f  + g p−1  f  + g p−1 p p p p (1.4.10) = f pp + gpp . f p gp We have equality precisely when f (x)/t = g(x)/(1 − t) μ-almost everywhere. This simplifies to f p + gp f p + gp f (x) = g(x) f p gp 1 1 f (x) = g(x) f p gp gp f (x) = f p g(x). This incorporates all of the previous conditions for equality, if we permit f p = 0 or gp = 0. The proof is completed by simplifying and taking pth roots in (1.4.10).  To be sure, consider e0 = {1, 0, 0, 0, . . .} and e1 = {0, 1, 0, 0, 0, . . .} in p . Then e0 + e1 p = (1p + 1p )1/p > 2 = e1 p + e2 p , and thus the inequality in (1.4.9) could be strict. But not all is lost. We can interpret (1.4.4) as saying that the Triangle Inequality holds with an extra multiplicative constant. This makes  · p a quasinorm on Lp when p ∈ (0, 1). Furthermore, from (1.4.5) we see that d(f, g) := f − gpp

1. THE BASICS OF p

8

satisfies the Triangle Inequality. Clearly, d is also symmetric and positive definite. Hence Lp endowed with d is a metric space. In fact, it is a complete metric space, and the proof is very similar to that for the case p ∈ [1, ∞]. When p ∈ (0, 1), H¨ older’s inequality also reverses for nonnegative functions. Proposition 1.4.11. Let p ∈ (0, 1) and let  ative functions, and g p dμ > 0, then X





f g dμ 

(1.4.12)

+

1 

p

X

1 p

= 1. If f and g are nonneg-



p

f dμ

X

1 p

g p dμ

 1 p

,

X

with equality precisely if there exists a nonnegative constant γ such that f (x)g(x)1/(1−p) = γ

μ-almost everywhere.



Proof. Let us note that p must be negative, and so by the hypothesis that





g p dμ > 0, g must be positive μ-almost everywhere. If f is the zero function, X

then there is nothing more to prove. Thus suppose that f is nonvanishing on a set of positive measure. If the left hand side of (1.4.12) is infinite, then again we are done. Therefore we may assume that f g ∈ L1 (μ) and define g(x)

G(x) :=  X

g p



Observe that

dμ

f (x)G(x) , K(x) := f G dμ X

1/p ,

x ∈ X.



p

G dμ = 1 and X

K dμ = 1.

X 1/(1−p)

Furthermore, K p ∈ L1/p , and 1/Gp ∈ L , since  p  p (K p )1/p dμ = K dμ = 1 X



and

X

(1/Gp )1/(1−p) dμ



1−p

X

Notice that

1 p



Gp dμ

=

1−p = 1.

X

> 1, and 1 1 p

+

1 1 1−p

= 1.

In other words, 1/p and 1/(1 − p) are H¨older conjugates. We may therefore apply H¨older’s Inequality (Theorem 1.2.4), to obtain  p  1−p 1 1 (K/G)p dμ  (K p ) p dμ (1/Gp ) 1−p dμ = 1. X

X

X

Expressed in terms of f and g, this says   2p  p p p p p [f g /g ] dμ g dμ X X  1.  p   p p dμ p f g dμ g X X After performing cancellations, clearing the denominator, and taking pth roots, we obtain (1.4.12).

1.5. THE EMBEDDING OPERATOR

9

The condition for equality, according to Theorem 1.2.4, is the existence of nonnegative constants α and β, not both zero, such that αK(x) = βG(x)

p − 1−p

μ-almost everywhere. Expressed in terms of f and g, this is equivalent to p   1 1−p 1−p αf (x)g(x) =β f g dμ gp X



Now consolidate the constants. 1.5. The Embedding Operator

Any sequence that belongs to some p with p ∈ (0, ∞) must converge to zero, and hence it also belongs to ∞ . More can be said. Let a = {a0 , a1 , a2 , . . .} ∈ r for some r ∈ (0, 1). Repeated application of (1.4.2) shows that for any nonnegative integer N ,

1  |a0 | + |a1 | + |a2 | + · · · + |aN |  |a0 |r + |a1 |r + |a2 |r + · · · + |aN |r r  ar . Since this is true for all N , it follows that (1.5.1)

|a0 | + |a1 | + |a2 | + · · ·  ar

and a ∈ 1 . If equality holds, then equality must also hold in each of the intermediate estimates  r |a0 |r + |a1 |r + · · · + |aN −1 |r + |aN | + |aN +1 | + |aN +2 | + · · ·  r  |a0 |r + |a1 |r + · · · + |aN −1 |r + |aN |r + |aN +1 | + |aN +2 | + · · · , where we have applied (1.4.2) N times to the left side of (1.5.1). This implies that for each N , at most one of the two expressions |aN | and   |aN +1 | + |aN +2 | + · · · is nonzero. If any aN is nonzero, then it must be the only nonzero term in the entire sequence. If no aN is nonzero, then of course a is the zero sequence. Next, suppose 0 < p1 < p2 < ∞, so that the role of r in the above analysis is played by p1 /p2 < 1, and let each |ak | be replaced by |ak |p2 . Then the assertion is that 1

1

(|a0 |p2 + |a1 |p2 + |a2 |p2 + · · · ) p2  (|a0 |p1 + |a1 |p1 + |a2 |p1 + · · · ) p1 whenever a ∈ p1 . Equality holds when at most one of the terms a0 , a1 , a2 , . . . is nonzero for each N . This proves the following embedding property. Proposition 1.5.2. If 0 < p1 < p2  ∞, then p1 ⊆ p2 , and ap2  ap1 for all a ∈ p1 . Equality holds if and only if a is the zero vector, or is a multiple of eN for some nonnegative integer N . In the case 1  p1 < p2  ∞, the inclusion operator from p1 to p2 has norm one.

1. THE BASICS OF p

10

1.6. Schauder Basis A sequence {vk }k0 of vectors in a Banach space X constitutes a Schauder basis for X if for every x ∈ X there exists a unique sequence of scalars {ak }k0 such that  ak vk , x= k0

where the above sum converges in the norm of X . For example, in Section 1.1 we encountered the vectors ek := {δ0k , δ1k , δ2k , . . .},

(1.6.1)

where δjk is the Kronecker delta, and k  0. If p ∈ [1, ∞), then {ek }k0 forms a Schauder basis for p . When p = 2 the basis from (1.6.1) is also an orthonormal basis. Any Banach space that has a Schauder basis must be separable. Indeed, consider the countable dense set consisting of finite sums of the form N 

(rk + isk )ek ,

k=0

where rk and sk are rational numbers. However, ∞ is another story. The collection of sequences E := {x = {x0 , x1 , . . .} : xn = 0 or xn = 1} is uncountable, and has the disjointness property x − y∞ = 1,

x = y.

∞

Thus  is not separable, and hence cannot have a Schauder basis. We shall see that ∞ is exceptional in a number of other ways as well. 1.7. The Dual of p Recall that if X is a Banach space, then a linear functional λ on X is a continuous linear mapping from X into C. The collection of linear functionals on X is called the dual space of X , and it is denoted by X ∗ . We are reminded that X ∗ is itself a Banach space when given the norm λ = sup{|λ(x)| : x = 1}. The Riesz Representation Theorem tells us that for p ∈ [1, ∞), the dual space  (Lp )∗ of Lp is isometrically isomorphic to Lp where as usual p1 + p1 = 1. That is, 

every λ ∈ (Lp )∗ can be identified with a unique g ∈ Lp such that λ(f ) = f, g = f (x)g(x) dμ, f ∈ Lp . X 1 ∗

∞

In particular, (L ) L . Remark 1.7.1. We emphasize that the notation f, g represents the canonical  pairing of f ∈ Lp with g ∈ Lp , and not an inner product. In the case p = p = 2, the inner product would contain the complex conjugate g in place of g, and the inner product is denoted by (f, g). Let us further take note that the correspondence is not unique. Throughout the literature, other representations exist and have their advantages and disadvantages.

1.7. THE DUAL OF p

11

In particular, here is a description of (p )∗ when p ∈ [1, ∞). Theorem 1.7.2 (Riesz Representation Theorem). If p ∈ [1, ∞) then (p )∗   . That is, every λ ∈ (p )∗ can be identified with a unique b ∈ p such that  ak bk , a ∈ p . (1.7.3) λ(a) = a, b = p

k0 

Conversely, if b ∈ p then the mapping λ(a) := a, b,

a ∈ p ,

is a linear functional on p . In either case, λ = bp . 

Proof. First, suppose that p ∈ (1, ∞), so that p ∈ (1, ∞). Let b ∈ p , and define  λ(a) = ak bk k0

for finite sequences a. There is no harm in assuming that b = 0. H¨ older’s Inequality (Theorem 1.2.5) then shows that







|λ(a)| =

ak bk  |ak bk |  ap bp . k0

k0

From this we deduce that λ can be extended continuously to all of p , and λ  bp . Let us set u = {u0 , u1 , . . .}, where  0, bk = 0, (1.7.4) uk := p −1 p −2 |bk | bk /bp , bk = 0, and observe that up =



p(p −1)



|bk |p(p −1) /bp

=

k0







|bk |p /bpp = 1.

k0 



In the above, note the use of p(p − 1) = p . Then from     uk bk = |bk |p /bpp −1 = bp , λ(u) = k0

k0

we conclude that λ = bp .

(1.7.5)

Conversely, suppose that λ ∈ (p )∗ . Let bk := λ(ek ), We now let (1.7.6)

 vk :=

k  0.

0,  |bk |p −2 bk ,

bk = 0, bk = 0,

and define the sequence a(n) for each n  1 by  vk , 0  k  n, (n) ak := 0, k > n.

1. THE BASICS OF p

12

Then |λ(a(n) )|  λa(n) p , n n 1     p |bk |p  λ |bk |p , k=0 n 

k=0

 1  1−

p

|bk |p

k=0 n 



|bk |p

 1 p

 λ,  λ.

k=0 

Because this holds for every n, it must be that b ∈ p , and bp  λ. Equality holds since, by (1.7.5), λ(u) = bp and up = 1, with u as given in (1.7.4). This proves the claim when p ∈ (1, ∞). With p = 1 and thus p = ∞, let b ∈ ∞ , and define  ak bk λ(a) = k0

for a ∈  . This is permissible, since  |ak bk |  a1 b∞ . 1

k0 1 ∗

It follows that λ ∈ ( ) , with λ  b∞ . From sup{|λ(ek )| : k  0} = sup{|bk | : k  0} = b∞ , we see that λ = b∞ . Conversely, let λ ∈ (1 )∗ . If bk := λ(ek ) then |bk |  λek  = λ ∞

for all k; that is, b ∈  , with b∞  λ. Now for any a ∈ 1 ,     λ(a) = λ ak e k = ak λ(ek ) = ak bk . k0

Moreover, |λ(a)| 

 k0

k0

|ak ||bk |  b∞

k0



|ak | = a1 b∞

k0

which shows that λ  b∞ . But as we saw above, b∞  λ and so equality follows.  Applying Theorem 1.7.2 twice shows that p is reflexive when p ∈ (1, ∞). In other words, (p )∗∗ is equal to the canonical embedding of p in this case. But 1 is not reflexive, however, as is seen from the following construction. Proposition 1.7.7. The dual space (∞ )∗ contains elements not belonging to 1

 . Proof. Let c ⊆ ∞ be the set of convergent complex sequences. One can check that c is a closed linear subspace of ∞ . Indeed, suppose that the sequence a(n) in c converges to a in ∞ . We will be done if we show that a is itself a convergent sequence of numbers, i.e., a ∈ c. To that end, let > 0, and let n be an

1.7. THE DUAL OF p

13 (n)

index sufficiently large that a(n) − a∞ < /3. The sequence a(n) = {ak }k0 is (n) (n) convergent; therefore there exists K such that |aj −ak | < /3 whenever j, k  K. And now (n)

(n)

|aj − ak |  |aj − aj | + |aj

(n)

(n)

− ak | + |ak − ak |

(n)

 a(n) − a∞ + |aj < .

(n)

− ak | + a(n) − a∞

This shows that a = {a0 , a1 , a2 , . . .} is a Cauchy sequence, and thus belongs to c. Define λ : c −→ C, λ(a) = lim ak , a = {ak }k0 ∈ c. k→∞

Since taking limits respects addition and scalar multiplication, λ is a linear functional. Furthermore, since |λ(a)| = | lim ak |  sup{|ak | : k  0} = a∞ , k→∞

it is bounded (hence continuous). The Hahn-Banach Extension Theorem furnishes an extension Λ of λ to all of ∞ , with Λ = λ. Suppose, for the sake of argument, that there is a sequence b ∈ 1 such that  ak bk Λ(a) = k0 ∞

for all a ∈  . Then we have 0 = λ(ek ) = Λ(ek ) = bk ,

k  0,

which implies that Λ ≡ 0. However Λ(a) = 1, where a := {1, 1, 1, . . .} ∈ c. Therefore, no such b ∈ 1 can exist.  We saw in Section 1.4 that when p ∈ (0, 1) the quantity d(a, b) = a − bpp defines a metric on p . Thus, it makes sense to speak of the continuous linear functionals on such p . That is, we can define the dual space (p )∗ , even though p is not a Banach space. ∞



Theorem 1.7.8. For each p ∈ (0, 1), the dual space (p )∗ can be identified with via the bilinear pairing  an bn , a ∈ p , b ∈ ∞ .

a, b = n0 ∞

Proof. Let b ∈  . Observe that for each a ∈ p we have



 



an bn  |an ||bn |  b∞ |an |  b∞ ap , (1.7.9)

n0

n0

n0

where the last inequality above follows from (1.5.1). This means that the linear functional  λ(a) = an bn n0

is continuous on p . Now suppose λ ∈ (p )∗ . Let us first show that the sequence {λ(ek )}k0 is bounded. If it is not, then we can find indices {nk }k1 and unimodular constants

1. THE BASICS OF p

14

ξnk so that λ(ξnk enk ) > 2k . Consider the sequence a = {aj }j0 defined to be zero whenever j = nk and ank = ξnk 2−k . In other words,  a= ξnk 2−k enk . k0 p

Note that a certainly belongs to  . However,   λ(a) = λ ξnk 2−k enk k0

=



2−k λ(ξnk enk )

k0

>



2−k 2k

k0

= 1 + 1 + 1 + ··· , which clearly diverges. Thus {λ(ek )}k0 is a bounded sequence of complex numbers. Now define b ∈ ∞ by bn = λ(en ) and observe that for any a ∈ p we have  

 λ(a) = λ an e n = an λ(en ) = an bn . n0

n0

n0

This means that the linear functional λ can be identified with b via the above pairing.  1.8. Norming Functionals If X is a Banach space and x ∈ X \ {0}, then there exists a λ ∈ X ∗ such that λ = 1

(1.8.1)

and λ(x) = x.

To see this, let M be the one-dimensional subspace of X spanned by x, and define κ(cx) = cx for all scalars c. Then κ is a linear functional on M and  |κ(cx)|   |c|x  κ = sup : c = 0 = sup : c = 0 = 1. cx cx We now invoke the Hahn-Banach Extension Theorem to obtain an extension λ of κ to all of X , with λ = κ = 1. This λ is a norming functional for x. Let us stress that in general such a λ is not unique. We have already encountered norming functionals for members of p when p ∈ (1, ∞). Suppose that a ∈ p \ {0}. Borrowing from (1.7.4), we define b by ⎧ ⎪ if ak = 0 ⎨0 (1.8.2) bk = |ak |p−2 ak . ⎪ if ak = 0 ⎩ ap−1 p Then 

bpp =

 |ak |(p−1)p (p−1)p k0 ap

and

a, b =

=

 |ak |p k0

app

 |ak |p = ap . ap−1 p k0

=1

1.9. NOTES

15



Therefore b ∈ p is a norming functional for a. Later we will see that when p ∈ (1, ∞), the norming functional for any a ∈ p \ {0} is unique. When p = 1 or p = ∞, however, norming functionals need not be unique. For example, the vector e0 ∈ 1 is normed by both e0 and e0 + e1 in ∞ (1 )∗ . Similarly, the vector e0 + e1 ∈ ∞ is normed by both e0 and e1 ∈ 1 ⊆ (∞ )∗ . Definition 1.8.3. Anticipating further uses of the formula (1.8.2), let us introduce the signed power notation  0, a = 0, ar := (1.8.4) |a|r−1 a, a = 0, r r r ar := {a0 , a1 , a2 , . . .}

(1.8.5)

for any a ∈ C, sequence a, and r > 0. In particular, a1 = a, and hence a1 = {a0 , a1 , a2 , . . .}. Thus if a ∈ p \ {0} and p ∈ (1, ∞) \ {2}, then the norming functional of a is λa :=

(1.8.6)

ap−1 p p−1 ∈  . ap

If a ∈ 2 \ {0}, then the norming functional of a is  a  a a1 a2 0 = , , , . . . ∈ 2 . (1.8.7) λa := a2 a2 a2 a2 Below are some properties concerning the signed power notation which are straightforward to check. They will be useful when norming functionals and related devices come into play in the chapters to come. Lemma 1.8.8. Let p ∈ (1, ∞). Then for w, z ∈ C, n  0, and s > 0, we have (i) (zw)s = z s ws , (ii) |z|p = z p−1 z, (iii) (z s )n = (z n )s ,  (iv) (z p−1 )p −1 = z. 

Furthermore, ap−1 ∈ p whenever a ∈ p . 1.9. Notes For the classical inequalities covered in Sections 1.2, 1.4, and 1.5, we broadly follow the text [24]. There are alternate proofs of H¨older’s Inequality, along with sharper versions, in a paper of Wu [168]. Theorem 1.3.1 is known as the RieszFischer Theorem in many sources, for example, [16]. As we have seen in this chapter, much of what can be said about p follows from Lp (μ) for any measure μ. However, the identification of the dual of p for p ∈ (0, 1) with ∞ does not. Day showed that the dual of Lp [0, 1] is equal to zero when p ∈ (0, 1) [46]. The proof of (p )∗ ∞ when p ∈ (0, 1) relies on the fact that 1/p   |an |  |an |p , a ∈ p , p ∈ (0, 1). n0

n0

This inequality was used in (1.7.9). The corresponding inequality for Lp [0, 1], 1  1 1/p |f (x)|dx  |f (x)|p dx , f ∈ Lp [0, 1], p ∈ (0, 1), 0

0

16

1. THE BASICS OF p

fails, e.g., f (x) = 2χ[0, 14 ] + 2χ[ 12 , 34 ] . The papers of Shapiro [152, 153] and Stiles [159] demonstrate several other fascinating properties of p when p ∈ (0, 1).

10.1090/ulect/075/02

CHAPTER 2

Frames In this chapter we will discuss the concept of a frame in p . In a Hilbert space, a frame extends the idea of an orthonormal basis. As in the case of an orthonormal basis, a vector x can be recovered from a frame {xn }n1 via the quantities { x, xn }n1 . By relaxing the requirement of orthogonality, or even linear independence, we introduce some redundancy in representing vectors. In associated applications, such as signal processing, image processing, and data compression, this redundancy lends some numerical stability and resilience to noise and missing pieces of data. Though frames were initially conceived for Hilbert spaces, they can be extended in a natural way to certain Banach spaces, including p . 2.1. Frames in Hilbert Spaces Let us introduce frames in Hilbert spaces and point out a few of their main properties. In this section we will only outline the results since we will provide detailed proofs of the corresponding results for p spaces in the sections that follow. Definition 2.1.1. A sequence of vectors {xn }n1 in a Hilbert space H , with inner product (·, ·) and corresponding norm  · , is a frame if there are positive constants A and B such that  1 2 2 (2.1.2) Ax  |(x, xn )|  Bx, x ∈ H . n1

The constants A and B are called frame bounds. When A = B, the frame is called a tight frame, while when A = B = 1, the frame is called a Parseval frame. By Parseval’s Theorem for Hilbert spaces, any orthonormal basis for H is a frame (in fact a Parseval frame). It is also true that the closed linear span of a frame is all of H . Indeed, if the closed linear span of the frame {xn }n1 is not H , there must be an x ∈ H \ {0} with x ⊥ {xn }n1 . Then the basic (lower bound) property of a frame from (2.1.2) would imply x = 0, a contradiction. A frame must also be a norm bounded sequence. To see this, we have from (2.1.2) that |(x, xn )|  Bx,

x∈H,

n  1.

Taking the supremum over all x = 1 yields xn   B for all n. For a frame {xn }n1 define the analysis operator Θ : H :→ 2 (N),

Θx = {(x, xn )}n1

and observe from Definition 2.1.1 that Θ is bounded (and bounded below). The adjoint Θ∗ of Θ satisfies 

 an xn Θ∗ : 2 (N) → H , Θ∗ {an }n1 = n1 17

18

2. FRAMES

and is called the synthesis operator. Their composition  Θ∗ Θ : H → H , Θ∗ Θx = (x, xn )xn n1

is a positive invertible operator on H called the frame operator. When {xn }n1 is a Parseval frame, then Θ∗ Θ is the identity operator, which yields the important recovery formula  x= (x, xn )xn , x ∈ H . n1

As it turns out, any frame {xn }n1 is isomorphic to a Parseval frame in that there is a bounded invertible operator R on H such that {R xn }n1 is a Parseval frame. We point out this Hilbert space detail since it contrasts greatly with Parseval frames for p when p = 2 (see Corollary 2.3.8). This brings us to the related topic of a Riesz basis. Definition 2.1.3. A sequence of of vectors {xn }n1 is a Hilbert space H is called a Riesz basis if  {xn : n  1} = H and there are positive constants C1 and C2 such that for all finite sequences a1 , . . . , aN of complex numbers, we have C1

N 

|an |2

1 2

N N    1 2    an xn   C 2 |an |2 .

n=1

n=1

n=1

Theorem 2.1.4. For a sequence {xn }n1 of vectors in a Hilbert space H the following are equivalent. (i) {xn }n1 is a Riesz basis. (ii) There is an orthonormal basis {un }n1 for H and a bounded invertible operator T on H such that T un = xn for all n  1. A Riesz basis is a frame but not always the other way around. In fact, for a Riesz basis {xn }n1 , the analysis operator  an xn {an }n1 → n1

is a bounded invertible operator from  (N) onto H , while for a frame this operator is bounded and surjective but not necessarily injective. 2

2.2. Frames in p In the previous section, the vectors {xn }n1 in the definition of a frame could be viewed as elements of the dual of H , which, by the Riesz Representation Theorem for Hilbert spaces, can be identified with H itself. This observation suggests a way to extend the concept of frames to p and to certain other Banach spaces. With this in mind, we have the following definition. First recall that for p ∈ (1, ∞) the  dual of p can be identified with p , where p1 + p1 = 1, via the bilinear dual pairing  

a, b = aj bj , a ∈ p , b ∈ p . j0

2.2. FRAMES IN p

19 

Definition 2.2.1. For p ∈ (1, ∞) a sequence {bn }n0 ⊆ p is a frame for p if there are positive constants A and B such that (2.2.2)

Aap 



| a, bn |p

1

p

 Bap ,

a ∈ p .

n0

The bounds A and B are called the frame bounds. As in the Hilbert space case (Definition 2.1.1), when A = B, the frame is called a tight frame and when A = B = 1, the frame is called a Parseval frame. It is important to note that the elements bn of the frame belong to the dual  space p but satisfy the frame inequalities for vectors in p . Note how this differs from the Hilbert space case where the frame vectors lie in the space for which they are a frame. Here are some examples of frames. 

Example 2.2.3. The sequence {en }n0 ⊆ p of standard basis vectors is a frame for p . For any a = {a0 , a1 , a2 , . . .} ∈ pA we have a, en  = an and so   | a, en |p = |an |p = app . n0

n0

In fact, as one can see from the above formula, {en }n0 is a Parseval frame for p . Example 2.2.4. The sequence  −1/p  (2.2.5) 2 e0 , 2−1/p e0 , 2−1/p e1 , 2−1/p e1 , 2−1/p e2 , 2−1/p e2 , . . . is a frame since for any a ∈ p     | a, 2−1/p e0 |p + | a, 2−1/p e0 |p + | a, 2−1/p e1 |p + | a, 2−1/p e1 |p + · · · becomes 1 − p ·p

2·2

|a0 |p + 2 · 2

1 − p ·p

|a1 |p + 2 · 2

1 − p ·p

|a2 |p + · · · ,

which is equal to app . As with the previous example, (2.2.5) is a Parseval frame. This example also shows that a frame need not be a linearly independent set. Example 2.2.6. A similar calculation as in the previous two examples will show that the sequence   e0 , 2−1/p e1 , 2−1/p e1 , 3−1/p e2 , 3−1/p e2 , 3−1/p e2 , . . . is a Parseval frame for p . Remark 2.2.7. When p = 2 it seems like we might have two different definitions of a frame for 2 . If we regard 2 as a Hilbert space with the sesquilinear pairing  an bn , (a, b) = n0

then the criterion for {an }n1 ⊆ 2 to be a frame from Definition 2.1.1 would be  |(b, an )|2  C2 b22 , b ∈ 2 , (2.2.8) C1 b22  n1

20

2. FRAMES

for some positive constants C1 , C2 . If we regard 2 as a Banach space whose dual is paired with 2 via the bilinear pairing 

a, b = an bn , n0

then the criterion for a frame from Definition 2.2.1 would be  | b, an |2  D2 b22 , b ∈ 2 , (2.2.9) D1  b22  n1

for some positive constants D1 , D2 . One can see that these two criteria are the same since 2 = {b : b ∈ 2 } and b, an  = (b, an ). Thus {an }n1 is a frame in both senses. For a frame with bounds A and B, any A with 0 < A < A and B  with B > B are also frame bounds. However, we can define the optimal frame bounds by setting   1  p p A = inf | a, bn | : ap = 1 

n0

and B = sup

 

| a, bn |p

1

p

 : ap = 1 .

n0

As was already observed for Hilbert spaces, frames for p are norm bounded. 

Proposition 2.2.10. If {bn }n0 ⊆ p is a frame for p , then sup{bn p : n  0} < ∞. Proof. From (2.2.2) | a, bn |  Bap for all a ∈ p and all n  0. By the Riesz Representation Theorem (Theorem 1.7.2) we get bn p  B for all n.  

Again, similar to the Hilbert space case, a frame for p spans p . 

Proposition 2.2.11. If {bn }n0 ⊆ p is a frame for p , then   {bn : n  0} = p . Proof. If this were not true then, by the Hahn-Banach Separation Theorem, there would be an a ∈ p \ {0} for which a, bn  = 0 for all n  0. The lower bound inequality in Definition 2.2.1 would imply that a = 0, which is a contradiction.  Similar to the analysis operator Θ for the Hilbert space case, we have, for a  frame {bn }n0 ⊆ p for p , the analysis operator (2.2.12)

Θ : p → p , 

Θ(a) = { a, bn }n0 , 

and the synthesis operator Θ∗ : p → p . To compute these operators explicitly, we need a detail from functional analysis. Lemma 2.2.13. For a bounded operator T from a Banach space X to a Banach space Y , its adjoint T ∗ : Y ∗ → X ∗ is onto if and only if T has a bounded inverse on Ran T .

2.3. PARSEVAL AND TIGHT FRAMES

21

Proposition 2.2.14. The analysis operator Θ is a bounded injective operator with closed range. Moreover, Θ∗ is bounded and surjective, and satisfies 

 (2.2.15) Θ∗ {cn }n0 = c n bn . n0

Proof. The fact that Θ is bounded and injective with closed range follows from the upper and lower inequalities in Definition 2.2.1. By the definition of the adjoint, Θ∗ will satisfy  

a, Θ∗ en  = Θa, en  =

a, bk ek , en = a, bn . k0 ∗

This shows that Θ en = bn and the identity in (2.2.15) follows. The surjectivity of Θ∗ follows from Lemma 2.2.13 and the first part of the proposition.  This leads us to an equivalent criterion for being a frame. 

Theorem 2.2.16. A sequence {bn }n0 ⊆ p is a frame for p if and only if the mapping  (2.2.17) {cn }n0 → c n bn , n0

defined on finitely supported sequences {cn }n0 , extends to a bounded operator from   p onto p with the extension satisfying (2.2.17). 

Proof. Suppose that {bn }n0 ⊆ p is a frame for p . Then the analysis operator Θ is bounded on p , and the lower bound from Definition 2.2.1 says that Θ is an bounded invertible operator (that is to say, an isomorphism) onto its range Ran Θ. Lemma 2.2.13 shows that Θ∗ must be onto. From Proposition 2.2.14, the operator in (2.2.17) is equal to Θ∗ and hence onto.   Conversely, suppose that Y : p → p defined by (2.2.17) is onto. Then Y ∗ : p → p is bounded and bounded below (i.e., for some positive constant C we have Y ∗ ap  Cap for all a ∈ p ). By (2.2.17)

Y ∗ a, en  = a, Y en  = a, bn ,

a ∈ p .

Thus { a, bn }n1 ∈ p and moreover, 

| a, bn |p

1

p

= Y ∗ ap  Y ∗ ap = Y ap ,

n1

which shows that the upper bound frame condition is satisfied. Since Y is a surjective mapping, its adjoint Y ∗ , which is Θ from Proposition 2.2.14, will have bounded inverse. This will yield the lower frame bound condition.  2.3. Parseval and Tight Frames In the previous section we gave some examples of Parseval frames. Here is a description of all of them. 

Theorem 2.3.1. For p ∈ (1, ∞) \ {2} a sequence {bn }n0 ⊆ p is a Parseval frame for p if and only if (2.3.2)

bn = cn eσ(n) ,

n  0,

22

2. FRAMES

where {cn }n0 is a sequence of complex numbers for which  (2.3.3) |cn |p = 1, k  0, n∈σ −1 ({k})

and σ : N0 → N0 is a surjective function. Proof. Suppose bn = cn eσ(n) with a surjective σ : N0 → N0 such that (2.3.3) holds. Then for  a= a n e n ∈ p n0

we have



| a, bn |p =

n0



|cn a, eσ(n) |p

n0

=

 

|cn |p | a, ek |p

k0 σ(n)=k

=



n∈σ −1 ({k})

k0

=



 |cn |p |ak |p



1 · |ak |p

k0

= app and so {bn }n0 is a Parseval frame for p . Now assume that {bn }n0 is a Parseval frame for p . If bj = 0 we can define σ(j) = 1 and set the corresponding constant cj from (2.3.2) to be equal to zero. For each bj = 0 define suppt(bj ) := {n  0 : en , bj  = 0}. We will now show that when bj = 0 we have suppt(bj ) = {i} for some i  0. In other words, the support of each (nonzero) bj is a singleton. Towards a contradiction, suppose there are distinct integers m and n in suppt(bj ). From the assumption that {bj }j0 is a Parseval frame, we get p     | em , bs |p = 

em , bs es  , 1 = em pp = s0

and similarly for n. Thus (2.3.4)

Xm :=



em , bs es ,

p

s0

Xn :=

s0



en , bs es

s0

define unit vectors in p . Again, using the fact that {bj }j0 is a Parseval frame for p , we get   | em , bs  + en , bs |p = | em + en , bs |p = em + en pp = 2. Xm + Xn pp = s0

s0

A similar calculation shows that Xm − Xn pp = em − en pp = 2, and so Xm + Xn pp + Xm − Xn pp = 2(Xm pp + Xn pp ).

2.3. PARSEVAL AND TIGHT FRAMES

23

By a version of Clarkson’s Inequalities (see Theorem 3.1.16 below), and the important restriction that p = 2, it follows that suppt(Xm ) ∩ suppt(Xn ) = ∅. By the formulas from (2.3.4), this contradicts the conditions

Xm , ej  = em , bj  = 0 and

Xn , ej  = en , bj  = 0.

Thus suppt(bj ) = {i} for some i  0 and so bj = ei , bj ei . To summarize what we have done so far, we have produced a map σ : N0 → N0 defined by  1 if bj = 0 σ(j) := i if bj = 0 and a sequence {cj }j0 of complex numbers defined by  0 if bj = 0 (2.3.5) cj :=

ei , bj  if bj = 0 such that bj = cj eσ(j) . The map σ must be onto since otherwise some ek would be missing from the linear  span of {bj }j0 , contradicting the fact that {bj }j0 must a spanning set for p (Proposition 2.2.11). To finish, we need to show that the sequence {cj }j0 in (2.3.5) satisfies (2.3.3). Towards this end we have, for each k  0, 1 = ek pp  = | ek , bn |p n0

=



| ek , cn eσ(n) |p

n0

=



|cn |p | ek , eσ(n) |p

n0



=

|cn |p

n∈σ −1 ({k})



which is precisely (2.3.7).

When p = 2 any orthonormal basis for 2 is a Parseval frame and not all of these take the form (2.3.2). Thus the hypothesis that p = 2 is an indispensible hypothesis of this theorem. We leave it to the reader to adapt the above proof to show the following theorem regarding tight frames. 

Theorem 2.3.6. For p ∈ (1, ∞)\{2} a sequence {bn }n0 ⊆ p is a tight frame for p with bound A if and only if bn = cn eσ(n) ,

n  0,

where {cn }n0 is a sequence of complex numbers for which  (2.3.7) |cn |p = Ap , k  0, n∈σ −1 ({k})

and σ : N0 → N0 is a surjective function.

24

2. FRAMES

As mentioned earlier, for the Hilbert space case every frame is isomorphic to a Parseval frame. For p frames the situation is much different. Corollary 2.3.8. For each p ∈ (1, ∞) \ {2} there is a frame {bn }n0 for p  for which there is no bounded invertible operator R on p such that {Rbn }n0 is a Parseval frame for p . Proof. Consider the sequence {e0 , e0 + e1 , e1 , e1 + e2 , e2 , e2 + e3 , e3 , e3 + e4 , . . .}. This forms a frame for p . To be sure, for each a ∈ p and each j  0 the quantity | a, ej |p + | a, ej + ej+1 |p satisfies | a, ej |p  | a, ej |p + | a, ej + ej+1 |p  2p−1 (| a, ej |p + | a, ej+1 |p ). Now sum over j  0 to get 

| a, ej |p + | a, ej + ej+1 |p  2p app , app  j0 

which shows this sequence is a frame. Suppose R is an invertible operator on p such that {Re0 , Re0 + Re1 , Re1 , Re1 + Re2 , Re2 , Re2 + Re3 , Re3 , Re3 + Re4 , . . .} is a Parseval frame for p . Then Theorem 2.3.1 says that Rej = cj eσ(j) ,

Rej+1 = cj+1 eσ(j+1) ,

R(ej + ej+1 ) = dj ekj .

In particular, cj = 0 for all j  0. Since Rej + Rej+1 = R(ej + ej+1 ) this yields the identity cj eσ(j) + cj+1 eσ(j+1) = dj ekj 

which is an obvious contradiction. 2.4. Riesz Bases 

Let us next develop a notion of Riesz basis for p (the dual space of p ). We will do this in the dual space setting since, as we will see in a moment, it will make connections to our earlier treatment of frames. 



Definition 2.4.1. A sequence {bn }n0 ⊆ p is a Riesz basis for p if   {bn : n  0} = p and there are positive constants C1 and C2 such that for every finite sequence c0 , c1 , . . . , cN of complex numbers, we have C1

N  n=0



|cn |p

 1 p

N      c n bn  n=0

p

 C2

N 



|cn |p

 1 p

.

n=0

The numbers C1 and C2 are called the Riesz basis bounds for {bn }n0 .

2.4. RIESZ BASES

25





If {bn }n0 ⊆ p is a Riesz basis then every b ∈ p can be written as  c n bn , b= n0

where the convergence is unconditional (every rearrangement of the series also converges – and to the same value). As with frames, there are optimal Riesz basis constants C1 and C2 obtained by setting      cn bn p : c = {cn }n0 ∈ p , cp = 1 C1 = inf  n0

and

     C2 = sup  cn bn p : c = {cn }n0 ∈ p , cp = 1 . n0

There is a relationship between frames and Riesz bases. 

Proposition 2.4.2. If {bn }n1 ⊆ p is a Riesz basis, then it is a frame for p  and the optimal Riesz basis and frame bounds coincide. Proof. The criterion for a Riesz basis shows that the mapping  {cn }n0 → c n bn n0 



defines a bounded surjective operator from p onto p . By Theorem 2.2.16, {bn }n1 is a frame. By Theorem 2.2.16 the optimal bounds coincide.  Every Riesz basis is a frame. However, the converse holds only under certain conditions. 

Proposition 2.4.3. Let {bn }n0 ⊆ p be a frame for p . Then the following are equivalent. (i) (ii) (iii) (iv) (v)



{bn }n0 ⊆ p is a Riesz basis;   If {cn }n0 ∈ p and n0 cn bn = 0, then cn = 0 for all n  0; Θp = p ; There exists a sequence {an }n0 ⊆ p with an , bk  = δn,k ; {bn }n0 is minimal in that for each n  0,  bn ∈ {bk : k = n}.

Proof. We leave it to the reader to use the Hahn-Banach separation theorem to show that the equivalence of (iv) and (v). The claim (i) =⇒ (ii) follows from the norm inequalities in the definition of a Riesz basis (Definition 2.4.1). The claim (ii) =⇒ (i) uses the synthesis operator 

 Θ∗ {cn }n0 = c n bn n0 



which is bounded from p onto p , that is,     c n bn   n0

∗

p



 Θ 

n0

|cn |

p

 1 p

.

26

2. FRAMES

The assumption (ii) shows that Θ∗ is also injective and thus has a bounded inverse   cn bn = {cn }n0 , (Θ∗ )−1 n0

in other words, 



|cn |p

     (Θ∗ )−1  c n bn   .

 1 p

n0

p

n0

The above two inequalities show that {bn }n0 is a Riesz basis. To see that (i) =⇒ (iii) use the fact that {bn }n0 is a Riesz basis to conclude  that Θ∗ has a bounded inverse on Θ∗ p . By Lemma 2.2.13, Θ∗∗ = Θ : p → p is surjective. The proof of (iii) =⇒ (i) uses the assumption that Θp = p (and thus   Θ is bijective), along with another use of Lemma 2.2.13 to get that Θ∗ : p → p is bijective. This verifies that {bn }n0 is a Riesz basis. This leaves us to prove that (iii) ⇐⇒ (iv). Assuming (iii), choose an ∈ p such that Θan = en . From here, the definition of Θ from (2.2.12) yields an , bk  = δn,k . Now assuming (iv), the existence of the sequence {an }n0 will show that en ∈ Θp for all n. But since {en : n  0} has dense linear span in p and Θ has closed range (Proposition 2.2.14), (iii) follows.  The sequence {an }n0 satisfying the condition in (iv) is called the biorthogonal sequence to {bn }n0 . The following assertion identifies the relationship between a Riesz basis {bn }n0 and its dual basis from the previous proposition. 

Theorem 2.4.4. Suppose {bn }n1 ⊆ p is a Riesz basis with Riesz basis bounds C1 and C2 . Then there is a unique Riesz basis {an }n0 ⊆ p for which 

a, bn an , a ∈ p , (2.4.5) a= n0

and (2.4.6)

b=



an , bbn ,



b ∈ p .

n0

Furthermore, {an }n0 has Riesz basis bounds 1/C1 and 1/C2 . Proof. By Proposition 2.4.3(iii), the analysis operator Θ : p → p ,

Θa = { a, bn }n0 ,

is invertible. Thus we can choose an ∈ p with an = Θ−1 (en ). Consequently, for any a ∈ p we have    a = Θ−1 Θa = Θ−1

a, bn en =

a, bn Θ−1 en =

a, bn an , n0

n0

n0

which verifies the identity in (2.4.5). For each {cn }n0 ∈ p use the fact that Θ is surjective to obtain an a ∈ p for which cn = a, bn  for all n  0. Then by

2.5. AN ANALOGUE OF THE FEICHTINGER CONJECTURE FOR p

27

Proposition 2.4.2      c n an  p = 

a, bn an p = ap n0

n0

  1  1  p p  | a, bn |p = |cn |p . C1 C1 1

n0

1

n0

This shows that {an }n0 satisfies the upper bound for the Riesz basis condition  with bound 1/C1 . Furthermore, for each b ∈ p we see that bp = sup{| a, b| : a ∈ p , ap = 1}   

= sup

an , b a, bn  : a ∈ p , ap = 1 n0

 C2





| an , b|p

 1 p

.

n0

Thus {an }n0 satisfies the lower bound condition for a frame with bound 1/C2 .  Since a Riesz basis is also a frame, it follows that {an }n0 is a frame for p . From the construction of an we see that an , bk  = δn,k and so by Proposition 2.4.3, {an }n0 is a Riesz basis for p and is the dual basis for {bn }n0 . Since   {an }n0 is a frame for p we can use Theorem 2.2.16 to produce, for each b ∈ p ,  a sequence {dn }n0 ∈ p such that  dn bn . b= n0

Now use the biorthogonality condition an , bk  = δn,k to see that dn = a, bn  and so the formula in (2.4.6) is verified. The Riesz basis bounds follow from the above discussion along with Proposition 2.4.2.  2.5. An Analogue of the Feichtinger Conjecture for p 



We say that {bn }n0 ⊆ p is a Riesz basic sequence for p if there are positive constants C1 and C2 such that for every finite sequence c0 , c1 , . . . , cN of complex numbers, we have N N N     1   1   p p   C1 |cn |p  c n bn    C 2 |cn |p . n=0

n=0

p

n=0

Notice how the definition of a Riesz basicsequence is similar to that of a Riesz  basis, but here we are not assuming that {bn }n0 = p . A theorem in Hilbert spaces says that if {xn }n0 is a frame that also satisfies inf{xn  : n  1} > 0, then {xn }n1 can be written as a finite union of Riesz basic sequences (analogously defined as in p ). At one time this fact was known as the Feichtinger conjecture. Here is a version of this theorem for p . 

Theorem 2.5.1. Suppose p ∈ (1, ∞) \ {2} and {bn }n0 ⊆ p is a tight frame for p such that inf{bn p : n  0} > 0. Then {bn }n0 is a finite union of Riesz basic sequences.

28

2. FRAMES

Proof. By Theorem 2.3.6 bn = cn eσ(n) for all n  0, where {cn }n0 is a sequence of complex numbers for which  |cn |p = Ap , k  0, n∈σ −1 ({k})

and σ : N0 → N0 is a surjective function. If C = inf{bn p : n  0} > 0 then C  bn p = cn eσ(n) p = |cn |. From here we see that Ap =



|cn |p

n∈σ −1 ({k})





min{|cn |p : n ∈ σ −1 ({k})}

n∈σ −1 ({k})

 card(σ −1 ({k}))C p . We conclude that for each k  0 we have card(σ −1 ({k})) 

Ap . Cp

Thus we can decompose our frame {bn }n0 as    cj e2 : j ∈ σ −1 ({2}) ··· {bn }n0 = cj e1 : j ∈ σ −1 ({1}) and each one of the sets in the union above has at most Ap /C p elements. Take one vector from each of those sets in the union to form a Riesz basic sequence. Then take another element from (the remaining elements of) each set in the above union to obtain another Riesz basic sequence. Continue this process until all of the sets in the above union are exhausted (which will occur after finitely many steps) to decompose the frame {bn }n0 into finite many Riesz basic sequences.  The above proof, note the use of Theorem 2.3.6 for which the hypothesis that p = 2 is indispensable. Also make note of the needed hypothesis of a tight frame (see Corollary 2.3.8). 2.6. Notes An important initial paper on frames in Hilbert spaces is one of Duffin and Schaeffer [51], while some good texts are [38, 77]. Much of the discussion and proofs of frames and Riesz bases for p spaces in this chapter follows the papers of Christensen and Stoeva [39] and Liu, Liu, and Zheng [117]. The Feichtinger conjecture for Hilbert spaces was solved by Marcus, Spielman, and Srivastava [120]. Frames will appear again at the end of Chapter 9 when we discuss sampling.

10.1090/ulect/075/03

CHAPTER 3

The Geometry of p In this chapter we will examine certain geometric properties of p , including convexity and smoothness. In addition, we cover a notion of orthogonality on a general normed linear space, called Birkhoff-James orthogonality. This concept of orthogonality is connected to norming functionals, extremal problems, and metric projection in a direct manner. Consequently, it will play an important role in the later chapters when we discuss the function theory for a space of analytic functions related to p . 3.1. Convexity Definition 3.1.1. A subset S of a Banach space X is said to be convex if tx + (1 − t)y ∈ S whenever x, y ∈ S and t ∈ [0, 1]. That is, whenever S contains two points, then it also contains the line segment joining them. For example, the closed unit ball X1 := {x ∈ X : x  1} of X is convex. To see this, let x and y belong to X1 and let t ∈ [0, 1]. Then tx + (1 − t)y  tx + (1 − t)y  1. In some instances, the closed unit ball X1 is more than just merely convex. Definition 3.1.2. A Banach space X is said to be strictly convex if the conditions x = y, x = 1 and y = 1 in X imply that x + y < 2. Geometrically speaking, strict convexity means that whenever we connect two distinct points on the unit sphere ∂X1 = {x ∈ X : x = 1} with a line, the midpoint lies strictly inside the unit ball. That is, the unit sphere does not have any “flatness.” The space 1 is not strictly convex, since the vector {t, 1 − t, 0, 0, 0, . . .} lies on the unit sphere for all t ∈ [0, 1]. Similarly, ∞ is not strictly convex since {1, t, 0, 0, 0, . . .} lies on the unit sphere for all t ∈ [−1, 1]. When p ∈ (1, ∞), the space p is strictly convex, but in fact even more can be said. Definition 3.1.3. A Banach space X is said to be uniformly convex if for any ∈ (0, 2] there exists a δ > 0 such that the conditions x  1, y  1, and x − y  imply that  21 (x + y)  1 − δ. Geometrically speaking, uniform convexity means that at every point of the unit sphere there is at least a minimum amount of “roundness.” With strict convexity, there is roundness, but not a uniformly minimum amount. A uniformly convex space is strictly convex, as is clear from the definitions. However, there are strictly 29

3. THE GEOMETRY OF p

30

convex spaces that are not uniformly convex. For example, the space of summable sequences endowed with the norm a := a1 + a2 is strictly, but not uniformly, convex. See [47] for more on this subject. Here is another useful fact about uniform convexity. Proposition 3.1.4. Let {xn }n1 and {yn }n1 be two sequences of vectors in a uniformly convex Banach space X . Then the conditions xn   1, yn   1, and lim  12 (xn + yn ) = 1 imply that n→∞

lim xn − yn  = 0

n→∞

and

lim xn  = lim yn  = 1.

n→∞

n→∞

Proof. Let ∈ (0, 2]. By uniform convexity there exists a δ > 0 such that the conditions x  1, y  1 and x − y  imply that  12 (x + y)  1 − δ. Furthermore, by our assumptions, there exists a positive integer N such that  12 (xn + yn ) > 1 − δ whenever n  N . This means that for such an n we have xn − yn  < and thus lim xn − yn  = 0. Since n→∞

xn = 12 (xn + yn ) + 12 (xn − yn ) then xn    12 (xn + yn ) −  12 (xn − yn ) and so lim inf xn   lim inf  12 (xn + yn ) − lim inf  12 (xn − yn ) = 1 − 0 = 1. n→∞

n→∞

n→∞

But since xn   1, it follows that lim xn  = 1. A similar proof shows that n→∞

lim yn  = 1.

n→∞



In this section we will show, via a theorem of Clarkson (Theorem 3.1.9), that Lp is uniformly convex whenever p ∈ (1, ∞). The following lemmas will help lay the groundwork for this. Lemma 3.1.5. Let a and b be any complex numbers. (i) If p ∈ (1, 2] then |a + b|p + |a − b|p  2(|a|p + |b|p ). (ii) If p ∈ [2, ∞) then |a + b|p + |a − b|p  2(|a|p + |b|p ). In either case, when p = 2, equality holds only when either a or b is zero. Proof. The p = 2 case is the well-known and easily verified identity (3.1.6)

|a + b|2 + |a − b|2 = 2(|a|2 + |b|2 ).

Suppose that p ∈ (1, 2). Then |a + b|p + |a − b|p = 1 · (|a + b|p ) + 1 · (|a − b|p )  (12/(2−p) + 12/(2−p) )1−p/2 [(|a + b|p )2/p + (|a − b|p )2/p ]p/2 = 21−p/2 (|a + b|2 + |a − b|2 )p/2 by H¨older’s Inequality, using the conjugate exponents 2/p and (2 − p)/2. Now from the identity in (3.1.6) we get |a + b|p + |a − b|p  21−p/2 · 2p/2 · (|a|2 + |b|2 )p/2 = 2(|a|2 + |b|2 )p/2 .

3.1. CONVEXITY

31

Finally, apply Proposition 1.5.2, taking into account that p ∈ (1, 2), to obtain |a + b|p + |a − b|p  2(|a|p + |b|p ).

(3.1.7)

Equality holds in the above estimates precisely when ab = 0. The case p ∈ (2, ∞) is similarly handled, with H¨ older’s Inequality applied using the conjugate exponents p/2 and p/(p − 2): 2(|a|p + |b|p )2/p  2(|a|2 + |b|2 ) = |a + b|2 + |a − b|2 = |a + b|2 · 1 + |a − b|2 · 1  ([|a + b|2 ]p/2 + [|a − b|2 ]p/2 )2/p (1p/(p−2) + 1p/(p−2) )(p−2)/p = 2 · 2−2/p (|a + b|p + |a − b|p )2/p . Now tidy up the constants and take p/2 powers. Again, equality holds precisely when ab = 0.  Lemma 3.1.8. Let a and b be complex numbers. (i) If p ∈ (1, 2], then |a + b|p + |a − b|p  2p−1 (|a|p + |b|p ). (ii) If p ∈ [2, ∞), then |a + b|p + |a − b|p  2p−1 (|a|p + |b|p ) If p = 2, then equality holds in the above if and only if a = ±b. Proof. We have already done the hard work. In the p ∈ (1, 2) case, Lemma 3.1.5 says that |u + v|p + |u − v|p  2(|u|p + |v|p ) for any complex numbers u and v. Substitute u = a + b and v = a − b to obtain |2a|p + |2b|p  2(|a + b|p + |a − b|p ) 2p−1 (|a|p + |b|p )  |a + b|p + |a − b|p . The p ∈ (2, ∞) case is handled the same way. The condition for equality is uv = 0, which is equivalent to a = ±b. Once again, when p = 2 the claim is elementary.  By replacing the constants a and b with functions and then integrating, we have the following fundamental inequalities. Theorem 3.1.9 (Clarkson). Let p ∈ (1, ∞) and f, g ∈ Lp . 

(i) If p ∈ (1, 2], then f + gpp + f − gpp  2p−1 f pp + gpp . 

(ii) If p ∈ [2, ∞), then f + gpp + f − gpp  2p−1 f pp + gpp . If p = 2, then equality holds if and only if f (x) = ±g(x) μ-almost everywhere. Theorem 3.1.10. Lp is uniformly convex when p ∈ [2, ∞). Proof. Let ∈ (0, 2] and suppose that f p  1,

gp  1,

f − gp  .

Then by Theorem 3.1.9 we have the two inequalities f + gpp + p  2p−1 · 2, Uniform convexity now follows.

1

f + gp  (2p − p ) p . 

To prove that L is uniformly convex when p ∈ (1, 2] we need a few additional technical lemmas. p

3. THE GEOMETRY OF p

32

Lemma 3.1.11. If p ∈ (1, 2] and x ∈ [0, 1], then 





(1 + x)p + (1 − x)p  2(1 + xp )p −1 .

(3.1.12)

When p ∈ [2, ∞) the inequality reverses. In either case, when p = 2, equality holds precisely when x = 0. Proof. Let (3.1.13)









f (t, x) := (1 + t1−p x)(1 + tx)p −1 + (1 − t1−p x)(1 − tx)p −1 ,

where 0  t  1 and 0  x  1. Notice that f (1, x) is the left hand side of (3.1.12), while f (xp−1 , x) is the right hand side of (3.1.12). Here we rely on the identity (p − 1)(p − 1) = 1. Suppose that p ∈ (1, 2]. Then our goal is achieved if we can show that ∂f /∂t  0. But this follows from      ∂f (3.1.14) = (p − 1)x(1 − t−p ) (1 + tx)p −2 − (1 − tx)p −2  0, ∂t  as the factor (1 − t−p ) is nonpositive. When p ∈ [2, ∞), the factor     (1 + tx)p −2 − (1 − tx)p −2 is nonpositive as well, showing that the derivative is nonnegative, and again we are done. Equality holds in (3.1.13) when the derivative in (3.1.14) vanishes identically, and when p = 2 that happens precisely when x = 0.  From Lemma 3.1.11 we immediately obtain

p −1    (3.1.15) |a + b|p + |a − b|p  2 |a|p + |b|p when p ∈ (1, 2] and a, b ∈ C, and the reverse when p ∈ [2, ∞). Equality occurs when a = 0 or b = 0. Clarkson’s name is also attached to the following inequalities. Theorem 3.1.16 (Clarkson). Let p ∈ (1, ∞) and f, g ∈ Lp .

p −1    (i) If p ∈ (1, 2] then f + gpp + f − gpp  2 f pp + gpp .

  p p p p p −1 (ii) If p ∈ [2, ∞), then f + gp + f − gp  2 f p + gp . If p = 2, equality holds if and only if f and g are disjointly supported. Proof. First consider the case when p ∈ (1, 2]. We will use the fact that 0 < p − 1 < 1 and thus the Triangle Inequality flips for Lp−1 (Proposition 1.4.8). First,   p   1    p p−1 p p f p = |f | dμ = |f |p (p−1) dμ = |f |p p−1 . X

X

And now by the Reverse Triangle Inequality and (3.1.15),       f + gpp + f − gpp = |f + g|p p−1 + |f − g|p p−1     |f + g|p + |f − g|p p−1  

p −1     2 |f |p + |g|p  p−1

p −1  = 2 f pp + gpp .

3.2. METRIC PROJECTION

33

When p ∈ [2, ∞), the usual Triangle Inequality applies to Lp−1 and the chain of inequalities above goes through in the reverse sense.  Theorem 3.1.17. Lp is uniformly convex when p ∈ (1, 2]. Proof. Suppose that f p  1, gp  1, and f − gp  for some ∈ (0, 2]. Theorem 3.1.16 yields the inequalities 





f + gpp + p  2(1 + 1)p −1





1

and f + gp  (2p − p ) p

which imply uniform convexity.



3.2. Metric Projection Suppose that M is a subspace of a Banach space X and x ∈ X . The distance from x to M is defined to be (3.2.1)

inf{x − y : y ∈ M }.

When x ∈ M , then of course this distance is zero, as we can choose y = x in the above. Otherwise, the infimum is not necessarily attained, or attained uniquely, by some y0 ∈ M . If X is uniformly convex, however, then existence and uniqueness of the minimizing y0 are assured. Theorem 3.2.2. If a Banach space X is uniformly convex and M is a subspace of X , then for any x ∈ / M the infimum in (3.2.1) is uniquely attained by some y0 ∈ M . Proof. By the uniform convexity hypothesis there is a positive function δ( ) on (0, 2] such that the conditions u  1, v  1 and u − v  imply that 1 2 u + v  1 − δ( ). We may insist that δ be a monotone increasing function of . Let d := inf{x − y : y ∈ M }, and for all integers n  1 define Cn := {y ∈ M : x − y  d + n1 }. We see that C1 ⊇ C2 ⊇ C3 ⊇ · · · . If w = z ∈ Cn , then x − w  d + n1 and x − z  d + n1 . Therefore, by uniform convexity,    w − z   w + z   1 d + x −  1−δ n . 2 d + n1 But 12 (w + z) belongs to M and so x − 12 (w + z)  d. Putting these inequalities together, we see that   w − z   (3.2.3) d 1−δ d + n1 , n  1. 1 d+ n If diam Cn := sup{w − z : w, z ∈ Cn }, then (3.2.3) forces diam Cn → 0 as n → ∞. For each n, choose a yn from Cn . Then {yn }n1 is a Cauchy sequence. Since X is complete, the sequence converges to some y0 . But each Cn is closed, so it contains the limit point y0 , and consequently y0 ∈ n1 Cn . Evidently, x − y0  = d and

3. THE GEOMETRY OF p

34

y0 ∈ M . Uniqueness follows from the fact that diam Cn → 0, which forces the set  n1 Cn to be a singleton. Definition 3.2.4. If M is a subspace of a uniformly convex Banach space X , then we may define the metric projection mapping PM : X → M by PM (x) = y0 , where y0 is that unique element of M for which x − y0  = inf{x − y : y ∈ M }. In particular, metric projection mappings exist for the uniformly convex spaces p , p ∈ (1, ∞) (Theorems 3.1.10 and 3.1.17). Indeed, when p = 2, the metric projection mapping PM coincides with the orthogonal projection operator onto M . It is important to stress, however, that PM is generally not a linear mapping. Nevertheless, we shall soon see in Proposition 3.3.5 that the metric projection is continuous. Proposition 3.2.5. Let M be a subspace of a uniformly convex Banach space X . If x ∈ X , then PM x  2x. Proof. By the definition of PM , x − PM x  x − y,

y ∈ M.

In particular, with y = 0, x − PM x  x. Hence PM x  PM x − x + x  2x.



In section 4.6 we will improve this bound under certain circumstances. 3.3. Birkhoff-James Orthogonality We now define a notion of orthogonality in Banach spaces where there is no prima facie criterion for orthogonality arising from an inner product. Definition 3.3.1. For vectors x and y belonging to a Banach space X let us write x ⊥X y whenever (3.3.2)

x + cy  x

for any scalar c. We will say that x is orthogonal to y in the Birkhoff-James sense. In the special case X = p , we will use the notation ⊥p in place of the more cumbersome ⊥p . When X is a Hilbert space, such as 2 , this notion of orthogonality coincides with the usual one. However, in general, the relation ⊥X is neither symmetric nor linear. If S is a subset of X and x ⊥X y for every y ∈ S , then we write x ⊥X S . Notice that if M is any subspace of X , then according to the definition of metric projection (Definition 3.2.4), PM x must satisfy (3.3.3)

x − P M x ⊥X M ,

x∈X.

3.3. BIRKHOFF-JAMES ORTHOGONALITY

35

Lemma 3.3.4. Suppose that M is a subspace of a uniformly convex Banach space X and x ⊥X M . If ym ∈ M for all m  1, and x + ym  → x, then ym  → 0. Proof. The assertion is trivial when x = 0. Otherwise, put x x + ym , Ym = Xm = x + ym  x + ym  (we can neglect the at most finitely many indices incurring division by zero) and check that Xm   1 (by orthogonality — see (3.3.2)) and Ym  = 1 and Xm  → 1. Furthermore,  x + 12 ym   x   Xm  =  =  12 (Xm + Ym )  1. x + ym  x + ym  Thus  12 (Xm + Ym ) → 1. Therefore by Proposition 3.1.4, lim Xm − Ym  = 0.

m→∞

But Ym − Xm =

ym x − ym 

and so ym  = Ym − Xm  · x + ym  → 0 · x = 0.



We are now able to show that for a uniformly convex space, the metric projection onto a subspace is continuous. Theorem 3.3.5. Let M be a subspace of a uniformly convex Banach space X , and let x and {xn }n1 be vectors in X . If lim xn − x = 0, then n→∞

lim PM xn − PM x = 0.

n→∞

Proof. By repeated use of the property (3.3.3) and the Triangle Inequality, we get x − PM x  x − PM xm   x − xm  + xm − PM xm   x − xm  + xm − PM x  x − xm  + xm − x + x − PM x for all m. Letting m → ∞, we see that lim x − PM xm  = x − PM x.

m→∞

We may now apply Lemma 3.3.4, using the fact that x − PM x ⊥X M , along with PM x − PM xm ∈ M for all m. The conclusion is that lim PM x − PM xm  = 0,

m→∞

as was to be shown.



Since p is uniformly convex when p ∈ (1, ∞) (Theorems 3.1.10 and 3.1.17), we conclude the following. Corollary 3.3.6. If p ∈ (1, ∞) and M is a subspace of p , then the metric projection mapping PM is continuous.

36

3. THE GEOMETRY OF p

3.4. Smoothness Let x be a nonzero vector in a Banach space X . In Section 1.8 we encountered the norming functional for x, that is, a unit norm λx ∈ X ∗ such that λx (x) = x. We saw that the existence of such a linear functional is guaranteed, but it is generally not unique. Definition 3.4.1. If a Banach space X has the property that every nonzero vector has a unique norming functional, then X is said to be smooth. Remark 3.4.2. The traditional definition of smoothness of a Banach space involves the differentiability of the norm: a space X is smooth if the Gˆ ateaux derivative x + ξy − x lim ξ→0 ξ exists for all nonzero vectors x and y in X (here ξ is a complex variable). That this criterion for smoothness is equivalent to the one in Definition 3.4.1 is verified in [17, p. 177–182]. We will have occasion to exploit the uniqueness of norming functionals for some spaces. We will also encounter certain directional derivatives of the norm in Theorem 3.4.6 and Lemma 4.6.5. The following theorem ensures the existence of smooth spaces and shows that smoothness and strict convexity are nearly dual concepts. Theorem 3.4.3. Let X be a Banach space. If X ∗ is strictly convex, then X is smooth. If X ∗ is smooth, then X is strictly convex. Proof. Suppose that X is not smooth. Then some unit vector x ∈ X has distinct norming functionals λ1 and λ2 ∈ X ∗ . Then λ1 + λ2   λ1  + λ2  = 2 by the Triangle Inequality. On the other hand, (λ1 + λ2 )(x) = λ1 (x) + λ2 (x) = x + x = 2. This shows that λ1 + λ2  = 2 and so X ∗ cannot be strictly convex (Definition 3.1.2). Next suppose that that X is not strictly convex. Then for some distinct unit vectors x and y in X we have x+y = x+y = 2. Let x+y be normed by the functional λ ∈ X ∗ . Then λ(x + y) = λ(x) + λ(y) = 2. But |λ(x)|  λx = 1 and |λ(y)|  λy = 1, which forces λ(x) = 1 = λ and λ(y) = 1 = λ. That is, λ, viewed as a vector in the Banach space X ∗ , is normed by the distinct functionals in X ∗∗ given by κ → κ(x) and κ → κ(y) for all κ ∈ X ∗ . These are the canonical embeddings of x and y into X ∗∗ . Thus X ∗ fails to be smooth. This completes the proof of both claims by contraposition.  Corollary 3.4.4. The space p is smooth when p ∈ (1, ∞). 

Proof. By Theorem 1.7.2, p can be viewed as (p )∗ . Theorems 3.1.10 and  3.1.17 ensure that p is uniformly convex, and hence it is strictly convex. The p smoothness of  then follows from Theorem 3.4.3.  This confirms that if a is a nonzero sequence in p for some p ∈ (1, ∞), then the norming functional of a is uniquely given by the sequence λa =

ap−1 p p−1 ∈  , ap

3.4. SMOOTHNESS

37

as previously defined in (1.8.6). There is an important relationship between norming functionals and BirkhoffJames orthogonality in p . In what follows, (z) and (z) stand for the real and imaginary parts, respectively, of a complex number z. Lemma 3.4.5. If a and b are complex numbers with b = 0, and p ∈ (1, ∞), then the function ψ(t) := |a + bt|p , t ∈ R, is (real) differentiable for all t, and 

ψ  (t) = p|a + bt|p−2  [a + bt]b , where |0|p−2 0 is interpreted as zero. More specifically, whenever a/b ∈ R, we have ψ  (−a/b) = 0. Proof. First, observe that



p|a + bt|p−2  [a + bt]b  p|a + bt|p−1 |b| for all t; since p > 1, this is well behaved for all t, and in particular it converges to zero as t → −a/b, that is, as a + bt approaches zero. Next, write a = a1 + ia2 and b = b1 + ib2 for real numbers a1 , a2 , b1 and b2 . Then p  ψ(t) = |a + bt|p = (a1 + b1 t)2 + (a2 + b2 t)2 2 ,  p −1 

p ψ  (t) = (a1 + b1 t)2 + (a2 + b2 t)2 2 2[a1 + b1 t]b1 + 2[a2 + b2 t]b2 2 = p|a + bt|p−2 (a1 b1 + a2 b2 + t[b21 + b22 ]) 

= p|a + bt|p−2  [a + bt]b .  Theorem 3.4.6. Let p ∈ (1, ∞). Suppose that a and b ∈ p , with a = 0. Then a + tbpp − app = p  b, ap−1 , t→0 t where t is a real variable. lim

Proof. Define the real valued function f (t) := a + tbpp , for t ∈ R. Put Ψn (t) := |an + bn t|p . By Lemma 3.4.5,

 Ψn (t) := p|an + bn t|p−2  [an + bn t]bn for all n  0. Note that for |t|  1,



p−1 |Ψn (t)|  p|bn | |an | + |bn |

for all n. Then the bounding expression on right side is summable over n, since a, b ∈ p , and then H¨older’s Inequality yields 1     

p−1  

p (p−1)  p1 p |an | + |bn | |bn | |an | + |bn |  |bn |p n0

n0

2

p−1

n0 

bp (ap + bp )p/p .

We have thus checked that sufficient conditions have been met for “differentiation under the integral” (see, for example, [103, Section 6.3]), where integration here is summation with respect to n. That is, we have found that f is differentiable, and  d  d f  (t) = a + tbpp = |an + bn t|p = Ψn (t). dt dt n0

n0

3. THE GEOMETRY OF p

38

Substituting t = 0, we obtain  a + tbpp − app = f  (0) = Ψn (0) t→0 t n0  p|an |p−2 (an bn ). = lim



n0

The next theorem provides a practical test for Birkhoff-James orthogonality in  p . Recall from Lemma 1.8.8 that ap−1 ∈ p whenever a ∈ p . Theorem 3.4.7. Let p ∈ (1, ∞) and suppose that a, b ∈ p . Then a ⊥p b ⇐⇒ b, ap−1  = 0. Proof. If a = 0 then both conditions hold trivially and there is nothing more to prove. Otherwise, let λa be the unique norming functional for a. We already know that λa can be identified with ap−1 /ap−1 . Consequently, the condition p

b, ap−1  = 0 says that λa (b) = 0. It follows that for any scalar c, a + cbp = λa a + cbp  |λa (a + cb)| = |λa (a) + c · 0| = |λa (a)| = ap , that is to say, a ⊥p b. Conversely, assume that a ⊥p b. Then the real valued function f (t) := a + tbpp ,

t ∈ R,

attains a minimum value when t = 0. Theorem 3.4.6 ensures that f is differentiable, and provides the value of f  (t). By the First Derivative Test, it must be that f  (0) = 0, or  p|an |p−2 (an bn ) = 0. n0

Repeating the above argument with f (it) in place of f (t) yields  p|an |p−2 (an bn ) = 0. n0

This shows that b, ap−1  = 0.



We have previously remarked that Birkhoff-James orthogonality is generally neither symmetric nor linear. However, the smoothness of p , when p ∈ (1, ∞), gives us the following. Corollary 3.4.8. If p ∈ (1, ∞), then the relation ⊥p on p is linear in its second argument.

3.4. SMOOTHNESS

39

Proof. If a ⊥p b1 and a ⊥p b2 , then for any scalars c1 and c2 ,

c1 b1 + c2 b2 , ap−1  = c1 b1 , ap−1  + c2 b2 , ap−1  = c1 b1 , ap−1  + c2 b2 , ap−1  = c1 · 0 + c2 · 0 = 0. In conclusion, a ⊥p (c1 b1 + c2 b2 ). This establishes linearity.



For the next proposition we will need the following Generalized Dominated Convergence Theorem. Lemma 3.4.9. Suppose that g and {gn }n1 are nonnegative functions belonging to L1 (μ) such that gn → g μ-almost everywhere and gn dμ → g dμ. Suppose further that f and {fn }n1 belong to L1 (μ) with |fn |  |gn |

fn → f

and

Then

μ-almost everywhere.

fn dμ →

f dμ.

Proposition 3.4.10. If p ∈ (1, ∞), then the relation ⊥p on p is continuous in both its arguments. Proof. Suppose that an ⊥p b for all n, and an − ap → 0. Let us write an = {an,0 , an,1 , an,2 , . . .}, and so an,k → ak for each k. Furthermore, for each n and k we have p−1

|ak

p−1









− an,k |p  2p (|ak |p (p−1) + |an,k |p (p−1) ) 

= 2p (|ak |p + |an,k |p ). Since for each k,





lim 2p (|ak |p + |an,k |p ) = 2p +1 |ak |p

n→∞

and lim

n→∞







2p (|ak |p + |an,k |p ) = 2p +1

k0



|ak |p < ∞,

k0

then by the Generalized Dominated Convergence Theorem (Lemma 3.4.9),  p−1 p−1  |ak − an,k |p = 0. lim n→∞

k0

p−1

This says that ap−1 − an With that, we have

p → 0.

| b, ap−1 | = | b, ap−1  + b, ap−1 − ap−1 | n n = |0 + b, ap−1 − bp−1 | n  bp ap−1 − ap−1 p → 0. n This confirms that ⊥p is continuous in its first argument.

40

3. THE GEOMETRY OF p

For continuity in the second argument, assume that a ⊥p bn for all n, and bn − bp → 0. Then | b, ap−1 | = | b, ap−1  − 0| = | b, ap−1  − bn , ap−1 |  b − bn p ap−1 p . Now let n → ∞ to conclude that b, ap−1  = 0, or a ⊥p b.



In light of Corollary 3.4.8 and Proposition 3.4.10, it makes particular sense to speak of the orthogonality of a vector to a subspace of p . We have seen that metric projection, Birkhoff-James orthogonality, and norming functionals are intimately related. We will make use of these concepts in later chapters. 3.5. Notes The uniform convexity of the Lp spaces when p ∈ (1, ∞) was obtained by Clarkson [42], who derived the inequalities bearing his name along with other extensions of the Parallelogram Law. In fact, Theorems 3.1.9 and 3.1.16 are known as Clarkson’s Inequalities. Hanner [79] furnished another approach, using his eponymous inequality, appearing as Theorem 4.5.14. The proof presented here is made possible by Friedrichs [63]. Dozens of other proofs exist in the literature. For example, a proof of Meir [123] is based on an extension of Hanner’s Inequality. More recently, Hanche-Olsen [78] delivered another proof, notable for its simplicity and directness. Yet another proof of uniform convexity was supplied by Shioji [155], who also presents an elementary derivation of the Riesz Representation Theorem. The notion of orthogonality covered here is due to Birkhoff [20] and James [91]. For a survey on this relation on normed linear spaces, and on isosceles orthogonality, see Alonso, Martini, and Wu [10]. There is also a version of orthogonality due to Roberts [137]. The reflexivity of the Lp spaces when p ∈ (1, ∞) could also be deduced from their uniform convexity as a result of the Milman-Pettis theorem [124, 134]. Lemma 3.3.4 and Theorem 3.3.5, on the continuity of metric projection in a uniformly convex space, are from a paper of Cheng, Miamee, and Pourahmadi [25]. In a Hilbert space, the orthogonal projection operator is well defined for any subspace. However, in a general Banach space, and indeed any linear metric space, the existence, uniqueness and continuity of metric projections are delicate issues. We also point put a deep result of James [92], which characterizes reflexive Banach spaces. See [17] for an exposition on convexity and smoothness in general Banach spaces. The Generalized Dominated Convergence Theorem (Lemma 3.4.9) comes from the text [142].

10.1090/ulect/075/04

CHAPTER 4

Weak Parallelogram Laws The Parallelogram Law is a distinguishing feature of Hilbert spaces. In this chapter we partially extend this notion to Banach spaces, thereby arriving at the weak parallelogram laws. In particular, we will show that the p spaces satisfy the weak parallelogram laws when p ∈ (1, ∞). Connections are then established with convexity, smoothness and duality. Though the results in this chapter are lengthy and technical, our patience will be amply rewarded in Section 4.6 when we derive an extension of the classical Pythagorean Theorem to weak parallelogram spaces. As in the Hilbert space case, the Pythagorean Theorem for weak parallelogram spaces relates the lengths of orthogonal vectors and their sum, where orthogonality is now in the Birkhoff-James sense (Definition 3.3.1). The extended version of the Pythagorean Theorem, comprising a system of inequalities, will have numerous applications and far reaching consequences. They will play a crucial role in the development of function theory in later chapters. 4.1. The Parallelogram Law A Hilbert space H satisfies the Parallelogram Law:

 (4.1.1) x + y2 + x − y2 = 2 x2 + y2 , x, y ∈ H . Verifying this is a simple matter of expressing the norm in H in terms of its inner product. As it turns out, the Parallelogram Law distinguishes Hilbert spaces from Banach spaces. Theorem 4.1.2 (Jordan-von Neumann). Suppose that X is a Banach space and (4.1.1) holds for all x, y ∈ X . Then there exists an inner product ·, · on X satisfying ! x = x, x, x ∈ X . Proof. Temporarily, let us view X as a vector space with real scalars and set 

[x, y] := 14 x + y2 − x − y2 , x, y ∈ X . We need to show that the mapping [·, ·] : X × X → R satisfies the properties of an inner product (at least for real scalars). We immediately see that [x, y] = [y, x] and

[x, x] = x2 .

Next, let us establish that [·, ·] is additive in its first argument, that is to say, [x + y, z] = [x, z] + [y, z], 41

x, y, z ∈ H .

42

4. WEAK PARALLELOGRAM LAWS

By the Parallelogram Law we have 

2 x + z2 + y2 = x + y + z2 + x − y + z2

 2 y + z2 + x2 = x + y + z2 + y − x + z2 . Rearranging, we find that x + y + z2 = 2x + z2 + 2y2 − x − y + z2 x + y + z2 = 2y + z2 + 2x2 − y − x + z2 . The left sides of the two equations above could thus be expressed as the average of the two right sides x + y + z2 = x2 + y2 + x + z2 + y + z2 (4.1.3)

− 12 x − y + z2 − 12 y − x + z2 .

Replacing z with −z in this last equation results in x + y − z2 = x2 + y2 + x − z2 + y − z2 (4.1.4)

− 12 x − y − z2 − 12 y − x − z2 .

This enables us to conclude that 

[x + y, z] = 14 x + y + z2 − x + y − z2 



= 14 x + z2 − x − z2 + 14 y + z2 − y − z2 (4.1.5) = [x, z] + [y, z]. Thus [·, ·] is additive in the first slot. Our next task to confirm that [·, ·] satisfies the properties of an inner product is to verify that [cx, y] = c[x, y], c ∈ R. We have already shown in (4.1.5) this holds when c = 2. One can readily see it is also true when c = 0 and when c = −1. Repeated application of additivity shows that it is true for all c ∈ Z. Now if c = m/n, where m, n ∈ Z and n = 0, then 1   1  n x, y = n · x, y = [x, y] n n and thus m  1  m x, y = m x, y = [x, y]. n n n To extend this result to all real c, let us start by noting that for any t, t0 ∈ R we have



tx + y − t0 x + y  |t − t0 |x, via the Triangle Inequality. This implies that for any fixed vectors x and y, the function tx + y is continuous in the real variable t. It follows that the mapping [tx, y] t is continuous on R \ {0}. Since this mapping takes the constant value [x, y] on the nonzero rationals, it must take the same value for all nonzero reals as well (we have already checked the c = 0 case above). Finally, to extend this to complex scalars, define ·, · : H × H → C by 

(4.1.6)

x, y := 14 x + y2 + ix + iy2 − x − y2 − ix − iy2 . t →

4.2. BASIC PROPERTIES

43

The change of the inner product from [·, ·] to ·, · ensures that

x, x = x2 ,

ix, x = i x, x,

x, y = y, x.

To establish additivity, apply the real case above to the real and imaginary parts of ·, · separately, add the results, and use (4.1.3) and (4.1.4): 

x + y, z = 14 x + y + z2 + ix + y + iz2

− x + y − z2 − ix + y − iz2 

= 14 x + y + z2 − x + y − z2 

+ 4i x + y + iz2 − x + y − iz2 



= 14 x + z2 − x − z2 + 14 y + z2 − y − z2 



+ 4i x + iz2 − x − iz2 + 4i y + iz2 − y − iz2 = x, z + y, z. Note the use of (4.1.5) in the second to last step. In particular, for real numbers a and b we have

(a + ib)x, y = ax, y + ibx, y = a x, y + ib x, y = (a + ib) x, y. We have shown that ·, · from (4.1.6) satisfies the properties of an inner product and thus H is a Hilbert space.  Remark 4.1.7. Any Banach space satisfying the inequality

 (4.1.8) x + y2 + x − y2  2 x2 + y2 , x, y ∈ H , must be a Hilbert space (see [48] for more on this result and its consequences). Indeed, if the inequality above holds for all x and y, then replacing x and y with 1 1 2 (x + y) and 2 (x − y) will show that the reverse inequality holds for all x and y and so (4.1.6) is satisfied. 4.2. Basic Properties In this section we will explore the manner in which a Banach space departs from the Parallelogram Law, or from the inequality (4.1.8). This motivates the following definitions. Definition 4.2.1. For fixed C > 0 and r ∈ (1, ∞), we say that a Banach space X satisfies the r-lower weak parallelogram law with constant C, if 

x + yr + Cx − yr  2r−1 xr + yr , x, y ∈ X . In shorthand, we say X is r-LWP(C). Similarly, X satisfies the r-upper weak parallelogram law with constant C, if 

x + yr + Cx − yr  2r−1 xr + yr , x, y ∈ X . This is also denoted by X is r-UWP(C). Remark 4.2.2. Perhaps the use of the word “weak” in this terminology is a bit inaccurate, since these conditions are generally not directly comparable to the Parallelogram Law unless r = 2.

44

4. WEAK PARALLELOGRAM LAWS

It will often be convenient to suppress the parameter r or the constant C, and speak simply of weak parallelogram conditions or spaces. We will sometimes refer to C as the weak parallelogram constant. In the Parallelogram Law for Hilbert spaces the associated exponent is r = 2, and the corresponding constant is C = 1. Thus, for a Banach space X satisfying a weak parallelogram law, the values of r and C measure how far a Banach space X deviates from being a Hilbert space. With the substitutions X = x+y and Y = x−y, we can express the r-LWP(C) law equivalently as (4.2.3)

X + Yr + X − Yr  2(Xr + CYr ),

and the r-UWP(C) law as (4.2.4)

X + Yr + X − Yr  2(Xr + CYr ).

These alternate expressions for the weak parallelogram laws will be helpful in calculations to follow. Example 4.2.5. The Parallelogram Law proper is the special case where both 2-LWP(1) and 2-UWP(1) hold, and we know from Theorem 4.1.2 that this characterizes inner product spaces. Example 4.2.6. Clarkson’s Inequalities (Theorem 3.1.9) tell us that p is pUWP(1) when p ∈ (1, 2], and p is p-LWP(1) when p ∈ [2, ∞). In Section 4.5 we will derive all of the weak parallelogram laws satisfied by p when p ∈ (1, ∞). The precise form of the factor 2r−1 in the weak parallelogram laws is motivated in part by Clarkson’s Inequalities (Theorem 3.1.9). In addition, this choice of 2r−1 makes possible a sort of Pythagorean Theorem for weak parallelogram spaces. This will be presented in Section 4.6. Let us proceed by establishing some basic properties of weak parallelogram spaces. This will include the behavior of the parameters r and C. As it turns out, a weak parallelogram space always satisfies weak parallelogram laws for a wide range of parameter values. This will help us later with the proof of Theorem 4.4.1, connecting the weak parallelogram laws with duality. Proposition 4.2.7. Suppose that X is a Banach space, C > 0, and p ∈ (1, ∞). Then we have the following: (i) If X is p-LWP(C), then C ∈ (0, 1], p ∈ [2, ∞), and X is r-LWP(B) whenever r  p and 0 < B  C r/p . (ii) If X is p-UWP(C), then C ∈ [1, ∞), p ∈ (1, 2], and X is r-UWP(B) whenever 1 < r  p and B  C r/p . Proof. Assume that X is p-LWP(C). Then for any x ∈ X we have 2p Cxp = x − xp + Cx + xp  2p−1 (xp + xp ) = 2p xp , which forces C  1.

4.2. BASIC PROPERTIES

45

Next suppose that r  p, and apply Proposition 1.5.2, the p-LWP(C) condition, H¨older’s inequality, and (4.2.3) to get    r p 2 xr + C r/p yr  2 xp + Cyp  r p p p  2 [1/2] · [x + y + x − y ]  r p = 2 · 2−r/p x + yp + x − yp p  r r r p· p p· p p · r x + y 2 + x − y      (r/p)/(r/p) × 1(r/p) + 1(r/p) 1−r/p

= x + yr + x − yr . Here (r/p) = [1 − 1/(r/p)]−1 and we used the identity

  1 − (r/p) + (r/p)/(r/p) = 1 − (r/p) + (r/p) 1 − 1/(r/p) = 0.

This confirms that X is r-LWP(C r/p ). Next, note that if X is p-LWP(C), then we can replace C with a smaller positive constant B in the weak parallelogram inequality, and it remains true for all vectors x and y in X . Finally, consider the inequality x + yr + Cx − yr  2r−1 (xr + yr ), where x is a unit vector in X , and y = bx for some constant b ∈ (0, 1). Evidently, C

2r−1 (1 + br ) − (1 + b)r . (1 − b)r

We can take the limit of the right hand side as b increases to 1 via two applications of L’Hˆopital’s rule to obtain C  lim

b→1−

2r−1 r(r − 1)br−2 − r(r − 1)(1 + b)r−2 . r(r − 1)(1 − b)r−2

When r < 2 the right hand side is zero, which means that X cannot be r-LWP. Therefore, the condition p-LWP(C) requires that p  2. This proves statement (i). Statement (ii) is handled in a similar way.  We now turn to the existence of optimal weak parallelogram constants. Proposition 4.2.8. Let X be a Banach space. (i) If X such (ii) If X such

is p-LWP(B) for some B, then there exists a unique constant C  1 that X is p-LWP(C), and B  C whenever X is p-LWP(B). is p-UWP(B) for some B, then there exists a unique constant C  1 that X is p-UWP(C), and B  C whenever X is p-UWP(B).

46

4. WEAK PARALLELOGRAM LAWS

Proof. The collection of constants B for which X is p-LWP(B) is nonempty and bounded above by 1. Therefore it has a supremum C which is also bounded above by 1. By definition, x + yp + Bx − yp  2p−1 (xp + yp ),

x, y ∈ X .

Fix x and y and take a supremum over B to get x + yp + Cx − yp  2p−1 (xp + yp ). Thus X is p-LWP(C). The UWP case is analogous.



Continuing with the matter of optimal constants, suppose that a Banach space X is given, and it satisfies r-LWP(Cr ), where Cr is the optimal constant associated with r. We already know that X must satisfy s-LWP(C s/r ) for all s > r. Let us s/r write Cs for the optimal constant in that case. It follows that Cs  Cr , or 1/s 1/r 1/r Cs  Cr . Consequently, the expression Cr , viewed as a function of r, is nonincreasing as r decreases. Thus lim Cr1/r

r→p+

exists, which implies

 C(p) :=

lim Cr1/r +

p

r→p

is well defined. Since, for fixed x and y, x + yr + Cr x − yr  2r−1 (xr + yr ) we can pass to a limit to get x + yp + C(p) x − yr  2p−1 (xp + yp ). Thus if C(p) = 0, then X is p-LWP(C(p) ). Naturally, a similar thing holds for the upper weak parallelogram case. Let us summarize this discussion as follows. Lemma 4.2.9. Let X be a Banach space. (i) Suppose X satisfies a lower weak parallelogram law. Let p be the infimum of the set of parameters r for which X satisfies an r-lower weak parallelogram law, and let Cr be the optimal constant for each r-LWP condition. If  p C := lim Cr1/r > 0, + r→p

then X is p-LWP(C). (ii) Suppose X satisfies an upper weak parallelogram law. Let p be the supremum of the set of parameters r for which X satisfies an r-upper weak parallelogram law, and let Cr be the optimal constant for each r-UWP condition. If  p C := lim Cr1/r < ∞, − r→p

then X is p-UWP(C).

4.3. GEOMETRIC CONSEQUENCES OF WEAK PARALLELOGRAM LAWS

47

4.3. Geometric Consequences of Weak Parallelogram Laws The weak parallelogram laws tell us something about the geometry of a space. Here is how they relate to convexity and smoothness. For a Banach space X , the modulus of convexity is defined by δ( ) := inf{1 − (x + y/2) : x, y ∈ X , x  1, y  1, x − y  }. It is evident from Definition 3.1.3 that X is uniformly convex if δ( ) > 0 for all

∈ (0, 2). The modulus of smoothness of X is ρ(t) :=

1 2

sup{x + ty + x − ty − 2 : x, y ∈ X , x = y = 1}.

We say that X is uniformly smooth if the right derivative of ρ(t) at t = 0 is zero [115]. Proposition 4.3.1. Let p ∈ (1, ∞) and C ∈ (0, 1]. If a Banach space X satisfies p-LWP(C), then X is uniformly convex. Proof. Assume that X satisfies p-LWP(C), ∈ (0, 2C −1/p ), x  1, y  1 and x − y > . Then x + yp + Cx − yp  2p−1 (xp + yp ),  x + y p   2p   + C p  2p−1 (1 + 1), by (4.2.4) 2  x + y p  p   .   1−C 2 2 It follows that X is uniformly convex, with modulus of convexity   p  1 p δ( )  1 − 1 − C > 0. 2



Proposition 4.3.2. Let p ∈ (1, ∞) and C ∈ [1, ∞). If a Banach space X satisfies p-UWP(C), then X is uniformly smooth. Proof. Assume that X satisfies p-UWP(C), t ∈ (0, C −1/p ), x = 1, and y = 1. Then 1





1

x + ty + x − ty  (x + typ + x − typ ) p (1p + 1p ) p 1

1

1

x + ty + x − ty − 2  2 p 2 p (xp + Ctp yp ) p − 2  1 1 x + ty + x − ty − 2  (1 + Ctp ) p − 1. 2 It follows that the modulus of smoothness ρ(t) satisfies 1

ρ(t)  (1 + Ctp ) p − 1. Thus, for t sufficiently small, 2C p−1 ρ(t)  t , t p which vanishes as t decreases to zero.



48

4. WEAK PARALLELOGRAM LAWS

4.4. Duality of the Weak Parallelogram Laws In this section we uncover the relationship between duality and the weak parallelogram laws. The main theorem here is the following. Theorem 4.4.1. Let X be a Banach space, p ∈ (1, ∞), and C > 0. 

(i) X is p-LWP(C) if and only if X ∗ is p -UWP(C −p /p );  (ii) X is p-UWP(C) if and only if X ∗ is p -LWP(C −p /p ). Thus UWP and LWP are dual notions, and the associated parameters enjoy a very pleasing symmetry. For the proof, most of the hard work is done in the following lemma. The method of the lemma fails when p = p = 2 and thus additional steps are needed. Lemma 4.4.2. Let X be a Banach space, p ∈ (1, ∞) \ {2}, and C > 0. 

(i) If X is p-LWP(C), then X ∗ is p -UWP(C −p /p );  (ii) If X is p-UWP(C), then X ∗ is p -LWP(C −p /p ). Proof. Let u and v be positive numbers, and define the function f1 on the two points {1, 2} by f1 (1) = u and f1 (2) = v. Similarly define f2 (1) = u and f2 (2) = −v. Let the measure μ be the mass v 2 at the point {1}, and the mass u2 at the point {2}. It is readily seen that f1 f2 dμ = 0 and f12 dμ = f22 dμ = 2u2 v 2 . {1,2}

{1,2}

{1,2}

For ease of reading the formulas below, we write all integrals as g dμ = g dμ. {1,2} ∗

For any x1 , x2 in X , λ1 , λ2 in X , and a1 , a2 ∈ C; we have [f1 (t)λ1 + f2 (t)λ2 ][f1 (t)a1 x1 + f2 (t)a2 x2 ] dμ(t) (4.4.3)

= (2u2 v 2 )[a1 λ1 (x1 ) + a2 λ2 (x2 )].

Furthermore, standard estimates confirm that







[f1 (t)λ1 + f2 (t)λ2 ][f1 (t)a1 x1 + f2 (t)a2 x2 ] dμ(t)





 [f1 (t)λ1 + f2 (t)λ2 ][f1 (t)a1 x1 + f2 (t)a2 x2 ] dμ(t)  f1 (t)λ1 + f2 (t)λ2 f1 (t)a1 x1 + f2 (t)a2 x2  dμ(t)  1/p   f1 (t)λ1 + f2 (t)λ2 p dμ(t) (4.4.4)  1/p × f1 (t)a1 x1 + f2 (t)a2 x2 p dμ(t) (4.4.5)

       1/p  up v 2 λ1 + λ2 p + u2 v p λ1 − λ2 p  1/p × up v 2 a1 x1 + a2 x2 p + u2 v p a1 x1 − a2 x2 p .

4.4. DUALITY OF THE WEAK PARALLELOGRAM LAWS

49

Now, suppose that X satisfies p-LWP(C). By Proposition 4.2.7 we know that p > 2 > p . Apply the above bound, taking v = 1 and u = C −1/(p−2) . The calculation then yields





(2u2 ) a1 λ1 (x1 ) + a2 λ2 (x2 )

      1/p  up λ1 + λ2 p + u2 λ1 − λ2 p  × up a1 x1 + a2 x2 p + u2 a1 x1 − a2 x2 p ]1/p       1/p  u2 · λ1 + λ2 p + u2−p λ1 − λ2 p 1/p  × a1 x1 + a2 x2 p + Ca1 x1 − a2 x2 p     1/p    1/p  u2 · λ1 + λ2 p + u2−p λ1 − λ2 p × 2p−1 (a1 x1 p + a2 x2 p )     1/p     1/p = 21/p u2 · λ1 + λ2 p + u2−p λ1 − λ2 p · (a1 x1 p + a2 x2 p ) . 

Let us cancel the common factor of u2 from both sides. Also, we check that u2−p =  C −p /p . Thus we may deduce that





21−1/p a1 λ1 (x1 ) + a2 λ2 (x2 )

   1/p    1/p  λ1 + λ2 p + C −p /p λ1 − λ2 p (a1 x1 p + a2 x2 p ) . Choose x1 and x2 to be unit vectors that are approximately norming for λ1 and λ2 , respectively. Then, take the supremum of the left hand side over the condition |a1 |p + |a2 |p = 1. The conclusion is 















21−1/p (λ1 p + λ2 p )1/p  [λ1 + λ2 p + C −p /p λ1 − λ2 p ]1/p . 

In other words, X ∗ is p -UWP(C −p /p ). This proves assertion (i). Note that the case p = p = 2 cannot be handled this way, as then either u or v ends up being zero, and the associated calculation tells us nothing. To get (ii), assume that X is p-UWP(C), and take u = v = 1 in (4.4.3) and (4.4.5). We cite the weak parallelogram laws in their alternate form (4.2.3) to get 2|a1 λ1 (x1 ) + a2 λ2 (x2 )|      1/p  λ1 + λ2 p + λ1 − λ2 p      1/p  λ1 + λ2 p + λ1 − λ2 p

1/p  · a1 x1 + a2 x2 p + a1 x1 − a2 x2 p  1/p · 2(a1 x1 p + Ca2 x2 p ) .

Proceeding as before, we take x1 and x2 to be unit vectors that are approximately norming for λ1 and λ2 , respectively. Then take the supremum of the left hand side over the condition |a1 |p + C|a2 |p = 1. The left hand side can be expressed as





2 a1 λ1 (x1 ) + (C 1/p a2 )C −1/p λ2 (x2 ) .

50

4. WEAK PARALLELOGRAM LAWS

The conclusion is           1/p   1/p  λ1 + λ2 p + λ1 − λ2 p , 21/p λ1 p + C −p /p λ2 p 

which is to say that X ∗ is p -LWP(C −p /p ).



We are now in a position to complete the proof of Theorem 4.4.1. First let us handle the p = p = 2 case omitted from Lemma 4.4.2. Suppose that X is 2-LWP(C), with C being the optimal constant. Then X is r-LWP(C r/2 ) for every  r > 2. It follows from Lemma 4.4.2 that the dual space X ∗ is r  -UWP(C −r /2 ), where r  is the conjugate index to r. Clearly the supremum of all the values of r  that arise in this way is 2. Furthermore, 2      2 1 1/r   lim C (−r /2)/r = . lim Cr + + C r→2 r→2 ∗ Therefore, by Lemma 4.2.9, X satisfies 2-UWP(C2 ) for some constant C2  1/C. A second application of Lemma 4.4.2 now reveals that the second dual X ∗∗ satisfies 2-LWP(1/C2 ). But X is isometrically isomorphic to a subspace of X ∗∗ , and so X is 2-LWP(1/C2 ). Since C was chosen to be the optimal constant for X satisfying the condition 2-LWP(·), and C  1/C2 , it must be that C2 = 1/C. (Actually, a LWP space is already uniformly convex, hence reflexive; however, this does not hold for UWP spaces, and thus it does not help the other half of the proof.) This verifies that if X is 2-LWP(C), then X ∗ is 2-UWP(1/C). As ever, a similar argument shows that if X is 2-UWP(C), then X ∗ is 2-LWP(1/C). As another corollary to Lemma 4.2.9, we have the following statements. If  X ∗ is p-LWP(C), then X is p -UWP(C −p /p ); if X ∗ is p-UWP(C), then X is  p -LWP(C −p /p ). As before, this is because X is isometrically isomorphic to a subspace of X ∗∗ . At last the proof of Theorem 4.4.1 is complete. 4.5. Weak Parallelogram Laws for p In this section we will identify all of the weak parallelogram laws satisfied by the p spaces. This will be accomplished for the parameter values p ∈ (1, 2] through a series of lemmas. For the case p ∈ [2, ∞), we use the duality result from Theorem 4.4.1. No weak parallelogram laws are satisfied by 1 or ∞ since they are neither uniformly convex nor uniformly smooth. We begin with a calculation that explores the optimal weak parallelogram constant in this setting. Lemma 4.5.1. Suppose that p ∈ (1, 2] and r ∈ [2, p ]. Then the constant defined by (4.5.2)

2r−r/p (1 + tp )r/p − (1 + t)r 0t0

Thus the infimum in (4.5.2) must be positive when r = 2. Next, for general values of r in the interval [2, p ], fix p ∈ (1, 2], and t ∈ [0, 1), and consider " # d 2r−r/p (1 + tp )r/p − (1 + t)r dr (1 − t)r    = (1 − t)r 2r−r/p (1 + tp )r/p log[21−1/p (1 + tp )1/p ] − (1 + t)r log(1 + t)  

− 2r−r/p (1 + tp )r/p − (1 + t)r (1 − t)r log(1 − t) /(1 − t)2r . The final expression is nonnegative. To see this, we use the negativity of the factor log(1 − t), the previously derived inequality (4.5.3), and the fact that the function ψ(s) = sr log s is increasing on at least the interval [1, ∞). This shows that the expression 2r−r/p (1 + tp )r/p − (1 + t)r (1 − t)r from the defining equation (4.5.2) is a nondecreasing function of r for each p and t. It follows that 2r−r/p (1 + tp )r/p − (1 + t)r  (p − 1) > 0 (1 − t)r t→1− lim

for all p and r. This forces Cp,r to be positive for all parameter values.



We are now able to state all of the weak parallelogram laws satisfied by the p spaces. Theorem 4.5.5. If p ∈ (1, 2] then p is: (4.5.6)

r-UWP(1) when r ∈ (1, p];

(4.5.7)

r-LWP(Cp,r ) when r ∈ [2, p ]; and

(4.5.8)

r-LWP(1) when r ∈ [p , ∞).

52

4. WEAK PARALLELOGRAM LAWS

If p ∈ [2, ∞) then p is: (4.5.9)

r-LWP(1) when r ∈ [p, ∞);

(4.5.10)

r-UWP(Cp ,r ) when r ∈ [p , 2]; and

(4.5.11)

r-UWP(1) when r ∈ (1, p ]

−p/p

The weak parallelogram constants are optimal. Remark 4.5.12. The alert reader will notice how there are pairs of values (r, p) that are unaccounted for in the statement of Theorem 4.5.5. We will account for these missing parameters values in Proposition 4.7.1 below. The proof will proceed in stages, by which (4.5.6), (4.5.7) and (4.5.8) are verified. The remaining three cases are their dual statements, in the sense that “lower” and “upper” are interchanged, each exponent is switched with its H¨ older conjugate, and the weak parallelogram constants do just the right thing. We may then invoke the duality theorem (Theorem 4.4.1). The case (4.5.7) relies on Hanner’s Inequality (see Theorem 4.5.14 below), which in turn rests on the following pillars. Lemma 4.5.13. Let p ∈ (1, 2]. If a and b are complex numbers, then

p (|a| + |b|)p + |a| − |b|  |a + b|p + |a − b|p . Proof. Let u and v be nonnegative real numbers, and consider the function ϕ(t) := (u2 + v 2 + 2uvt)p/2 + (u2 + v 2 − 2uvt)p/2 ,

t ∈ [−1, 1].

By elementary calculus we find that ϕ(t) attains minimum values at −1 and 1. Given complex numbers a and b we set u = |a| and v = |b|, then |a + b|p = [(a + b)(a + b)]p/2 = [|a|2 + |b|2 + 2(ab)]p/2 = [|a|2 + |b|2 + 2|a||b|t)]p/2 = (u2 + v 2 + 2uvt)p/2 for some t ∈ [−1, 1]. Similarly, |a − b|p = (u2 + v 2 − 2uvt)p/2 for the same value of t. Thus we have

p (|a| + |b|)p + |a| − |b| = (|a|2 + |b|2 + 2|a||b|)p/2 + (|a|2 + |b|2 − 2|a||b|)p/2 = ϕ(1)  ϕ(t) = (u2 + v 2 + 2uvt)p/2 + (u2 + v 2 − 2uvt)p/2 = |a + b|p + |a − b|p .



Hanner’s Inequality is another way to generalize the Parallelogram Law, and it can be used to deduce the uniform convexity of p for p ∈ (1, 2]. Theorem 4.5.14 (Hanner’s Inequality). If p ∈ (1, 2] and a and b ∈ p , then

p

p

 (4.5.15) ap + bp + ap − bp  a + bpp + a − bpp .

4.5. WEAK PARALLELOGRAM LAWS FOR p

53

Proof. Let An := |an | and Bn := |bn | for all n  0, so that A = {An }n0 and B = {Bn }n0 are nonnegative sequences in p . By summing over n and applying Lemma 4.5.13, we have A + Bpp + A − Bpp  a + bpp + a − bpp . Since it is also true that

p 

p 

p

p

ap + bp + ap − bp = Ap + Bp + Ap − Bp

(note that ap = Ap and bp = Bp ), it suffices to prove that (4.5.15) holds for A and B. To accomplish this, we introduce the function

p

g(u, v) := (u1/p + v 1/p )p + u1/p − v 1/p , u  0, v  0. Then

p 

d d  1/p g(t, 1) = (t + 11/p )p + t1/p − 11/p

dt dt = (t1/p + 1)p−1 t1/p−1 + |t1/p − 1|p−1 sgn(t1/p − 1)t1/p−1 for all t > 0, and  d2 1 p − 1 1/p−2  1 0 g(t, 1) = − t dt2 p |t1/p − 1|2−p (t1/p + 1)2−p for all t > 0, t = 1. Thus, g(t, 1) is a convex function of t. From this we deduce that for positive numbers a, b, c and d, (c + d)g((a + b)/(c + d), 1)  cg(a/c, 1) + dg(b/d, 1). And now from g(tu, tv) = tg(u, v) we obtain (4.5.16)

g(a + b, c + d)  g(a, c) + g(b, d).

A similar argument establishes that this holds if any of the numbers a, b, c and d is zero. Finally, repeated application of (4.5.16) enables us to say   p  g Apk , Bk  g(Apk , Bkp ), k0

k0

k0

which shows that (4.5.15) holds for A and B. This completes the proof.



Here is the last bit of legwork needed to prove (4.5.7). Lemma 4.5.17. Let 1 < p  2  r  p . For any real numbers u and v, |u + v|r + Cp,r |u − v|r  2r−r/p (|u|p + |v|p )r/p . Proof. For real values of t define (4.5.18)

h(t) := 2r−r/p (1 + |t|p )r/p − |1 + t|r − Cp,r |1 − t|r .

We can readily confirm that h(t) = |t|r h(1/t) when t =  0, and h(t)  h(|t|). Consequently, we will be done if we can show that h(t)  0 for t in the interval [0, 1]. But this follows from Lemma 4.5.1.  In general, the value of Cp,r is attained in (4.5.2) by an interior point t of [0, 1), or at t = 0. When r = 2, the infimum occurs in the limit as t increases to 1. Proposition 4.5.19. If p ∈ (1, 2], then Cp,2 = p − 1.

54

4. WEAK PARALLELOGRAM LAWS

Proof. From (4.5.18) we have h(t) := 22−2/p (1 + |t|p )2/p − (1 + t)2 − Cp,2 (1 − t)2 in this case. For x ∈ (0, 12 ], let k(x) := (2 − p)x2−2/p + (p − 1)x1−2/p . It is straightforward to check that k (x)  0, and so k is decreasing on (0, 12 ]. But h (t) = 23−2/p k(tp /(1 + tp )) − 2 − 2Cp,2 for all t ∈ (0, 1] and so h is decreasing on (0, 1]. Since h (1) = 2(p − 1 − Cp,2 )  0, h is increasing on [0, 1] to the value h (1) = 0. Thus h is decreasing on [0, 1], from which it follows that the infimum in (4.5.2) is attained in the limit at the endpoint 1. We previously showed in (4.5.4) that this limiting value is p − 1.  Lemma 4.5.20. If 1 < p  2  r  p then p is r-LWP(Cp,r ). Proof. Suppose that a and b ∈ p , and let 2u := a + bp + a − bp and 2v := a + bp − a − bp . By Lemma 4.5.17, Hanner’s Inequality, and H¨older’s Inequality, a + brp + Cp,r a − brp = (u + v)r + Cp,r (u − v)r  2r−r/p (|u|p + |v|p )r/p 

p 

p r/p = 2r−r/p 12 a + bp + 12 a − bp + 12 a + bp − 12 a − bp 

r/p  2r−r/p app + bpp 

(r/p)(p/r)  r/(r−p)

(r/p)([r−p]/r) 1  2r−r/p app(r/p) + bpp(r/p) + 1r/(r−p) 

= 2r−r/p arp + brp 2r/p−1 

= 2r−1 arp + brp . This verifies that the constant Cp,r is sufficiently small that p satisfies r-LWP(Cp,r ). To see that this constant is optimal, consider the special case a = te0 + e1 and b = e0 + te1 for some t ∈ [0, 1). Then 

2r−1 arp + brp − a + brp 2r−1 2(1 + tp )r/p − 2r/p (1 + t)r = a + brp 2r/p (1 − t)r =

2r−r/p (1 + tp )r/p − (1 + t)r . (1 − t)r

The infimum of the final expression is Cp,r . Thus there is a sequence of values of t for which equality to the infimum holds in the limit. Consequently, the lower weak parallelogram law 

a + brp + Ca − brp  2r−1 arp + brp cannot be valid for any value of C greater than Cp,r . This proves the case (4.5.7) in Theorem 4.5.5.



4.6. PYTHAGOREAN INEQUALITIES

55

Lemma 4.5.21. Suppose that p ∈ (1, 2]. If 2  p  r < ∞, then for all x and y in p , (4.5.22)

x + yrp + x − yrp  2r−1 (xrp + yrp ).

Proof. In this situation, 2(xrp + yrp )p/r 

 2(xrp + yrp )p (p−1)/r (4.5.23)





 2(xpp + ypp )p−1  x + ypp + x − ypp  (x + yrp + x − yrp )p/r 21−p/r ,

where the step from line (4.5.23) to the next employs Theorem 3.1.16. It follows that 2(xrp + yrp )  x + yrp + x − yrp . Now replace x and y with x+y and x−y, respectively, to see that this is equivalent to (4.5.25). The weak parallelogram constant is 1, and must be optimal. Indeed, equality holds in (4.5.22) if x = y.  This proves (4.5.8) in Theorem 4.5.5. Lemma 4.5.24. Suppose that p ∈ (1, 2]. If 1 < r  p  2, then for all x, y ∈ p , (4.5.25)

x + yrp + x − yrp  2r−1 (xrp + yrp ).

Proof. Again, by H¨older’s Inequality, Proposition 1.5.2, and Theorem 3.1.9,

r/p (p−r)/p  2 xrp + yrp  xp + yp  p−1

r/p (p−r)/p −r(p−1)/p = 2 [xp + yp ] 2 2 

r/p  [x + yp + x − yp ] 21−r 

 [x + yr + x − yr ] 21−r , which is (4.5.25). As before, the weak parallelogram constant 1 is optimal, and equality holds in (4.5.25) if x = y.  This confirms (4.5.6) in Theorem 4.5.5. We have therefore verified Theorem 4.5.5 for p ∈ (1, 2]. For the dual case p ∈ [2, ∞), we refer to Theorem 4.4.1. In Proposition 4.7.1, we will confirm that Theorem 4.5.5 is complete in the sense that no weak parallelogram laws hold when p and r lie outside of the listed ranges. 4.6. Pythagorean Inequalities In a Hilbert space, if two vectors x and y are orthogonal then x2 + y2 = x + y2 . The proof of this is a straightforward matter of writing out both sides in terms of the inner product. In this section we will derive something similar for smooth weak parallelogram spaces. The resulting Pythagorean laws for such spaces take

56

4. WEAK PARALLELOGRAM LAWS

the form of a family of inequalities. These inequalities will be useful for a variety of applications. Suppose that X is a smooth Banach space (recall Definition 3.4.1), so that for each nonzero x in X , there is a unique norm one functional λx satisfying λx (x) = x.

(4.6.1)

We establish below that weak parallelogram laws on smooth Banach spaces are expressible as bounding conditions on these norming functionals (UWP spaces, we recall from Proposition 4.3.2, are automatically smooth). Here z stands for the real part of a complex number z. Lemma 4.6.2. Let p ∈ (1, ∞) and let X be a smooth Banach space. (i) X is p-LWP if and only if for some positive constant K, and for all x = 0 and y in X , (4.6.3)

x + yp  xp + Kyp + pxp−1 (λx (y)).

(ii) X is p-UWP if and only if for some positive constant K, and for all x = 0 and y in X , (4.6.4)

x + yp  xp + Kyp + pxp−1 (λx (y)).

The proof of this lemma requires a detail called McShane’s Lemma. We have already seen a version of this for the p spaces in Theorem 3.4.6. Lemma 4.6.5 (McShane’s Lemma). Let X be a smooth Banach space, p ∈ (1, ∞), and x, y ∈ X . Then x + typ − xp t→0 pt lim

exists and is equal to xp−1 (λx (y)). Proof. The existence of the limit comes from the fact that X is a smooth Banach space. To obtain the value of limit, we start off with a derivative calculation. Let p ∈ (1, ∞) and t be a real variable. Suppose that h(t) = u(t) + iv(t), where u and v are differentiable functions of a real variable, not both vanishing on some interval. Then p/2 d d |h(t)|p = [h(t)p/2 h(t) ] dt dt p/2

=

ph(t)p/2 h(t)

= p|h(t)|

2h(t) p−2

h (t)

p/2

+

ph(t)

h(t)p/2 h (t) 2h(t)

[h (t)h(t)]

If h(t) = λx (x + ty), then an application of L’Hˆ opital’s Rule, along with the fact that λx (x) = x, yields

|λx (x + ty)|p − |λx (x)|p d 1 1 d

= lim |h(t)|p = |h(t)|p

lim t→0 tp p t→0 dt p dt t=0 p−1 = |λx (x)| (λx (y)) = xp−1 (λx (y)). Since λx  = 1 and

|λx (x + ty)|  x + ty,

4.6. PYTHAGOREAN INEQUALITIES

57

we have lim+

t→0

x + typ − xp |λx (x + ty)|p − |λx (x)|p  lim pt pt t→0+ p−1 = x (λx (y)) |λx (x + ty)|p − |λx (x)|p pt t→0− x + typ − xp .  lim− pt t→0

= lim

Since we already know that x + typ − xp t→0 pt lim

exists, the assertion is proved.



These results make possible the following extension of Theorem 3.4.7, characterizing Birkhoff-James orthogonality. Corollary 4.6.6. Let X be a smooth Banach space. Then x ⊥X y if and only if λx (y) = 0. Proof. That condition x ⊥X y holds either when x = 0 or when the function t → x + ty has a minimum at t = 0. An application of Lemma 4.6.5 completes the proof.  Proof of Lemma 4.6.2. Suppose that (4.6.3) holds. Then xp + Kyp + pxp−1 (λx (y))  x + yp , xp + K − yp + pxp−1 (λx (−y))  x − yp , 2xp + 2Kyp  x + yp + x − yp . Replace x with x + y, and replace y with x − y, to find that (4.6.7)

x + yp + Kx − yp  2p−1 (xp + yp ).

Thus X is p-LWP(K). Conversely, assume that X is p-LWP(C) for some C > 0, and apply u + vp + Cu − vp  2p−1 (up + vp ) to the pair of vectors u = x and v = x + 2−n y, for n  0. The first step, with n = 0, shows that 2x + 12 yp + 2C 12 yp − xp  x + yp and subsequent steps may take the form (4.6.8)

2x + 2−(n+1) yp + 2C2−(n+1) yp − xp  x + 2−n yp .

58

4. WEAK PARALLELOGRAM LAWS

We then apply (4.6.8) repeatedly, substituting the last x + 2−n yp term with the smaller quantity on the left. This yields the estimate x + yp  −(1 + 2 + 22 + · · · + 2n−1 )xp   1 1 1 1 + p−1 + p−1 2 + p−1 3 + · · · + p−1 n Cyp 2 (2 ) (2 ) (2 ) n −n p + 2 x + 2 y 1 − 2−(p−1)n Cyp + 2n x + 2−n yp 2p−1 − 1 1 − 2−(p−1)n x + 2−n yp − xp p Cy = xp + + . 2p−1 − 1 2−n Now take the limit as n tends toward infinity. Smoothness ensures that the directional derivative in the final term exists. To obtain its value, apply McShane’s Lemma (Lemma 4.6.5) to conclude that C x + yp  xp + p−1 yp + pxp−1 (λx (y)). 2 −1 This verifies (4.6.3) with C K = p−1 2 −1 and an analogous argument establishes (4.6.4).  = −(2n − 1)xp +

From this we derive a Pythagorean Theorem for weak parallelogram spaces. It relates the lengths of two orthogonal vectors to that of their sum, where orthogonality is in the Birkhoff-James sense. There are two inequalities, corresponding to the upper and lower weak parallelogram laws. For a Hilbert space, the inequalities reduce to the familiar Pythagorean Theorem. Theorem 4.6.9. Let p ∈ (1, ∞) and X be a smooth Banach space. (i) If X is p-LWP(C), then there exists a positive constant K such that whenever x ⊥X y in X , (4.6.10)

xp + Kyp  x + yp .

(ii) If X is p-UWP(C), then there exists a positive constant K such that whenever x ⊥X y in X , (4.6.11)

xp + Kyp  x + yp .

In either case, the constant K can be chosen to be C/(2p−1 − 1). Proof. By Corollary 4.6.6, the condition x ⊥X y is equivalent to λx (y) = 0 (this is Theorem 3.4.7 in the case of p when p ∈ (1, ∞)). The proof of Lemma 4.6.2 ensures that the constant K = C/(2p−1 − 1) suffices.  Let us refer to the constant K as a Pythagorean constant for the space X . The value K = C/(2p−1 − 1) is generally not optimal, however, since the bounds used in its derivation do not achieve equality under the same conditions. The p spaces certainly satisfy Pythagorean inequalities when p ∈ (1, ∞). These follow directly from the weak parallelogram laws for p . Corollary 4.6.12. Let r ∈ (1, ∞) and p ∈ (1, ∞).

4.7. LACK OF WEAK PARALLELOGRAM LAWS

59

(i) If 1 < p  2  r < ∞ or 2  p  r < ∞, then there is a K > 0 such that xrp + Kyrp  x + yrp whenever x, y ∈ p with x ⊥p y. (ii) If 1 < r  p  2 or 1 < r  2  p < ∞, then there is a K > 0 such that xrp + Kyrp  x + yrp whenever x, y ∈ p with x ⊥p y. It is important to mention here that K depends on r and p. These inequalities will play an important role in the development of function theory in later chapters. 4.7. Lack of Weak Parallelogram Laws Finally, let us use the Pythagorean inequalities to check that Theorem 4.5.5 indeed describes all of the r-weak parallelogram laws for p . That is, for values of r and p lying outside of the cases identified in the theorem, a weak parallelogram law fails to hold. Proposition 4.7.1. Let r ∈ (1, ∞), and p ∈ (1, ∞). (ii) If r > p or r > 2, then there is no C > 0 such that x + yrp + Cx − yrp  2r−1 (xrp + yrp ),

x, y ∈ p .

(ii) If r < p or r < 2, then there is no C > 0 such that x + yrp + Cx − yrp  2r−1 (xrp + yrp ),

x, y ∈ p .

Proof. Our strategy is to show that the respective Pythagorean inequalities fail, and hence the corresponding weak parallelogram laws fail. Consider the vectors x = ae0 + ae1 and y = e0 − e1 . Note that x ⊥p y. Assuming a is positive and large, we find that x + yrp − xrp (|a + 1|p + |a − 1|p )r/p − (|a|p + |a|p )r/p = yrp (1 + 1)r/p  r/p p(p − 1) 3 = ar 1 + + O(1/a ) − ar 2a2 1 = ar−2 r(p − 1) + O(ar−3 ), 2 where the estimates come from the binomial series:   1  p(p − 1)  1 2  1 3  1 p 1+ + . =1+p +O a a 2 a a As a → ∞ this tends to the limit 0 if r ∈ (1, 2); it diverges to ∞ if r ∈ (2, ∞). It follows that p fails to be r-LWP if r ∈ (1, 2), and that p fails to be r-UWP if r ∈ (2, ∞). Next, consider 2-dimensional p with vectors x = e0 and y = ae1 . As before, x ⊥p y. Assume that a is positive and small. Then x + yrp − xrp (1 + ap )r/p − 1 r = = ap−r + O(a2p−r ). r r yp a p As a → 0+ , this tends to 0 if p > r; it diverges to ∞ when r > p. We may conclude that p fails to be r-LWP when p > r, and it fails to be r-UWP when r > p. 

60

4. WEAK PARALLELOGRAM LAWS

4.8. Metric Projections onto Nested Subspaces The following results concerning metric projections onto nested subspaces demonstrate a practical application of the Pythagorean inequalities. There are more such applications to come in our exploration of function spaces. Proposition 4.8.1. Let X be r-LWP(C). Suppose that Xn is a subspace of X for each n ∈ N, such that X 1 ⊇ X2 ⊇ X3 ⊇ · · · . Define X∞ = n1 Xn . If Pn is the metric projection operator from X to Xn , for all n ∈ N ∪ {∞}, then for any x ∈ X , Pn x converges to P∞ x in norm. Proof. By assumption, X is LWP, which implies that it is uniformly convex (and hence has unique nearest points), and satisfies the lower Pythagorean inequality, that is, x + yr  xr + Kyr ,

x ⊥X y.

Here one can take the constant K to be equal to C/(2r−1 − 1). Select any x ∈ X \ {0}. By the definition of metric projection, whenever m < n, we have x − Pm x = inf{x − z : z ∈ Xm }  inf{x − z : z ∈ Xn } = x − Pn x  x − P∞ x.

(4.8.2)

Thus, as a sequence indexed by n, x − Pn x is monotone nondecreasing, and bounded above. Accordingly, it converges. Next, for m < n, the vector Pm x − Pn x lies in Xm (the larger space), and hence the co-projection x − Pm x must satisfy x − Pm x ⊥p Pm x − Pn x. Consequently, the Pythagorean inequality says that x − Pn xr  x − Pm xr + KPm x − Pn xr . Since the (positive) difference x − Pn xr − x − Pm xr can be made arbitrarily small by choosing m sufficiently large, it follows that {Pm x} is a Cauchy sequence in norm, and converges to some vector z. We can see that z ∈ X∞ (Pm x ∈ Xj for m  j and thus z ∈ Xj for all j). From (4.8.2) we deduce that x − z  x − P∞ x, and thus equality must hold, whence z = P∞ x.



Proposition 4.8.3. Let X be a Banach space satisfying r-LPW(C). Suppose that Xn is a subspace of X for each n ∈ N, such that $∞

X1 ⊆ X2 ⊆ X3 ⊆ · · · .

Define X∞ = n=1 Xn . If Pn is the metric projection operator from X to Xn , for all n ∈ N ∪ {∞}, then for any x ∈ X , Pn x converges to P∞ x in norm.

4.9. NOTES

61

Proof. By the LWP condition, X is uniformly convex (and hence has unique nearest points), and satisfies the lower Pythagorean inequality x + yr  xr + Kyr whenever x ⊥X y. Let x ∈ X \ {0}. By the definition of metric projection, whenever m < n, we have x − Pm x = inf{x − z : z ∈ Xm }  inf{x − z : z ∈ Xn } = x − Pn x (4.8.4)

 x − P∞ x.

Thus, as a sequence indexed by n, x − Pn x is monotone nonincreasing, and bounded below. Therefore it converges. Next, for m < n, the vector Pm x − Pn x lies in Xn (the larger space), and hence the co-projection x − Pn x must satisfy x − Pn x ⊥p Pm x − Pn x. The Pythagorean inequality then says that x − Pm xr  x − Pn xr + KPm x − Pn xr . Since the (positive) difference x−Pm xr −x−Pn xr can be made arbitrarily small by choosing m sufficiently large, it follows that {Pm x}m1 is a Cauchy sequence in norm, and converges to some vector z. We see that z ∈ X∞ , and by (4.8.4) we have x − z  x − P∞ x. By (4.8.4), and letting n → ∞, we see that x − Pm x  x − z  x − P∞ x. This means that for every ym ∈ Xm x − ym   x − z  x − P∞ x. Now choose vectors ym ∈ Xm with ym → P∞ x. Thus by letting m → ∞ we see that x − z  x − P∞ x. Equality holds in these norms, so finally uniqueness of nearest points forces z = P∞ x.  The last two propositions seem to suggest that the projections Pn converge in the strong operator topology, and this is indeed the case for a Hilbert space. For Banach spaces, however, we remind the reader that the projections are metric projections and need not be linear mappings. 4.9. Notes Theorem 4.1.2 was proved by Jordan and von Neumann in [95]. Like Clarkson’s Inequalities (Theorem 3.1.9) and Hanner’s Inequality (Theorem 4.5.14), the weak parallelogram laws are a way to generalize the Parallelogram Law from Hilbert spaces to Banach spaces. Our presentation of Hanner’s Inequality follows his original paper [79]. The r = 2 case of the weak parallelogram law was first explored by Bynum and Drew [22] for p and in Bynum [21] for general Banach spaces. In

62

4. WEAK PARALLELOGRAM LAWS

a paper of Cheng, Miamee, and Pourahmadi [25] the r = p case is handled, and the corresponding Pythagorean inequalities are derived. The paper of Cheng and Ross [26] investigates the weak parallelogram laws with arbitrary parameters and identifies their basic properties. The duality property for the weak parallelogram laws is from Cheng and Harris [32]. Theorem 4.4.1 is from Cheng and Harris [31] while Theorem 4.5.5 is from Cheng, Mashreghi, and Ross [35]. Cheng and Ross [26] is our source for the Pythagorean inequalities in Theorem 4.6.9 and its subsequent corollaries. The optimal weak parallelogram constants for Lp are identified in a paper of Cheng, Mashreghi, and Ross [35]. Our treatment of McShane’s Lemma (Lemma 4.6.5) is borrowed from the text [24]. Weak parallelogram spaces are quite numerous. They include not only the Lp spaces, for p ∈ (1, ∞), but also many classes of spaces built from them. Indeed, the weak parallelogram properties are inherited by subspaces and quotient spaces; they are also preserved by Cartesian products and direct sums, as long as care is taken in defining the norms. Thus, for example, certain Sobolev spaces and Besov spaces, and other mixed-norm families of spaces, would be subject to the analysis of this chapter. See Cheng and Harris [31] for the details. The parameters arising from the weak parallelogram laws are just one avenue to describe how a Banach space might differ geometrically from a Hilbert space. There are other such characteristic constants under investigation. For example, the von Neumann-Jordan constant CN J for a Banach space X , first appearing in a paper of Clarkson [41], is the smallest value of C for which 1 x + y2 + x − y2  C C 2(x2 + y2 ) for all x and y ∈ X , not both zero. The greater C is, the further the Banach space X departs from a Hilbert space. One can show that if X is 2-LWP(C), then CN J  1/C; if X is 2-UWP(C), then CN J  C. The James constant J, discussed in a paper of Gao and Lau [65], for X is J := sup{min(x + y, x − y)} as x and y vary over the unit sphere of X . There is an industry in establishing connections between these parameters and the structure and behavior of the space [35, 41, 94, 96, 147, 167, 169, 170]. Propositions 4.8.1 and 4.8.3 on nested projections will play an important role in our treatment of the zero sets for analytic functions (Chapter 9).

10.1090/ulect/075/05

CHAPTER 5

Hardy and Bergman Spaces Beginning in Chapter 6 we will explore the space pA of analytic functions on the disk whose sequence of Taylor coefficients belong to p . We will see that this space, in terms of its function theoretic complexity, lies somewhere between two well-studied spaces of analytic functions, the Hardy space and the Bergman space. The latter two spaces will be the subject of the present chapter. Our understanding of the Hardy and Bergman spaces will further help to identify the types of questions to ask about pA . We will also touch upon the important shift and backward shift operators that will be explored in detail for pA . Since much of this material is standard and appears in a variety of places, we will just state the necessary results and direct the reader to the end notes for the references. 5.1. The Hardy Space The Hardy spaces are classes of analytic functions on the open unit disk D = {|z| < 1} that enjoy certain growth properties near the boundary T = {ξ : |ξ| = 1}. This feature has deep consequences which yield a complete understanding of their zero sets, canonical factorization, shift-invariant subspaces, and various interpolation problems. Thus the Hardy spaces provide a conceptual baseline for the study of other spaces. Let Hol(D) denote the vector space of analytic functions on D and dm(ξ) = |dξ|/2π denote normalized Lebesgue measure on T. Definition 5.1.1. For p ∈ (0, ∞) define the Hardy space H p by   H p := f ∈ Hol(D) : sup |f (rξ)|p dm(ξ) < ∞ , 01 A .

Another class of examples is the following. Proposition 6.2.2. If f is analytic in a neighborhood of D, then f ∈

p p>0 A .

If f is analytic in a neighborhood of D, then the power series f =  Proof. k a z has a radius of convergence R > 1. By the Cauchy-Hadamard formula k k0 for the radius of convergence of a power series [143, Ch. 10], we have 1 1 = lim |ak | k . R k→∞

Fix any ρ ∈ (1, R). Then |ak |  C

1 , ρk

k  0,

for some positive constant C. This implies f ∈

p p>0 A .



The pA classes contain various types of functions that are H¨ older continuous in the mean-square sense. Here is one such result that generalizes a theorem of Bernstein (see Proposition 13.1.3). Proposition 6.2.3. Let 0 < a < b < 1. Suppose that f ∈ H 2 satisfies (6.2.4) 0

2π

i[θ+t]

2 dθ

f (e  C|t|2b , ) − f (eiθ )

2π

for some C > 0. Then f ∈ pA where p =

t ∈ R,

2 2a+1 .

Proof. By Parseval’s identity, the condition (6.2.4) can be expressed as (6.2.5)





(eikt − 1)fk 2  C|t|2b ,

t ∈ R,

k0

where fk is the kth Taylor coefficient of f . We will also make use of the elementary bound

(6.2.6)

eis − 1

2





, π s

0 < s  π.

6. p AS A FUNCTION SPACE

76

Fix a nonnegative integer n. With t = 2−(n+1) and 2n + 1  k  2n+1 , we have kt < 1 and so (6.2.6) holds with s = kt. It follows that for any u > 0, n+1 2

ku |fk |2 

k=2n +1

n+1 2

k=2n +1

 π 2 2

eikt − 1 2



ku

|fk |2 kt

π 2 2(n+1)u  2 n 2 (2 + 1)2 |t|2 

n+1 2

ikt

(e − 1)fk 2

k=2n +1

2 π 2 2(n+1)u 

ikt (e − 1)fk

2 2 + 1) |t|

22 (2n

k1



|t| Cπ 2 22 (2n + 1)2 |t|2

=

Cπ 2 2(n+1)(u+2−2b) 22 (2n + 1)2

2 (n+1)u

2b

 Cπ 2 2(n+1)(u−2b) ,

(6.2.7)

where in the second step we have used (6.2.5). With the choice u = a + b > 0, it turns out that u − 2b = a − b < 0, and then the last expression (6.2.7) is summable in n. One can also check that for such u up a+b = > 1. 2−p 2a By definition, p must satisfy 23 < p < 2, and hence 1 < p2 < 3. We may therefore apply H¨ older’s inequality using the pair of conjugate exponents 2/p and 2/(2 − p). The above observations and estimates say that   1 |fk |p = |fk |p kup/2 · up/2 k k1 k1 p/2   (2−p)/2  1  |fk |2 ku up/(2−p) k k1 k1 

n+1    2

n0





|fk |2 ku

p/2  

k=2n +1

Cπ 2 2(n+1)(u−2b)

n0

1

(2−p)/2

kup/(2−p) 1/2   (2−p)/2 1 k1

k1

kup/(2−p)

. 

which is finite. This proves the result.

Let us now identify some other interesting functions for which the power series expansions are not readily available, yet we still can determine if they belong to pA . Newman and Shapiro proved some results about the growth of the Taylor coefficients of certain inner functions (recall the discussion following Theorem 5.1.9). Here are two of these theorems and their relationship to pA . Any finite Blaschke product B(z) =

N % zn − z , 1 − zn z n=1

z1 , . . . , zN ∈ D,

6.2. SOME INHABITANTS OF p A

77

is a rational function that is analytic in a neighborhood of D and thus belongs to pA (Proposition 6.2.2). Can an infinite Blaschke product % zn zn − z , B(z) = z N |zn | 1 − zn z n1  where zn ∈ D \ {0} and n1 (1 − |zn |) < ∞, belong to pA ? Theorem 6.2.8 (Newman and Shapiro). There is an infinite Blaschke product  B(z) = an z n with |an | = O( n1 ) and thus B ∈

&

n0

pA .

p>1

We will follow the original proof of Newman and Shapiro which requires a few technical details. We say that a Blaschke sequence {zn }n1 ⊆ D is uniformly separated if % z −z



j n

δ := inf

: n  1 > 0. 1 − zj zn j1 j =n

The constant δ is called the uniform separation constant. Lemma 6.2.9. Let {zn }n1 ⊆ D \ {0} be uniformly separated with separation constant δ and let % |zn | zn − z B(z) := . zn 1 − zn z n1

Then |B  (zn )| 

δ/2 , 1 − |zn |

n  1.

Proof. The logarithmic derivative of B is B  (z)  −1 + |zn |2 = , B(z) (zn − z)(1 − zn z) n1

which implies B  (z) =

 1 − |zn |2 −|zn | % |zj | zj − z . (1 − zn z)2 zn j1 zj 1 − zj z

n1

j =n

Therefore, B  (zn ) =

−|zn | % |zj | zj − zn 1 . 2 1 − |zn | zn j1 zj 1 − zj zn j =n

Taking the absolute value of both sides of the above, we obtain %

zj − zn

1 |B  (zn )| =

. 1 − |zn |2 j1 1 − zj zn j =n

Hence, |B  (zn )| 

δ/2 1 − |zn |

n  1.



6. p AS A FUNCTION SPACE

78

Definition 6.2.10. A sequence {zn }n1 ⊆ D is called exponential if there is a constant c ∈ (0, 1) such that 1 − |zn+1 |  c(1 − |zn |),

n  1.

Lemma 6.2.11. An exponential sequence is uniformly separated. Proof. Let {zn }n1 ⊆ D be an exponential sequence. By induction, 1 − |zj |  cj−k (1 − |zk |),

j > k  1.

This inequality yields several conclusions. First, when k = 1, we see that {zj }j1 is a Blaschke sequence and thus the Blaschke product with these zeros is well defined. Second, for j > k, |zj | − |zk | = (1 − |zk |) − (1 − |zj |)  (1 − cj−k )(1 − |zk |). Third, again for j > k, 1 − |zj zk | = (1 − |zj |) + |zj |(1 − |zk |)  (1 + cj−k )(1 − |zk |).



α − β |α| − |β|



1 − αβ  1 − |αβ| ,

Since

we see that for j > k,

Therefore,

α, β ∈ D,



zj − zk |zj | − |zk | 1 − cj−k



1 − zj zk  1 − |zj zk |  1 + cj−k .

% 1 − cj−k % 1 − cm %

zj − zk



= .

1 − zj zk  j−k 1+c 1 + cm

jk+1

m1

jk+1

By the same token, k−1 %

zj − zk

k−1 % |zk | − |zj | k−1 % 1 − ck−j % 1 − cm

   .

1 − zj zk

1 − |zj zk | 1 + ck−j 1 + cm j=1 j=1 j=1 m1

Combining these two estimates yields ∞ % |zj − zk | j1

|1 − zj zk |



 % 1 − cm 2 . 1 + cm m1

j =k

Since c ∈ (0, 1), the last product above is convergent to a number in (0, 1).



By the above two lemmas, we conclude that if {zn }n1 ⊆ D is an exponential sequence and B is the Blaschke product with those zeros, then there is a constant b > 0 such that b (6.2.12) |B  (zn )|  , n  1. 1 − |zn | The next detail needed to prove Theorem 6.2.8 is this real analysis lemma. Lemma 6.2.13. If u is a nonnegative C 1 function on [0, ∞) with lim tu(t) = lim tu (t) = 0,

t→∞

then

 k1

t→∞

u(k)  0

∞





∞

|u (t)| dt +

u(t) dt. 0

6.2. SOME INHABITANTS OF p A

79

Proof. Let t denote the integer part of t. By Riemann-Stieltjes integration, along with integration by parts, we have ∞ ∞  u(k) = u(t) dt = − tu (t) dt 0

k1



0 ∞





∞

(t − t)u (t) dt − tu (t) dt 0 0 ∞ ∞  |u (t)| dt + u(t) dt. =

0



0

Proof of Theorem 6.2.8. Let B be the Blaschke product whose zeros are zk = 1 − e−k , Then if B(z) =



k  1. an z n

n0

we can use Theorem 5.1.8 to see that n an = B(ξ)ξ dm(ξ). T

Let us now develop an another formula for an . If BN (z) :=

N % |zk | zk − z zk 1 − zk z

k=1

is the N th partial product of B, then lim BN (ξ) = B(ξ),

N →∞

ξ ∈ T \ {1}.

Hence, by the Dominated Convergence Theorem, n n an = B(ξ)ξ dm(ξ) = lim BN (ξ)ξ dm(ξ). N →∞

T

T

The Cauchy Residue Theorem yields ' N  zkn−1 1 ξ n−1 n dξ = BN (ξ)ξ dm(ξ) =  (z ) . 2πi T BN (ξ) BN k T k=1

Again the Dominated Convergence Theorem (discrete version) and the inequality  |BN (zk )| 

imply that an =

δ/2 , 1 − |zk |

 z n−1 k . B  (zk )

k1

Now bring in (6.2.12) to obtain the estimate 1 1  −k (1 − zk )zkn = e (1 − e−k )n , |an+1 |  b b k1

n  1.

k1

For a fixed integer n  1 let u(t) := e−t (1 − e−t )n ,

t  0.

Notice that the function u(t) will have one maximum at M (see Figure 1) and thus

6. p AS A FUNCTION SPACE

80 





 
















Figure 1. The graph of e−t (1 − e−t )n .

∞

|u (t)| dt =

0



M

u (t) dt −

0



∞

u (t) dt

M

= u(M ) − u(0) − (u(∞) − u(M )) = 2 max{u(t) : t  0} 1 n 2  1− = n+1 n+1 1 , =O n and



∞

1 . n+1 0 Now apply Lemma 6.2.13 to complete the proof. u(t) dt =



Remark 6.2.14. The construction above can be extended to any sequence {wk }k1 ⊆ D such that   1 − |w k+1 | :k1 0, the atomic inner function  z + 1  = f (z) = exp t an z n z−1 n0 & p satisfies |an | = O(n−3/4 ), and thus f ∈ A . p>4/3

The proof of this result requires some classical results on the coefficients of Laguerre polynomials which would take us somewhat far away from the focus of the book. Thus we refer the reader to the original paper for the details.

6.3. RELATIONSHIP TO H p SPACES

81

6.3. Relationship to H p Spaces As mentioned in our earlier discussions, there is a relationship between pA and the Hardy spaces H p from Chapter 5. Let us start with p = 2. Proposition 6.3.1. The space 2A is equal to H 2 with the same norm. Proof. Notice that f ∈ H 2 if and only if |f (rξ)|2 dm(ξ) < ∞. sup If f (z) =



0K

|ak |k

kK

 ACf,N + A

N



+A



|ak |p kN p |z|kp

k>K

|ak |p kN p |z|kp

k>K

   ACf,N + AC sup kN p |z|kp |ak |p k0

k0

 + AC sup kN p |z|kp . 

 ACf,N

k0

For fixed r ∈ (0, 1) we are left to estimate sup kN p r kp . k0

Observe that

d N p kp (k r ) = pkN p−1 r kp (k log r + N ). dk Setting the above equal to zero and solving for k (considering k to be a variable on (0, ∞)) yields N . k= log(1/r) Thus N p N p  N  N sup kN p r kp = r (N/ log(1/r))p = . log(1/r) e log(1/r) k0 Since log(1/r) =1 lim − 1−r r→1 we obtain  1 N p sup kN p r kp  B 1−r k0

for some B > 0. This yields f (N ) (z) = O



 1 , N p (1 − |z|)

z ∈ D.

We conclude from Proposition 6.6.3 that f (N −1) satisfies (6.6.10).



Remark 6.6.11. The previous result shows for example that if f ∈ pA for some p ∈ (0, 1) then there is an α > 0 such that |f (ξ) − f (ζ)|  C|ξ − ζ|α ,

ξ, ζ ∈ T.

In other words, the boundary function of f satisfies a Lipschitz condition on T. It turns out [55, Lemma 4] that |f (z) − f (w)|  C|z − w|α ,

z, w ∈ D.

That is, the Lipschitz criterion holds on all of D. Recall the evaluation functional Λw from (6.4.4).

6.7. NOTES

87

Proposition 6.6.12. For p ∈ (0, 1] we have Λw  = 1 for every w ∈ D, that is to say, sup{|f (w)| : f p  1} = 1. Proof. By the inequality in (6.6.2) we have sup{|f (z)| : f ∈ pA , f p  1}  1. Now use f ≡ 1 as a test function in the above to obtain equality.



pA ,

p ∈ (0, 1], is a complete vector space From Proposition 6.6.9 we see that that is contained in a class of functions that are smooth up to the boundary, of varying smoothness classes depending on p. We will now show that in addition, pA is an algebra, meaning that if a, b ∈ pA , then ab ∈ pA . Theorem 6.6.13. When p ∈ (0, 1] the space pA is an algebra. Furthermore, abp  ap bp for all a, b ∈ pA . Proof. If a(z) =



ak z k ,

b(z) =

k0

then a(z)b(z) =



bk z k ,

k0



ck z k ,

ck =

k 

aj bk−j .

j=0

k0

Repeated application of Proposition 1.4.1 yields k k k



p    p  



a b  |a ||b |  |aj |p |bk−j |p .

j k−j

j k−j j=0

j=0

j=0

Consequently, abpp =

k k

p     



aj bk−j  |aj |p |bk−j |p

k0 j=0

=

 j0

k0



j=0

|aj |p |bk−j |p = app bpp .



kj

We point out that when p = 1, the algebra 1A is called the Wiener algebra and has been studied quite extensively. We will cover selections from this area in Chapter 13. 6.7. Notes The proof of Lemma 6.2.11 follows [54, p. 155]. Further details of the Newman and Shapiro theorems (Theorem 6.2.8 and Theorem 6.2.15) are found in [129]. Theorem 6.2.15 is sharp since the singular inner function in that result belongs to pA for p ∈ ( 34 , ∞) but, by a paper of Gurari˘ı [72] does not belong to pA when p ∈ [1, 43 ]. See [54] for a proof of Littlewood’s (Proposition 6.5.2). The original paper of Littlewood is [116]. For other versions of the Hardy-Littlewood result (Proposition 6.6.3), the reader should consult [54]. See [99] for an extensive survey on the Wiener algebra as well as a selection of results in Chapter 13 of this book. In Section 6.6 we proved that when p ∈ (0, 1), every function in pA is smooth up to the boundary in that, depending on p, functions in pA have extensions to T which satisfy some Lipschitz smoothness condition (Proposition 6.6.9). Interestingly, however, the evaluational functional (f ) = f  (1) is not continuous, even when p is small

88

6. p AS A FUNCTION SPACE

enough for f  to be continuous on T. Observe that (z n ) = n but z n p = 1, n  0, which makes this functional unbounded. Thus, even though pA is contained in a smoothness (Lipschitz) class, the embedding is not continuous. We refer the reader to papers of Dyakonov [55], Havin and Shamoyan [80], and Shirokov’s book [156] for more on smoothness classes of analytic functions.

10.1090/ulect/075/07

CHAPTER 7

Some Operators on pA Operator theory on function spaces has been an important part of analysis ever since Beurling’s description of the invariant subspaces for the shift operator on the Hardy space. This subject is fascinating since it demonstrates the strong connection between the operator theory aspects of the linear transformations acting on the space (spectrum, norm, eigenvalues, invariant subspaces, commutant) and the complex analysis features of the elements in the ambient space of functions (zeros, poles, analytic continuation, boundary behavior). In this chapter, we begin to explore the operators which act on pA . Some of these operators such as the shift, backward shift, and multiplication operators will be examined in greater detail in later chapters. 7.1. The Shift Operator The shift operator plays an important role in both operator theory and complex analysis. With Beurling’s work as inspiration, the shift operator on various Hilbert spaces of analytic functions often serves as a model for many types of operators. Indeed, classes of subnormal operators and m-isometric operators are modeled by shifts on certain canonical Hilbert spaces of analytic functions. Then there are the masterpieces of operator theory, the Sz.-Nagy-Foia¸s and de Branges-Rovnyak machineries, which model large classes of contractions as compressions of shifts to corresponding analytic function spaces. For Banach spaces of analytic functions, the shift plays less of a representational role but still maintains an important presence. If the Banach space of analytic functions happens to be an algebra, then the invariant subspaces for the shift often become ideals and there is a large industry in characterizing these ideals and exploring their properties. Though we will have much more to say about the shift on pA in Chapter 10, for now we give some basic facts about this important operator. Definition 7.1.1. For p ∈ (1, ∞) define the shift operator S : pA → pA ,

(Sf )(z) = zf (z).

“shift” comes from the fact that multiplying the power series f (z) =  The name k a z by the independent variable z shifts its Taylor coefficients one step to k k0 the right: z(a0 + a1 z + a2 z 2 + · · · ) = 0 + a0 z + a1 z 2 + a2 z 3 + · · · . Recall the concepts of the spectrum σ(T ) and adjoint T ∗ of a bounded operator T on a Banach space. Proposition 7.1.2. For p ∈ (1, ∞) we have the following. (i) S is an isometry on pA . (ii) σ(S) = D. 89

90

7. SOME OPERATORS ON p A 

(iii) With the sesquilinear pairing between pA and pA given by (f, g) = lim f (rξ)g(rξ) dm(ξ), − r→1

the adjoint S ∗ of S is

T

 g(z) − g(0) , g ∈ pA . (S ∗ g)(z) = z  Proof. If f (z) = k0 ak z k ∈ pA , then p     Sf pp =  ak z k+1  = |ak |p = f pp

k0

p

k0

and so f is an isometry. This proves (i). To show that σ(S) = D, first observe that the Spectral Radius Formula, and the fact that S = 1 from (i), show that σ(S) ⊆ D. For the reverse inclusion, note that if λ ∈ D then (S − λI)pA ⊆ {f ∈ pA : f (λ) = 0}. Since point evaluations are continuous on pA (recall (6.1.3)), the closure of (S − λI)pA is a proper subspace of pA . Thus λ ∈ σ(S). Since σ(S) is a compact set, we must have D ⊆ σ(S). This proves (ii). The operator g(z) − g(0) (Bg)(z) = = b1 + b2 z + b3 z 2 + · · · , z   where g(z) = k0 bk z k , is a well defined contraction on pA , and for any f (z) =  p k k0 ak z ∈ A we have (f, Bg) = a0 b1 + a1 b2 + · · · . The power series (Sf )(z) = 2 a0 z + a1 z + a2 z 3 + · · · yields (Sf, g) = a0 b1 + a1 b2 + · · · 

and so (Sf, g) = (f, Bg) for all f ∈ pA , g ∈ pA . Thus B = S ∗ , verfying (iii). The matrix representation of S with ⎛ 0 0 ⎜1 0 ⎜ ⎜ (7.1.3) S = ⎜0 1 ⎜0 0 ⎝ .. .. . . while the matrix representation of S ∗ is ⎛ 0 ⎜0 ⎜ ⎜ S ∗ = ⎜0 ⎜0 ⎝ .. .

respect to the basis {1, z, z 2 , . . .} for ⎞ 0 0 ··· 0 0 · · ·⎟ ⎟ 0 0 · · ·⎟ ⎟ 1 0 · · ·⎟ ⎠ .. .. . . . . .

 pA

is



with respect to the basis {1, z, z 2 , . . .} for pA ⎞ 1 0 0 ··· 0 1 0 · · ·⎟ ⎟ 0 0 1 · · ·⎟ ⎟. 0 0 0 · · ·⎟ ⎠ .. .. .. . . . . . . Strictly speaking, the operators S and S ∗ depend on the parameter p but we will use the same symbol for all of then. There are several other operator theory concepts one can consider about the shift S and backward shift S ∗ . For example, there is the commutant {A ∈ B(pA ) : AS = SA} of S which we will take up in Chapter 12. There are also the subspaces M ⊆ pA for which SM ⊆ M . Beurling

7.2. THE DIFFERENCE QUOTIENT OPERATOR

91

described these S-invariant subspaces M when p = 2, but there does not seem to be a description of these subspaces when p = 2. All of this will be taken up in Chapter 10. We will discuss the invariant subspaces of S ∗ in Chapter 11. 7.2. The Difference Quotient Operator For f ∈ Hol(D) and fixed w ∈ D, the difference quotient f (z) − f (w) z−w defines an analytic function on D. Also notice that f → Qw f defines a linear transformation on Hol(D) denoted by Qw . With a bit more work, one can show this difference quotient operator is well defined on pA . (Qw f )(z) :=

Proposition 7.2.1. Let p ∈ [1, ∞] and w ∈ D. If f ∈ pA , then Qw f ∈ Moreover, Qw defines a bounded operator on pA whose operator norm satisfies Qw   (1 − |w|)−1 .

pA .

Proof. Recall that

f (z) − f (0) z is the backward shift on pA . In the discussion below, I will denote the identity operator on pA . Since σ(B) = D (Proposition 7.1.2) the operator I −wB is invertible on pA . We claim that for w ∈ D (Bf )(z) =

Qw = B(I − wB)−1 . To see this, suppose that f ∈ pA , and let g = B(I −wB)−1 f . Then Bf = (I −wB)g, that is, (Bf )(z) = g(z) − w(Bg)(z), z ∈ D.  This identity can be written in series form, with f (z) = k0 fk z k and g(z) =  k k0 gk z , as    fk+1 z k = gk z k − w gk+1 z k , k0

k0

k0

which yields fk+1 = gk − wgk+1 ,

k  0.

Accordingly we have gk = fk+1 + wfk+2 + w2 fk+3 + · · · ,

k  0.

Thus for every z ∈ D it must be that     g(z) = gk z k = fl wl−k−1 z k k0

=



k0

fl (z

l−1

+z

l1

=

 l1

fl

z l − wl z−w

f (z) − f (w) z−w = (Qw f )(z). =

lk+1

l−2

w + · · · + wl−1 )

92

7. SOME OPERATORS ON p A

This proves that Qw is a bounded operator on pA . Furthermore, from the fact that B = 1 (Proposition 7.1.2), the operator norm of Qw satisfies Qw  = B(I − wB)−1   B(I − wB)−1   |w|k Bk  B k0

1 .  1 − |w|



7.3. Hadamard Multipliers In linear algebra, one is familiar with the concept of Hadamard multiplication of matrices A B, where A and B are the same size matrix and the (j, k) entry of A B is the product of the corresponding entries of A and B. We can develop a similar notion for Taylor series. Definition 7.3.1. Let f, g ∈ Hol(D) with Taylor series expansions   f (z) = an z n and g(z) = bn z n . n0

n0

The Hadamard product of f and g is defined by  an bn z n . (f g)(z) = n0

It follows from the Cauchy-Hadamard Formula for the radius of convergence that f g ∈ Hol(D). Definition 7.3.2. We say that h ∈ Hol(D) is a Hadamard multiplier of pA if the linear transformation Mh : pA → pA ,

Mh f = h f

is well defined and continuous. Theorem 7.3.3. For each p ∈ (1, ∞) the Hadamard multiplier space of pA is ∞ isometrically isomorphic to ∞ A . More explicitly, for each h ∈ A , the operator Mh p is bounded on A and Mh  = h∞ . Conversely, if Mh  < ∞, then h ∈ ∞ A.  n ∞ Proof. First suppose that h(z) = n0 an z ∈ A . Then for each f (z) =  p n n0 bn z ∈ A , we have Mh f p =



|an bn |p

1

p

n0





 

sup |an | n0

|bn |p

1

p

n0

= h∞ f p . Thus h is a Hadamard multiplier and Mh   h∞ . To show that equality holds, consider the monomials fm (z) = z m , m  0. Then fm p = 1 and 1  p p |an bn | = |am |. Mh f p = n0

7.4. ISOMETRIES ON p A

93

Thus Mh   |am |, m  0, which implies the reverse inequality Mh   h∞ . Conversely, suppose that Mh  < ∞. Then Mh f p  Mh  f p ,

f ∈ pA .

If we apply the inequality above to the monomials fm , we obtain |am |  Mh , m  0. Taking a supremum in m shows that h ∈ ∞  A. The characteristics of the coefficient sequence for h are connected to the behavior of the operator Mh . For instance, we can also classify the compact Hadamard multipliers.  Proposition 7.3.4. Let p ∈ (1, ∞) and h(z) = n0 an z n ∈ ∞ A . Then Mh is a compact operator on pA if and only if an → 0. N Proof. Suppose an → 0 and define hN (z) = n=0 an z n . Note that MhN is a finite rank operator on pA (the nth Taylor coefficient of hN f vanishes when n > N ). By Theorem 7.3.3, the operator Mh is the limit of MhN in the operator norm topology. To see this, given > 0, there exists an index N such that |an | <

whenever n  N . Thus Mh − MhN  = h − hN ∞ = sup |an | < . n>N

Since Mh is the norm limit of a sequence of finite rank operators (which are compact) it follows that Mh is a compact operator. Conversely, suppose that Mh is compact. Note that the sequence of monomials z n converges weakly to zero in pA . Since pA is reflexive, the compact operator Mh is completely continuous and so Mh z n p → 0. But since Mh z n = h z n = an z n , we conclude that an → 0.  7.4. Isometries on pA The following theorem of Lamperti describes the isometries of pA for p ∈ (1, ∞)\ {2}. This result is really one about the sequence space p and analyticity plays no role. Theorem 7.4.1 (Lamperti). Suppose that p ∈ (1, ∞) \ {2}. An operator U on pA is an isometry if and only if for each k  0 there exist pairwise disjoint subsets Ik = {nk,1 , nk,2 , nk,3 , · · · } of N0 , and constants uk,1 , uk,2 , uk,3 , . . . satisfying  |uk,m |p , 1= nk,m ∈Ik

such that U (z k ) =



uk,m z nk,m .

nk,m ∈Ik

We will prove this via a series of steps. Lemma 7.4.2. Suppose that p ∈ (1, ∞) \ {2}. Let a, b ∈ pA . The condition (7.4.3)

a + bpp + a − bpp = 2(app + bpp )

holds if and only if ak bk = 0 for every k  0.

7. SOME OPERATORS ON p A

94

Proof. If ak bk = 0 for every k, then a ± bpp = app + bpp , and so (7.4.3) holds. For the converse, first assume that p ∈ (1, 2). By Lemma 3.1.5, (7.4.4)

|ak + bk |p + |ak − bk |p  2(|ak |p + |bk |p )

holds for every k  0. By summing over k, we find that a + bpp + a − bpp  2(app + bpp ). By hypothesis, equality holds here, so it must be that equality holds in (7.4.4) for every k. By Lemma 3.1.5 again, this implies that ak bk = 0 for all k. The p ∈ (2, ∞) case is treated in the same way.  The condition ak bk = 0 for all k is equivalent to saying that corresponding sequences a and b are disjointly supported, when viewed as functions on the index set N0 . Lemma 7.4.5. Suppose that p ∈ (1, ∞) \ {2}. If U is an isometry on pA , then U (z ) and U (z k ) are disjointly supported whenever j = k in N0 . j

Proof. When j = k, the vectors z j and z k are disjointly supported. Lemma 7.4.2 implies that z j + z k pp + z j − z k pp = 2(z j pp + z k pp ). Since U is an isometry, we have U z j + U z k pp + U z j − U z k pp = 2(U z j pp + U z k pp ). Lemma 7.4.2 now ensures that U z j and U z k are disjointly supported.



Proof of Theorem 7.4.1. Assume U is an isometry on p . For k  0 let Ik ⊆ N0 be the support of the sequence corresponding to U z k . Thus there are constants uk,1 , uk,2 , uk,3 , . . . and distinct indices nk,1 , nk,2 , nk,3 , · · · such that Ik = {nk,1 , nk,2 , nk,3 , · · · }  uk,m , j = nk,m ∈ Ik , k [U (z )]j = 0, j ∈ / Ik , where [U (z k )]j is the jth term in the Taylor expansion for U (z k ). The index subsets {Ik }k∈N0 are pairwise disjoint, according to Lemma 7.4.5. The condition of U being isometric requires 1 = z p = U z p = k

k

 

|uk,m |

nk,m ∈Ik

Thus U has the representation (7.4.6)

U (z k ) =

 nk,m ∈Ik

as claimed.

uk,m z nk,m ,

p

1

p

.

7.5. COMPOSITION OPERATORS

95

Conversely, if U is given by (7.4.6), and a ∈ pA , then   p p       U app = U ak z k  =  ak U (z k ) p

k0

  ak =



k0

nk,m ∈Ik

=





p

k0

p  uk,m z nk,m  p

|ak |p |uk,m |p

k0 nk,m ∈Ik

=



|ak |p · 1 = app ,

k0

verifying that U is an isometry. This completes the proof of Theorem 7.4.1.



Remark 7.4.7. It is worth mentioning that the sets Ik , though disjoint, need not be a partition of N0 . When p = 2, then 2A is a Hilbert space and there is a richer collection of isometries than those arising from Theorem 7.4.1. There is also the Wold decomposition for isometries on a Hilbert space which has no analogue in a general normed space. 7.5. Composition Operators If ϕ : D → D is analytic, when does the composition f ◦ ϕ belong to pA for all f ∈ pA ? In other words, when is the linear transformation Cϕ f = f ◦ ϕ a bounded operator on pA ? Here is a very basic result. Proposition 7.5.1. Suppose ϕ is an analytic self-map of D for which f ◦ϕ ∈ pA for every f ∈ pA . Then the composition operator Cϕ : pA → pA is bounded and ϕn p  Cϕ  for all n  0. Proof. If f ◦ ϕ ∈ pA for all f ∈ pA , an application of the Closed Graph Theorem will show that Cϕ defines a bounded composition operator on pA . Thus for any n  0 we have ϕn p = Cϕ z n p  Cϕ z n p = Cϕ .



When p = 2 we have 2A = H 2 (Proposition 6.3.1) and there is the following well-known theorem. Theorem 7.5.2 (Littlewood Subordination Theorem). If ϕ : D → D is analytic, then the composition operator Cϕ is a bounded operator on 2A . When p = 2, there seem to be severe restrictions on ϕ for the composition operator Cϕ to be bounded on pA . Here is one positive result which works for the Wiener algebra 1A and very much uses the multiplicative properties of 1A (Theorem 6.6.13). Proposition 7.5.3. Suppose that ϕ : D → D is analytic and there is a constant M > 0 such that ϕn 1  M for all n  0. Then the composition operator Cϕ is bounded on 1A with Cϕ   M .

7. SOME OPERATORS ON p A

96

 Proof. Let f (z) = k0 ak z k ∈ 1A . For any nonnegative integers m and n with n  m we have n n n       ak ϕk   |ak |ϕk 1  M |ak |,  1

k=m



k=m

k=m

and thus this series k0 ak ϕ is convergent. In other words, f ◦ ϕ ∈ 1A . On the other hand, the partial sums converge uniformly on compact subsets of D to f ◦ ϕ. Hence if we write n  (n)   ak ϕk (z) = ak z k and (f ◦ ϕ)(z) = bk z k , k

k=0

k0 (n) ak

then, for each fixed k  0, bound 

→ bk as n → ∞. Therefore, by Fatou’s Lemma, the

n   (n) |ak | =  ak ϕk 1  M f 1

k0

implies

k0



k=0

|bk | = f ◦ ϕ1  M f 1 .

k0

It follows that Cϕ   M .



Here is a specific example of when this happens. Corollary 7.5.4. Suppose ϕ ∈ 1A and ϕ is non-constant with ϕ1  1. Then ϕ maps D to itself and the composition operator Cϕ is a contraction on 1A . Proof. From (6.6.1) we see that |ϕ(z)|  ϕ1  1 for all z ∈ D and so ϕ is an analytic self-map of D. To conclude that Cϕ is a contraction on 1A , use the proof of the previous proposition, along with Theorem 6.6.13, to conclude that ϕn 1  ϕn1  1 for all integers n  0.  Certainly ϕ(z) = (1 + z)/K for any K  2 satisfies the hypothesis of the previous corollary. We leave it to the reader to use the embedding of 1A into pA for p ∈ (1, ∞) (Proposition 6.1.5) to fashion similar composition operator results for pA . One must be quite careful about trying to extend this result to general analytic self-maps ϕ of D, even very nice ones. Theorem 7.5.5 (Alp´ar). If ϕ is a disk automorphism that is not a rotation, that is to say, z−w ϕ(z) = ξ , ξ ∈ T, w ∈ D \ {0}, 1 − wz then Cϕ does not map 1A to itself. Proof. Let w ∈ D \ {0} be fixed, and let z−w , z ∈ D. ϕ(z) := 1 − wz In fact, ϕ extends continuously to the boundary of the disk, and we retain the name ϕ for the boundary function, which is evidently continuous. For nonnegative integers k and positive integers n let 2π dθ tk,n := . ϕ(eiθ )k e−inθ 2π 0

7.5. COMPOSITION OPERATORS

97

That is, tk,n is the nth Fourier coefficient of the kth power of ϕ. Our first objective will be to show that for some positive constant C C (7.5.6) |tk,n |  1/3 n holds for all k and n. To this end, we introduce the notation ϕ(eiθ ) = eiF (θ) , where F is monotone increasing, and maps the interval (0, 2π) onto some interval (ψ, ψ + 2π). Evidently, by writing r := |w| and θ0 := arg w, we have 1 − r2 . 1 − 2r cos(θ − θ0 ) + r 2 A change of variable leads us to θ0 +2π 2π dθ dt = , eikF (θ)−inθ eikt−inG(t) G (t) tk,n = 2π 2π 0 θ0 F  (θ) =

in which G is the inverse function of F . We may check that F  (θ) vanishes only when θ = θ0 and θ = θ0 ± π; consequently G (t) vanishes only at the points t = F (θ0 ) and t = F (θ0 ± π). In addition, F  (θ) fails to vanish at the points θ = θ0 or θ = θ0 ± π, and similarly G (t) fails to vanish when t = F (θ0 ) or t = F (θ0 ± π). Thus if we omit from (ψ, ψ + 2π) the three intervals  1 1   1 1  F (θ0 ) − 1/3 , F (θ0 ) + 1/3 , F (θ0 + π) − 1/3 , F (θ0 + π) + 1/3 , n n n n and  1 1  F (θ0 − π) − 1/3 , F (θ0 − π) + 1/3 , n n then at the points t of the remaining at most three intervals, the union of which we call Γ, we have |G (t)| > C1 /n1/3 for some positive constant C1 not depending on n or k. To such points we now apply the Van Der Corput lemma: If u (t) is continuous and |u (t)| > ρ in (a, b), then

b 8



eiu(t) dt  1/2 .

ρ a Take u(t) = kt − nG(t), so that C1 |u (t)| = n|G (t)|  n 1/3 = C1 n2/3 . n The conclusion is that



8 8



= 1/2 .

eikt−inG(t) G (t) dt  2/3 1/2 (C1 n ) Γ C1 n1/3 On the other hand, the omitted intervals have total length at most 4n−1/3 , and so a similar bound is trivially satisfied. Thus if we write ψ+2π 1 (7.5.7) ck,n := eikt−inG(t) dt, 2π ψ

7. SOME OPERATORS ON p A

98

including the possibility of negative k, we have obtained  1  |ck,n | = O 1/3 n uniformly in k. Finally, the corresponding bound remains valid when the integrand of (7.5.7) is multiplied by G (t), since the latter is twice continuously differentiable, and thus has absolutely summable Fourier coefficients {dk }. Indeed, ∞ ∞

  1    1 



|tk,n | =

ck−l,n dl = O 1/3 |dl | = O 1/3 . n n l=−∞ l=−∞ This verifies (7.5.6). Using this bound we see that for each k,   1  |tk,n |  |tk,n |2 |tk,n |−1  |tk,n |2 n1/3 . C n0

n1

Let K > 0. Then



n1

|tk,n | 

n0

K C



|tk,n |2 .

n1/3 K

If we take k → ∞, then each tk,n tends to zero for each n, particularly those indices for which n1/3 < K. On the other hand, Parseval’s identity assures that 2π  1 |tk,n |2 = |ϕ(eiθ )|2k dθ = 1. 2π 0 n0

Thus, lim



|tk,n | 

k→∞ n0

K . C

 Since K > 0 was arbitrary, the sum n0 |tk,n | diverges to ∞ as k → ∞. Accordingly, let us set  |tk,n |, Tk := n0

and choose M > 0 so that |tk,n |  M for all k and n. We have shown above that Tk → ∞. We now inductively define integers km and nm , and positive numbers Am , as follows. Take k0 = n0 = 0, and A0 = 1. Having defined km−1 , nm−1 and Am−1 , choose Am so that nm−1  1 0 < Am < Am−1 and 2M Am  1. 2 n=0 With Am thus determined, select the index km so that Tk m >

m−1 1  Ukl + m. Am l=0

This is possible since Tk → ∞. A similar thing holds for partial sums for Tkm . That is, select an index nm sufficiently large that nm  n=0

|tkm ,n | >

m−1 1  Ukl + m. Am l=0

7.6. NOTES

99

We may now define akm := Am , and ak := 0 if k is not among the indices km , m  0. Immediate by the definition of Am , we have   |ak | = Am < ∞. m0

k0

∞

k

That is, the function f (z) := k=0 ak z belongs to 1A . Fix m and let n  nm . Then  (7.5.8) tk,n ak bn := k0

=



tkm ,n Am

m0

=

 m −1 



ukm ,n Am + ukm ,n Am +

ukm ,n Am .

mm +1

m=0

Isolating the middle term on the right we have  m −1



|ukm ,n Am | = bn − ukm ,n Am −



ukm ,n Am

 mm +1

m=0 

 |bn | +

m −1 

|ukm ,n | + 2M Am +1 .

m=0

Consequently, nm 

|bn |  Am

n=0

nm 

|ukm ,n | −

n=0

 m −1 

Tk m −

m=0

nm 

2M Am +1 .

n=0

The sum in the first term exceeds the second by at least m , while the third term is less than unity. Hence nm  |bn |  m − 1. n=0

Therefore the function g(z) :=



bn z n

n0

fails to lie in But by (7.5.8), g(z) = f (ϕ(z)). That is, there is a function f ∈ 1A such that f ◦ ϕ lies outside of 1A . This proves the claim.  1A .

We will take up composition operators on 1A again in Section 13.3. 7.6. Notes A good reference for Hadamard multipliers is the text [93]. Generally speaking, if the norm on a Banach space of analytic functions is defined in terms of its Taylor coefficients (Dirichlet space, Hardy space H 2 , Bergman space A2 , and pA spaces), determining the Hadamard multipliers is straightforward. However, when the norm is defined differently, for example, as an integral in the Hardy spaces H p , the Bergman spaces Ap , or the harmonically weighted Dirichlet spaces Dμ , determining the Hadamard multipliers becomes more difficult [54]. Lamperti’s characterization of the isometries on Lp is from his paper [106]. Theorem 7.4.1 concerns the special

100

7. SOME OPERATORS ON p A

case of p . Specializing further, Li and So [114] characterized the isometries on Rn , endowed with the Lp norm. There is also a characterization, found in papers of de Leeuw, Rudin, and Wermer [49] and Forelli [60] of the isometries of H p , p = 2. See [54] for a proof of the Littlewood Subordination Theorem (Theorem 7.5.2). Theorem 7.5.5 comes from Alp´ar [11] and further pathologies are found in Hal´asz [74, 75]. The Van Der Corput Lemma, used in the proof of this theorem, is from the book [158]. See the work of Kahane [99] and Newman [127] for composition operator results of a more positive nature. Composition operators on other function spaces are discussed in the texts [45, 154].

10.1090/ulect/075/08

CHAPTER 8

Extremal Functions Extremal problems are a ubiquitous challenge throughout mathematics. They are driven by numerous applications and make contact with other important issues. In function theory, for example, extremal problems give us insight into the zero sets of various function spaces. This has certainly been true of the Hardy and Bergman spaces. Therefore, let us attend to several extremal problems for pA as we head towards our study of the zeros of these functions in the next chapter. Our treatment will be limited to the p ∈ (1, ∞) case as this ensures that the extremal problems have unique solutions, and that we may apply the tools from Banach space geometry that were developed in Chapters 3 and 4. 8.1. Zero Sets For an f ∈ Hol(D) \ {0}, consider the set {z ∈ D : f (z) = 0}, the zeros of f . From the Uniqueness Theorem for analytic functions, we know this set must be a sequence with no accumulation points in D. Since an individual zero z0 of f must be isolated, there is an open disk D(z0 , r) ⊆ D containing z0 and no other zeros of f . Furthermore, by the power series representation of f about z0 , there is a positive integer m0 and a nonvanishing analytic function g on D(z0 , r) for which f (z) = (z − z0 )m0 g(z),

z ∈ D(z0 , r).

This integer m0 is called the order of the zero z0 . Equivalently, z0 is a zero of f of order m0 if and only if f (z0 ) = f  (z0 ) = f  (z0 ) = · · · = f (m0 −1) (z0 ) = 0,

f (m0 ) (z0 ) = 0.

As is traditional in complex analysis, when we list the zeros W = {wk }k1 of an analytic function, we list the zero of order m in the sequence m times. The language often used here is to say that the zero is repeated according to multiplicity. Definition 8.1.1. Let f ∈ Hol(D) \ {0}. The zero set of f is the sequence W = {wk }k1 of points in D on which f vanishes, multiplicities taken into account. If F is a family of analytic functions on D, then we say that a nonempty W ⊆ D is a zero set of F if there is a function f ∈ F such that W is the zero set of f . Remark 8.1.2. The term “zero set” is a misnomer, since it refers not merely to a set of points, but also to the multiplicities attached to those points. We use the term “zero sequence” synonymously although it too is inaccurate, since listing the zeros and repeating them in accordance with multiplicity does not need to be done in any particular order. This abuse of language is standard and appears in most texts. 101

102

8. EXTREMAL FUNCTIONS

Ideally, we would like to characterize the zero sets of pA . However, the tools and methods that we have available connect to a weaker concept. This leads us to make the following nonstandard definition. Definition 8.1.3. Let F be a family of analytic functions on D. Suppose W = {wk }k1 ⊆ D and mk denotes the number of times wk appears in W . Then W is called a pre-zero set for F if there exists an f ∈ F \ {0}, such that f (j) (wk ) = 0,

0  j  mk − 1,

k  1.

Thus we say that W is a zero set for pA when there is a function from pA whose zeros, counting multiplicity, are precisely W . On the other hand, W is a pre-zero set if there exists some function in pA \ {0} that vanishes on at least W . This function could vanish at other points of the disk or, for a particular point in W , this function might have a zero of higher multiplicity than in W . A pre-zero set for pA is contained in a zero set for pA . Further toward our preparation for studying pA , we introduce a class of subspaces connected to prospective zero sets. Definition 8.1.4. If W = {wk }k1 ⊆ D and mk denotes the number of times wk appears in the sequence, define   (8.1.5) RW := f ∈ pA : f (j) (wk ) = 0, 0  j  mk − 1, k  1 . Since convergence in the norm of pA implies uniform convergence on compact subsets of D, via (6.1.3), RW is a subspace of pA . Let us emphasize that when we use the term “subspace” we are assuming not only that the set is closed under addition and scalar multiplication, but it is also norm closed. We readily see that W is a pre-zero set for pA if and only if RW = {0}. Our discussion of the pre-zero sets for pA , which will be taken up in the next chapter, involves the properties of solutions to several connected extremal problems. This will be the subject of the next few sections. 8.2. Solving an Extremal Problem For p ∈ (1, ∞) and a nonempty pre-zero set W ⊆ D \ {0} for pA , that is, RW = {0}, consider the following two extremal problems: inf {gp : g(0) = 1, g ∈ RW } ;

(8.2.1)

sup {|f (0)| : f p = 1, f ∈ RW } .

(8.2.2) Recall that if f ∈

pA

then Qw f ∈ pA , where

f (z) − f (w) z−w (Proposition 7.2.1). If f ∈ RW then, since 0 ∈ W , it must be that QN 0 f ∈ RW for some positive integer N . Thus the supremum in (8.2.2) is positive. A similar analysis shows that the set {g ∈ RW : g(0) = 1} is nonempty. Our first result shows that the infimum in (8.2.1) is attained and thus the infimum can be replaced by a minimum. (Qw f )(z) =

Proposition 8.2.3. Let p ∈ (1, ∞). For each pre-zero sequence W ⊆ D \ {0} for pA there is a unique gW ∈ RW with gW (0) = 1 and gW p = inf {gp : g(0) = 1, g ∈ RW } .

8.2. SOLVING AN EXTREMAL PROBLEM

103

Proof. The set {g ∈ RW : g(0) = 1} is nonempty, closed, and convex. Since we established earlier in Theorems 3.1.10 and 3.1.17 that pA is uniformly convex, there will be a unique element gW ∈ RW with gW (0) = 1 of minimal norm.  Next we establish that the supremum in (8.2.2) is attained and thus the supremum can be replaced by a maximum. Proposition 8.2.4. Let p ∈ (1, ∞) and let W ⊆ D \ {0} be a pre-zero sequence for pA . Then there is an fW ∈ pA with fW p = 1 such that |fW (0)| = sup{|f (0)| : f p = 1, f ∈ RW }. Furthermore, up to a multiplicative unimodular constant, fW is the unique solution to this extremal problem. Proof. Define M = sup {|f (0)| : f p = 1, f ∈ RW } ; I = inf {gp : g(0) = 1, g ∈ RW } . Let {fn }n1 ⊆ {f ∈ RW : f p = 1} be a sequence for which |fn (0)| → M . We may safely assume that each fn (0) is not zero and define gn (z) =

fn (z) . fn (0)

Notice that gn (0) = 1 and gn ∈ RW . By definition I  gn p =

1 fn p = , |fn (0)| |fn (0)|

I

1 1 = lim . n→∞ M |fn (0)|

n  1.

Thus (8.2.5)

On the other hand, let gW be the unique solution to the extremal problem in (8.2.1) (Proposition 8.2.3) and define k(z) =

gW (z) . gW p

Thus kp = 1 and k ∈ RW . Accordingly, 1 1 = = k(0)  M. I gW p Combine this inequality with the one in (8.2.5) to see that IM = 1. Moreover, if fW is a solution to (8.2.2) then gW (z) :=

fW (z) fW (0)

is a solution to (8.2.1). By taking the pA norm of both sides we get |fW (0)| =

fW p 1 = = M. gW p I

104

8. EXTREMAL FUNCTIONS

Let fW (0) = ξM (ξ ∈ T, M = |fW (0)|) be the polar decomposition of the complex number fW (0). Then by Proposition 8.2.3, gW is unique and fW (z) = fW (0)gW (z) = ξM gW (z) and thus fW is unique up to a multiplicative unimodular constant.



To give the concept of “inner” in the next section better context, let us work through the solution of the analogous extremal problems for the Hardy space H p . In this setting consider the extremal problem sup{|f (0)| : f H p = 1, f ∈ HW },

(8.2.6)

where W = {wn }n1 ⊆ D \ {0} is a zero set for H p and HW is the subspace of H p consisting of functions in H p which vanish on at least W , taking into account multiplicity. The advantage we have with H p over pA is the factorization from Theorem 5.1.9 which allows us to write any f ∈ H p with f (0) = 0 as (8.2.7)

f = ζBSμ G,

where ζ ∈ T, (8.2.8)

B(z) =

% an an − z , |an | 1 − an z

an ∈ D \ {0},

n1

is a Blaschke product,

  ξ+z dμ(ξ) Sμ (z) = exp − T ξ−z is a singular inner function, and  ξ + z  G(z) = exp log |f (ξ)| dm(ξ) T ξ−z

is an outer function. The factors ζ, B, Sμ , G are unique. If f ∈ HW and f H p = 1, then (8.2.9)

|f (0)| = |B(0)||Sμ (0)||G(0)|.

So if we want to maximize |f (0)|, we can do this by maximizing each of the values |B(0)|, |Sμ (0)|, and |G(0)| separately. Since |Sμ (0)| = e−μ(T) , the |Sμ (0)| factor in (8.2.9) is maximized by taking the measure μ to be zero, that is, the singular inner factor Sμ in (8.2.7) is a unimodular constant and so f = ζBG. To maximize the number |G(0)| we see that   p p |G(0)| = exp log |f (ξ)| dm(ξ)  |f (ξ)|p dm(ξ)  1. T

T

Furthermore, equality holds above precisely when |f (ξ)| = 1 almost everywhere. Thus the outer factor G in (8.2.7) is a unimodular constant. This says that f = ζB, i.e., f is Blaschke product. Since f ∈ HW , we know that W ⊆ {an }n1 . From (8.2.8) we know that % |B(0)| = |ak | k1

and this product will be maximized when the only |ak | appearing in the product above are the |wk |, where {wk }k1 is the given zero set for H p . Thus the solution

8.3. EXTREMAL FUNCTIONS AS INNER FUNCTIONS

105

to the extremal problem is a Blaschke product whose zeros are precisely W . This solution is also unique, modulo a multiplicative unimodular constant. The two main points to emphasize here are that the solution to the extremal function is inner, in fact a Blaschke product, and contains precisely the zeros W . With a much more complicated analysis, and with a different definition of inner, an analogous result is true for the Bergman space Ap . In the following sections of this chapter, we begin to explore what happens with pA . But first we need to develop a notion of “inner” for the pA spaces. 8.3. Extremal Functions as Inner Functions For a sequence W ⊆ D \ {0} with RW = {0}, let gW be the (unique) solution to the infimum problem (8.2.1). By definition, gW p  gW + S n Ψp ,

Ψ ∈ RW ,

n  1.

Birkhoff-James orthogonality (3.3.2) says that gW ⊥p S n gW ,

(8.3.1)

n  1,

where (Sf )(z) = zf (z) is the shift operator on pA (Definition 7.1.1). This motivates the following definition. Definition 8.3.2. Let p ∈ (1, ∞). A function f ∈ pA \ {0} is p-inner if f ⊥p S n f, Since f ⊥p g ⇐⇒



n  1. |ak |p−2 ak bk = 0

k0

 (Theorem 3.4.7), we see that a nontrivial f (z) = k0 ak z k is p-inner when  (8.3.3) |ak |p−2 ak an+k = 0, n  1. k0

This usage of the word “inner” has some precedent. In the H p case we can, as we did above, define f ∈ H p \ {0} to be H p -inner when f ⊥H p S n f for all n  1, where ⊥H p denotes the Birkhoff-James orthogonality in Lp = Lp (T, m). Since f ⊥H p g when |f |p−2 f g dm = 0, T

we see that f is H p -inner when |f (ξ)|p ξ n dm(ξ) = 0, T

n  1.

The above equation, along with its complex conjugate, shows that all the Fourier coefficients of |f |p vanish, except for the zeroth coefficient. This means that |f |p is a constant function on T. That is, apart from a normalizing condition, f is H p -inner if and only if f is inner in the traditional sense (see Proposition 5.1.14). Likewise for the Bergman space Ap , we know that f ⊥Ap g when |f |p−2 f g dα = 0. D

106

8. EXTREMAL FUNCTIONS

Thus f ∈ Ap \ {0} is Ap -inner when f ⊥Ap S n f for all n  1, that is, |f (z)|p z n dα(z) = 0, n  1. D

This was mentioned earlier in (5.2.4). Remark 8.3.4. From (8.3.3) the monomial z m , m  0, is p-inner. Furthermore, f ∈ pA is p-inner if and only if z m f (z) is p-inner for every m  0. For this reason we will often take the liberty of considering only prospective zero sets W that exclude the origin, and work with functions f that are normalized so that f (0) = 1. This does not constitute a meaningful loss of generality. At this point we should present some significant examples of p-inner functions. If f ∈ pA , p ∈ (1, ∞), and  [Sf ] := {S n f : n  1} is the S-invariant subspace of pA generated by Sf , we define f to be the metric projection of f onto [Sf ]. Recall from Theorem 3.2.2 (existence and uniqueness of metric projections onto subspaces) that f is the unique vector in [Sf ] for which f − fp = inf{f − gp : g ∈ [Sf ]}.

(8.3.5)

This leads us to the following result. Proposition 8.3.6. Let p ∈ (1, ∞). If f ∈ pA \ {0}, then J = f − f is p-inner, and every p-inner function arises in this manner. Proof. First observe that f is the unique vector in [Sf ] which satisfies (8.3.5). For any t ∈ C and n  1, use the fact that tS n (f − f) ∈ [Sf ] to get f − fp  (f − f) − tS n (f − f)p , which we write as Jp  J − tS n Jp . From the definition of Birkhoff-James orthogonality in (3.3.2), J ⊥p S n J for all n  1 and thus J is p-inner. Conversely, suppose that J is p-inner, and let P be the set of (analytic) polynomials. Then J ⊥p S n J for all n  1. But since the criterion for Birkhoff-James orthogonality in the pA setting is linear in the second argument (Corollary 3.4.8), we see that J ⊥p QJ for all Q ∈ P with Q(0) = 0. Thus J + QJp  Jp , which implies

   p = inf J + QJp : Q ∈ P, Q(0) = 0  Jp . J − J

The definition (and uniqueness) of J yields J = 0 and so the p-inner function J = J − J takes the desired form.  Remark 8.3.7. Henceforth, for any f ∈ pA , let us continue to use f to denote the metric projection of f onto the subspace [Sf ]. This function, and especially the associated co-projection function f − f, will make many appearances in the pages to follow. We stress that the symbol  will have this meaning rather than, for example, the Fourier transform or the canonical embedding into the second dual.

8.3. EXTREMAL FUNCTIONS AS INNER FUNCTIONS

107

We began this section with the observation that gW , the solution to the extremal problem from (8.2.1), is p-inner and thus takes the form f − f for some f ∈ pA . Finding f is a challenge but we can identify a choice of f with the following. Proposition 8.3.8. Let p ∈ (1, ∞) and suppose that W = {w1 , w2 , . . . , wn } is a finite sequence from D \ {0}. Let  z  z   z  (8.3.9) f (z) := 1 − 1− ··· 1 − w1 w2 wn  and J = f − f . Then J = gW , where gW is the unique solution to the extremal problem in (8.2.1). Proof. If g ∈ RW then, by n applications of Proposition 7.2.1, the function g/f also belongs to pA . The power series of g/f converges in norm (Proposition 6.1.6) and hence there are polynomials ϕk such that ϕk → g/f in pA . It follows that ϕk f → g in pA . This shows that RW is spanned by polynomial multiples of f . With that, we recall that f (0) = 1 to conclude gW p = inf{gp : g(0) = 1, g ∈ RW } = inf{ϕf p : ϕ ∈ P, ϕ(0) = 1} = inf{f + (ϕf − f )p : ϕ ∈ P, ϕ(0) = 1} = inf{f − Qf p : Q ∈ P, Q(0) = 0} = Jp . By the uniqueness of gW as the solution to the extremal problem in (8.2.1), along with the condition gW (0) = J(0) = 1, it must be the case that J = gW .  Remark 8.3.10. In the above proof we used the fact that if f is a polynomial then f pA ⊆ pA and the operator h → f h is continuous on pA . This follows from several applications of the continuity of the shift S. We also showed implicitly that RW = f pA whenever W is a finite subset of D and f is given by (8.3.9). We will also need the following circle of ideas which rely on the point evaluation functional kw , defined by 1 = 1 + wz + w2 z 2 + w3 z 3 + · · · , (8.3.11) kw (z) = 1 − wz   for each w ∈ D. Note that kw ∈ pA and, for any f (z) = n0 an z n ∈ pA , the usual 

bilinear pairing between pA and pA yields 

f, kw  = an wn = f (w). n0

Theorem 8.3.12. Suppose that p ∈ (1, ∞). Let {w1 , w2 , . . . , wM } be a set of distinct nonzero points of D, and let J := f − f, where  z   z  z  1− ··· 1 − . f (z) := 1 − w1 w2 wM Then −1  , Jp = inf 1 + b1 kw1 + b2 kw2 + · · · + bM kwM p where the infimum is taken over all complex coefficients b1 , b2 , . . . , bM .

108

8. EXTREMAL FUNCTIONS

Proof. First observe from the proof of Proposition 8.3.8 that {g ∈ pA : g(0) = g(wj ) = 0, 1  j  M } = [Sf ] = Sf pA . 

Next, suppose that λJ ∈ pA (pA )∗ is the norming functional for J, that is to say, λJ p = 1 and λJ (J) = Jp (recall (1.8.1)). From J ⊥p S j f,

j  1,

and Theorem 3.4.7, we see that λJ (S j f ) = 0, that is, λJ ∈ [Sf ]⊥ =

(8.3.13)

j  1,

 {1, kwj : 1  j  M }.

We may therefore write λJ as λJ = c0 + c1 kw1 + · · · + cn kwM , for some complex coefficients c0 , c1 ,. . . , cM . By the definition of norming functional we get J = λJ (J) = c0 J, 1 + c1 J, kw1  + · · · + cM J, kwM  = c0 , since J, 1 = J, k0  = J(0) = f (0) = 1.   The above implies that J/c0 ∈ pA (pA )∗ is a norming functional for λJ ∈ pA in that J/c0 p = 1 and (J/c0 )(λJ ) = 1 = λJ p . Using Theorem 3.4.7 again, we see that the condition J(kwj ) = 0, 1  j  M, yields λJ ⊥p kwj , 1  j  M. This is (8.3.13). Thus, by the definition of Birkhoff-James orthogonality, λJ solves the infimum problem inf c0 + c1 kw1 + · · · + cM kwM p where c0 = Jp is fixed, and the constants c1 , . . . , cM are varied. By renaming the constants, we have shown that 1 = λJ  = Jp inf 1 + b1 kw1 + · · · + bM kwM p , or equivalently, Jp =



inf 1 + b1 kw1 + · · · + bM kwM p

−1

.



Thus any particular selection of the constants b1 , b2 , . . . , bM gives rise to an upper bound for the infimum, and hence a lower bound for the norm of J. If the roots w1 , w2 , . . . , wM are not distinct, then a similar claim holds which involves the kernels for point evaluation of the derivatives. Another important fact to establish is that the p-inner function J = f − f has at least the zeros of f . Proposition 8.3.14. Let p ∈ (1, ∞). For any f ∈ pA \ {0}, the co-projection J = f − f has at least the same zeros as f with at least the same multiplicities.

8.4. A RELATED EXTREMAL PROBLEM

109

Proof. Without loss of generality, assume that f (0) = 1. If f has a zero at the origin of multiplicity m, then so does f and thus J. Thus we may carry out the argument below with Qm 0 f in place of f (where Qw is the difference quotient operator). Let w ∈ D be a zero of f of multiplicity n. We now argue that w is a zero of J with at least the same multiplicity. By the definition of the co-projection, there must be a sequence of polynomials hk such that hk (0) = 1 and hk f converges to J in norm. This is due to the fact that f belongs to the subspace [Sf ]. Since norm convergence implies pointwise convergence on D, we see that J(w) = lim hk (w)f (w) = 0. k→∞

Moreover, Qw (hk f ) → Qw J and so (Qw f )(w) = lim (Qw (hk f ))(w) = 0. k→∞



Thus J (w) = 0. Continue this m − 1 times to conclude that J(w) = J  (w) = · · · = J (m−1) (w) = 0, 

that is, w is a zero of J of order at least m.

Of course we need to explore whether J = f − f can have zeros that are not those of f . We will take up this crucial issue in Section 8.7. 8.4. A Related Extremal Problem We record the solution to a related extremal problem here and use it in the next chapter when we discuss the pre-zero sets for pA . For fixed p ∈ (1, ∞) and for a nonempty finite set W ⊆ D \ {0}, consider the extremal problem (8.4.1)

inf{1 − gp : g ∈ RW }.

By the nearest point property for uniformly convex spaces (Theorem 3.2.2), there is a unique G ∈ RW for which Φ = 1 − G satisfies (8.4.2)

Φp = inf{1 − gp : g ∈ RW }.

Proposition 8.4.3. The extremal problem in (8.4.1) has a unique solution Φ satisfying gW , Φ=1− 1 + (gW pp − 1)p −1 where gW is the unique solution to the extremal problem in (8.2.1). Proof. The infimum in (8.4.2) measures the nearest point of RW to the constant function 1 and hence, by the uniform convexity of pA , has a unique solution, that is, there is a G ∈ RW for which inf{1 − gp : g ∈ RW } = 1 − Gp . Taking g ≡ 0 as a test function in the above infimum, we see that 1 − Gp  1p = 1. Actually, we have that 1 − Gp < 1 since otherwise, via Birkhoff-James orthogonality 1 ⊥p RW . By Theorem 3.4.7 we conclude that g(0) = g, 1 = 0,

g ∈ RW .

110

8. EXTREMAL FUNCTIONS

But since we are assuming that 0 ∈ W and that RW = {0}, this is impossible. Since RW is closed, GW ∈ RW , and GW ≡ 1, then 1 − Gp > 0. Furthermore, if  G(z) = Gj z j , j0

we can see from the identity



1 − Gpp = |1 − G(0)|p +

|Gj |p

j1

that G(0) = 0. We can also prove that G(0) > 0. If G1 (z) = G(z) then

|G(0)| , G(0)



p  |Gj |p 1 − G1 pp = 1 − |G(0)| + j1

 |1 − G(0)| + p



|Gj |p

j1

= 1 − Gpp . Since G is the unique solution to the extremal problem in (8.4.1), it must be the case that G1 = G and so G(0) = |G(0)| > 0. We now argue that G(0) ∈ (0, 1). For this, note that   p t Gp : t ∈ R min 1 − G(0) occurs when t = G(0). On the other hand,  p  t   G = |1 − t|p + |t|p |Gj /G(0)|p . 1 − G(0) p j1

Taking derivatives with respect to t on both sides of the previous line and using a derivative calculation from the proof of Lemma 4.6.5, we get p  d t   G = −p(1 − t)p−1 + ptp−1 |Gj /G(0)|p . 1 − dt G(0) p j1

This vanishes when (1 − t)p−1 = tp−1



|Gj /G(0)|p ,

j1

or equivalently, when |1 − t|p−2 (1 − t) = |t|p−2 t



|Gj /G(0)|p .

j1

The above can only happen when t and (1 − t) have the same sign, which forces t ∈ (0, 1). However, by the above analysis, this minimum occurs when t = G(0), and so G(0) ∈ (0, 1).

8.5. ONE POINT EXTREMAL FUNCTION

111

 Finally, among all the functions g(z) = j0 gj z j ∈ RW with g(0) = G(0), G itself has the smallest norm. To see this, just consider the identity  |gj |p . 1 − gpp = |1 − G(0)|p + j1

Thus the function G/G(0) satisfies the extremal problem in (8.2.1). Hence by Proposition 8.3.8, G/G(0) = gW = J. From here we get inf{1 − gp : g ∈ RW } = 1 − Gp = 1 − G(0)gW p . To compute G(0), observe that G(0) is the value of x ∈ (0, 1) for which d 1 − xgW pp = 0. dx A computation similar to the one used above to show that G(0) ∈ (0, 1) yields

d d  1 − xgW pp = |1 − x|p + |x|p (gW pp − 1) dx dx = −p(1 − x)p−1 + pxp−1 (gW pp − 1) = −p(1 − x)p−1 + pxp−1 (gW pp − 1). Note the use of the fact that x and 1 − x are positive and Lemma 1.8.8 in the last step. Since gW (0) = 1, we have gW pp − 1 > 0. One can check that −p(1 − x)p−1 + pxp−1 (gW pp − 1) = 0 precisely when 1 . 1 + (gW pp − 1)p −1 This proves the desired formula for Φ = 1 − G. x=



8.5. One Point Extremal Function Let us work out the extremal function gW when W = {w} is a single point with w ∈ D \ {0}. We denote gW by gw . Recall the signed power notation  0, a = 0, s a := |a|s−1 a, a = 0, for any a ∈ C and s > 1 (Lemma 1.8.8). Proposition 8.5.1. For p ∈ (1, ∞) and w ∈ D \ {0} we have gw (z) =

1 − z/w . 1 − wp −1 z

Proof. Let f (z) = 1 − z/w and J = f − f, and recall from Proposition 8.3.8 that J = gw . Moreover, J ⊥p S n f for all n  1. Since f ∈ [Sf ] we have f(0) = 0 and so J(0) = 1. If   J(z) = Jk z k = 1 + Jk z k , k0

and f (z) =

 k0

k1

fk z k = f0 + f1 z = 1 −

1 z, w

112

8. EXTREMAL FUNCTIONS

we can use (8.3.3) to obtain 1 = 0, n  1. w Solutions to this type of recurrence relation are well known (for example see [58]). p−1 This has the solution Jn = Cwn , which can be checked directly by substitution. By using the identity p−1

p−1

Jnp−1 f0 + Jn+1 f1 = Jnp−1 · 1 − Jn+1 ·



ap−1p −1 = a

(8.5.2) from Lemma 1.8.8, we then have

J(z) = 1 +





(Cwk )p −1 z k .

k1

The constant C is uniquely determined by the requirement that J(w) = 0. This yields 0 = J(w)    C p −1 (wk )p −1 wk =1+ k0

=1+







C p −1 |wk |p −2 wk wk

k1 

= 1 + C p −1



|w|p . 1 − |w|p

Thus





C p −1 = −

1 − |w|p |w|p

and so J(z) = 1 −

 1 − |w|p k1

|w|p



wp −1k z k



=1−



1 − |w|p wp −1 z |w|p 1 − wp −1 z 



1 − wp −1 w wp −1 z wp −1 w 1 − wp −1 z   z 1 − wp −1 w 1 − wp −1 z − = w 1 − wp −1 z 1 − wp −1 z 1 − z/w . = 1 − wp −1 z =1−



Remark 8.5.3. The one point extremal function of Proposition 8.5.1 is special, so we will give it a name: (8.5.4)

Bw,p (z) :=

1 − z/w . 1 − wp −1 z

Thus Bw,p is p-inner. When p = 2, Bw,2 (z) =

1 w−z w 1 − wz

8.6. FINITE POINT EXTREMAL FUNCTIONS

113

which is a constant multiple of the Blaschke factor with a zero at w. Disappointingly, however, for general p the product of functions of the form Bw,p fails to be p-inner, as we shall see in the next section. This effect contributes to the difficulties in understanding the zero sets of pA . One can compute the one-point extremal function for other spaces. For example, in the Bergman space Ap , the one point extremal function is 2 − 1 w − z  w  p p w − z  p p 2 gw (z) = 1 + (1 − |w| ) 1+ 1−w . |w| 2 1 − wz 2 1 − wz The above formula shows that gw has a zero at w (of order one) and no others. 8.6. Finite Point Extremal Functions For a polynomial f of higher degree, the p-inner function f − f takes the following form. Proposition 8.6.1. Fix p ∈ (1, ∞). Suppose that s1 , s2 ,. . . , sd are distinct elements of D \ {0} and let n1 , n2 ,. . . , nd be positive integers. Let f be the polynomial   z n1  z n2 z nd f (z) = 1 − 1− ··· 1 − . s1 s2 sd Then J = f − f is of the form (8.6.2)

J(z) = 1 +

∞   d n m −1  k=1

Cj,m kj skm

p −1

zk ,

m=1 j=0

and the constants Cj,m are uniquely determined by the requirement that J (m) (sk ) = 0 for all k, 1  k  d, and all m, 0  m  nk − 1, where J (m) stands for the mth derivative of J. Before getting to the proof of this proposition, let us make the following two remarks. Remark 8.6.3. Before proceeding to the proof of this proposition, we note that J is analytic in a neighborhood of D. Indeed, each Taylor coefficient for J is a polynomial in sm raised to the p − 1 power. In that polynomial, one sees k times some skm . This decays geometrically, since the modulus of sm is less than 1. In fact, the root of largest modulus dictates the radius of convergence of J. From here, one sees that the radius of convergence of the Taylor series defining J  is R := r −(p −1) > 1, where r = max{|s1 |, |s2 |, . . . , |sm |} < 1. Remark 8.6.4. Let p = 4/3, so that p = 4. Choose two distinct, nonzero points r and s in D, and for simplicity assume both are real. Define, as usual,  z  z f (z) := 1 − 1− , r s and J := f − f. We know that J is of the form J(z) = 1 +

∞ 



(Cr k + Dsk )p −1 z k ,

k=1

where the coefficients C and D are uniquely determined by the conditions J(r) = z ) would also be a p-inner J(s) = 0. Moreover, C and D must both be real, or else J(¯

114

8. EXTREMAL FUNCTIONS

function with the prescribed zeros and unit constant term, violating uniqueness of metric projections. Thus the complex power p − 1 can be replaced by the simple power 3. Thus J(z) = 1 + =1+

∞ 

(Cr k + Dsk )3 z k

k=1 ∞ 

(C 3 r 3k + C 2 Dr 2k sk + CD2 r k s2k + D3 s3k )z k

k=1 3 3

=1+

C 2 Dr 2 sz CD2 rs2 z D 3 s3 z C r z + + + . 1 − r3 z 1 − r 2 sz 1 − rs2 z 1 − s3 z

On the other hand, the product of the corresponding one-point inner functions is Bp,r (z)Bp,s (z) =

1 − z/r 1 − z/s · . 1 − r 3 z 1 − s3 z

No choice of C and D will make them equal (on top of satisfying the requirement J(r) = J(s) = 0). Thus Bp,r (z)Bp,s (z) cannot be the p-inner function for the prescribed zero set. Proof of Proposition 8.6.1. By the definition of J we have J ⊥p S k f for all k  1. With N = n1 + · · · + nd , this gives rise to a recurrence relation (8.6.5)

p−1

Jk

p−1

p−1

f0 + Jk+1 f1 + · · · + Jk+N fN = 0,

k  1,

on the coefficients of J, which, again via [58], has the solution (8.6.2). Next, suppose that K is a function of the form (8.6.2), that also satisfies the condition K (m) (sk ) = 0 for all k, 1  k  d, and all m, 0  m  nk − 1. In other words, all of the roots of f are zeros of K to at least the same multiplicities. Then K is analytic in a neighborhood of D and its zero set contains those of f , multiplicities taken into account. It follows that K/f is also analytic in a neighborhood of D. Consequently, we have ϕn f → K in pA , where ϕn is the nth partial sum of K/f . This shows that K ∈ [f ]. Since K(0) = f (0) = 1, we also see that f − K belongs to [Sf ]. Finally, by virtue of K having the form (8.6.2), the function K satisfies K ⊥p [Sf ]. This forces f − K to be the metric projection f of f onto [Sf ]. We conclude that K = J. Finally, we turn to proving the uniqueness of the constants Cj,m defining J. Suppose that J has two representations of the form (8.6.2), namely, J(z) = 1 +

∞ d n m −1   k=1

=1+

p −1

zk

m=1 j=0

∞ d n m −1   k=1

Cj,m kj skm .j,m kj skm C

p −1

zk .

m=1 j=0

It must be that the corresponding Taylor coefficients coincide, or d n m −1  m=1 j=0

Cj,m kj skm

p −1

=

d n m −1  m=1 j=0

.j,m kj sk C m

p −1

8.7. EXTRA ZEROS

115

for all k  1. By the taking p − 1th powers of both sides, using (8.5.2) and transposing, we find that d n m −1  .j,m ]kj sk = 0 [Cj,m − C m m=1 j=0

for all k. The only way this can happen is if .j,m , Cj,m = C

1  m  d,

1  j  nm − 1,

since the sequences {kj skm }k1 constitute a complete, linearly independent set of solutions to the difference equation (8.6.5) underlying J [58, Corollary 2.24]. This shows that the constants Cj,m are uniquely determined by the conditions  J (m) (sk ) = 0 for all k, 1  k  d and all m, 0  m  nk − 1. In the Hardy space H p case the extremal function in (8.2.6) for a finite zero set W is just a constant multiple of the finite Blaschke product with precisely those zeros, repeated according to multiplicity, and no others. In the Bergman space Ap case, the extremal function for a finite zero set, though not a finite Blaschke product, still enjoys the property of having no extra zeros. For certain weighted Bergman spaces the extremal function for even a single point zero set can have an extra zero [82]. As we will see in the next section, in pA even extremal functions for finite zero sets can, under some circumstances, have extra zeros. 8.7. Extra Zeros Suppose that f ∈ and J = f − f. We know that the zeros of f must also be zeros of J. However, it could happen that J has zeros that are not zeros of f (multiplicities taken into account). Let us now exhibit the existence of such extra zeros. We will prove that an extra zero exists when p = 4/3. The existence of extra zeros has profound implications. It means that the zeros of a function f ∈ pA cannot always be exactly carried by the associated p-inner function J; that f cannot always be factored into JG for some nonvanishing analytic function G; that the S-invariant subspace [f ] generated by f need not coincide with [J]. These represent significant departures from the Hardy and Bergman space cases, and suggest that the ultimate structure of pA may be built from different ingredients. We begin with the following interesting observation. pA

Proposition 8.7.1. For n  1, r ∈ (0, 1), and p ∈ (1, ∞), the function B(z) :=

1 − z n /r n 1 − r n(p −1) z n

is p-inner. Proof. For n = 1 the function B is the one point extremal function for r (Proposition 8.5.1) and hence by (8.3.1) is p-inner. For other values of n, replacing z by z n merely spaces out the Taylor coefficients by n steps. This preserves the Birkhoff-James orthogonality of B to its forward shifts which makes B p-inner.  Our strategy for conjuring an extra zero entails the following steps. The zeros of B are the symmetrically placed points r, re2πi/n , re2πi·2/n , re2πi·3/n , . . . , re2πi·(n−1)/n .

116

8. EXTREMAL FUNCTIONS

These are the roots of the polynomial 1 − z n /r n . Plainly B(z) does not have any extra zeros. Now, if we remove the single point r from the zero set, we could expect the corresponding p-inner function J to be “close” to B(z). Furthermore, by manipulating our choices of r and n, we might hope to coax an extra zero into the unit disk. Our particular choice of roots will aid in the computation of J, with symmetries leading to cancellations and simplifications. Finally, the solution to an extremal problem, Theorem 8.3.12, will deliver the needed estimate for an extra zero. With that in mind, let p ∈ (1, ∞) and r ∈ (0, 1) be fixed. Let n  1, and consider the roots 0  j  n − 1.

wj := re2πij/n , Define the polynomial f (z) :=

1 − z n /r n 1 − z/r

and note that f has all of these roots wj except for w0 = r. As ever, define J := f − f in pA , the p-inner function arising from f (Proposition 8.3.6). From Proposition 8.6.1 we know that J(z) = 1 +

  n−1  k1

=1+

=1+

zk

Cj r k e2πikj/n

p −1

zk

j=1

  n−1  k1

p −1

j=1

  n−1  k1

Cj wjk

Cj e2πikj/n

p −1



r k(p −1) z k .

j=1

Since all of the Taylor coefficients of f are real, it must be that each Taylor coefficient of J is real as well (or else J(z) would also be a p-inner function corresponding to f , in violation of the uniqueness of metric projections). Thus the complex power p − 1 above is simply a signed power. The expression (8.7.2)



Dk := (C1 e2πik/n + C2 e2πik·2/n + · · · Cn−1 e2πik·(n−1)/n )p −1 ,

as k varies over the positive integers, takes at most n distinct values, repeating in cycles of n. Therefore, (8.7.3)

J(z) = 1 +

n 



Dk r (jn+k)(p −1) z jn+k

j0 k=1 

=1+





D2 r 2(p −1) z 2 Dn r n(p −1) z n D1 r (p −1) z .  −1) n +  −1) n + · · · + n(p n(p 1−r z 1−r z 1 − r n(p −1) z n

Imagine that we rearrange this last expression into the ratio of two polynomials in z, each of degree exactly equal to n. The constant coefficient of the numerator is 1, while the leading coefficient of the numerator is 

(Dn − 1)r n(p −1) .

8.7. EXTRA ZEROS

117

The numerator has n roots. We already know n − 1 of the them; they are the roots of f , namely w1 , w2 ,. . . , wn−1 . The product of these n − 1 roots is r n−1

  1 + 2 + · · · + [n − 1]  = (−r)n−1 . e2πij/n = r n−1 exp 2πi n j=1

n−1 %

Let us write w for the remaining root of the numerator. By comparing the two sides of  z   z  z z  , 1 + a1 z + a2 z 2 + · · · + an z n = 1 − 1− ··· 1− 1− w1 w2 wn−1 w we see that an = (Dn − 1)r n(q−1) =

(−1)n 1 = . (−w1 )(−w2 ) · · · (−wn−1 )(−w) w1 w2 · · · wn−1 w

Therefore the remaining zero w must be w=

(−1)n (Dn −

1)r n(q−1) w

1 w2

· · · wn−1

=

1 − Dn

1 . r n(q−1) r n−1

We can calculate Dn in the following manner. From the conditions C1 J(w1 ) = C2 J(w2 ) = · · · = Cn−1 J(wn−1 ) = 0, we see that 0 = C1 +



k C1 w1k + C2 w2k + · · · Cn−1 wn−1

k1

0 = C2 +



k C1 w1k + C2 w2k + · · · Cn−1 wn−1

p −1

p −1

C1 w1k C2 w2k

k1

.. . 0 = Cn−1 +



k C1 w1k + C2 w2k + · · · Cn−1 wn−1

p −1

k Cn−1 wn−1 .

k1

Adding all of these equations together yields 

p k

C1 w1k + C2 w2k + · · · Cn−1 wn−1

C1 + C2 + · · · + Cn−1 = − k1

=−



p  k

(C1 w1k + C2 w2k + · · · Cn−1 wn−1 )p −1

k1

= −Jpp + 1, where we have used (p − 1)p = p .  By definition, Dn = (C1 + C2 + · · · + Cn−1 )p −1 . But then, this quantity takes the value  Dn = −(Jpp − 1)p −1 . Thus we have (8.7.4)

w=

r n−1

·

r n(p −1)

1 . 1 + (Jpp − 1)p −1



If we can come up with a close estimate for Jp from below, and the resulting upper bound for w turns out to be less than unity, then we will be done.

118

8. EXTREMAL FUNCTIONS

To obtain a lower bound for Jp , we will use the formula (8.7.5)

  −1  , Jp = inf 1 + b1 k1 + b2 k2 + · · · + bn−1 kn−1 p

(Theorem 8.3.12), where the infimum is over the coefficients b1 , b2 , . . . , bn−1 . Let us call the function inside the infimum norm G, and notice p   Gpp = 1 + b1 k1 + b2 k2 + · · · + bn−1 kn−1 p p     j = (1 + b1 + b2 + · · · + bn−1 ) + (b1 w1j + b2 w2j + · · · + bn−1 wn−1 )z j   p

j1

p 



b1 wj + b2 wj + · · · + bn−1 wj p = 1 + b1 + b2 + · · · + bn−1 + 1 2 n−1 j1

p

= 1 + b1 + b2 + · · · + bn−1





b1 e2πi(1+jn)/n + · · · + bn−1 e2πi(1+jn)·(n−1)/n p r (1+jn)p + j0





b1 e2πi(2+jn)/n + · · · + bn−1 e2πi(2+jn)·(n−1)/n p r (2+jn)p + j0

+ ... 



b1 e2πi(n+jn)/n + · · · + bn−1 e2πi(n+jn)·(n−1)/n p r (n+jn)p . + j0

The calculation continues



1 + b1 + b2 + · · · + bn−1 p

p

+ b1 e2πi/n + b2 e2πi·2/n + · · · + bn−1 e2πi·(n−1)/n



rp 1 − r np

p r 2p

+ b1 e2·2πi/n + b2 e2·2πi·2/n + · · · + bn−1 e2·2πi·(n−1)/n

+ ··· 1 − r np

p r np + b1 en·2πi/n + b2 en·2πi·2/n + · · · + bn−1 en·2πi·(n−1)/n

. 1 − r np Any selection of the constants bj results in a valid lower bound for Jp . We can now proceed by selectively evaluating this quantity to obtain an estimate for Jp , and then using that in turn to see if the nth zero w lies in D. Let us carry out this estimation procedure when p = 43 , using n = 4 and r = 0.9. In this specific situation, the roots are ir, −r and −ir, and p = 4. Thus we need  to find the norm in pA of G(z) = 1 +

B C A + + 1 − irz 1 + irz 1 + rz

8.7. EXTRA ZEROS

119

for some choice of parameters A, B and C. By symmetry we may assume that C is real and that B = A. Now 4   A C  A 4   + + G4 = 1 + 1 − irz 1 + irz 1 + rz  ∞

4

4 

 m

= 1 + A + A + C +

Ai + A(−i)m + C(−1)m r 4m m=1

4

4 



= 1 + A + A + C +

A − A i − C r 4m m≡1

4

4 

 





+

− A + A + C r 4m +

− A − A i − C r 4m m≡2

4 



+

A + A + C r 4m ,

m≡3

m≡4

where in the sums, the congruences are modulo 4, and m  1. Since we want to minimize this quantity, we are aided by the assumption that A is real, so that the contributions from the expression (A − A)i are zero. We expand the geometric series to see that G44 is equal to 4 4 4 8 4 12 4 16



1 + 2A + C 4 + C r + (−2A + C) r + C r + (2A + C) r . 1 − r 16 1 − r 16 1 − r 16 1 − r 16 −1 Again, any choice of A and C will result in G4 being a lower bound for J4/3 . If we choose the parameter values A = −0.205683

and C = −0.202725,

G44

then the resulting value of is approximately 0.0574716. This provides the following bound for the norm of J: J4/3  2.042381. Substituting this into (8.7.4) gives us the bound for the extra zero w w  0.965699. We already know that w > 0, and hence w is an extra zero for J in D. We have proved the following. Theorem 8.7.6. Let p = 43 . There exists a polynomial f such that the p-inner function J := f − f has an extra zero in D. 

With the choice p = 43 , r = 0.9 and n = 4, the complex power ap −1 , applied to a real number a, is just a3 . In this situation we can calculate J numerically, by solving for the coefficients C1 , C2 and C3 in  (C1 w1 + C2 w2 + C3 w3 )3 z k J(z) = 1 + k1

that satisfy J(w1 ) = J(w2 ) = J(w3 ) = 0. This approach gives the estimate (0.075587)z + (0.0839856)z 2 + (0.0933173)z 3 − (1.13804)z 4 , 1 − (0.28243)z 4 and J has an extra zero at J(z) ≈ 1 +

w ≈ 0.965694580489323.

120

8. EXTREMAL FUNCTIONS

This confirms our findings above. Corollary 8.7.7. Let p = 43 . There exists a polynomial f such that [f ] = [h] for any p-inner function h. Proof. Let f and g satisfy [f ] = [g] in pA . Without loss of generality, we may assume that f (0) = 1 and g(0) = 1, by dividing out a common power of z if necessary. By our assumption that [f ] = [g], given any polynomial P , and any

> 0, there exists a polynomial Q such that   (f + Sf P ) − (g + SgQ) < . Likewise, given Q and , there exists a P such that the above holds. It follows inf f + Sf P p = inf g + SgQp . P

Q

Consequently, f − f = g − g. In particular, if f is a polynomial whose associated inner function has an extra zero in D (Theorem 8.7.6) and [f ] = [h] for some p-inner function h, then necessarily h = f − f. This is impossible if f − f has an extra zero.  This Corollary will appear again when we discuss a possible version of Beurling’s Theorem for pA spaces in Chapter 10. Remark 8.7.8. A similar program can be pursued for other values of p. By careful selection of r and n in the above construction, one can obtain extra zeros when p ∈ [1.03, 1.80]. The above method does not appear to yield extra zeros when p is closer to 1 or 2, or when p > 2. The existence of extra zeros in these cases remains open. Other mechanisms for pursuing an extra zero may need to be developed. 8.8. Bounds for Extra Zeros We have established that extra zeros can exist. We now turn to the question of their location. The construction from the previous section suggests that any extra zeros must lie close to the boundary of the unit disk. Indeed that turns out to be the case, and the Pythagorean inequalities for pA (Corollary 4.6.12) deliver the essential ingredient. Theorem 8.8.1. Suppose that f is a non-constant function in pA , with f (0) = 1. Let J = f − f be the associated p-inner function. Let w ∈ D be a zero of J that is not a zero of f (multiplicities taken into account). (i) If p ∈ (1, 2), then |w|  p2 . (ii) If p ∈ [2, ∞), then |w|p − (1 − |w|)p 

1 . 2p−1 − 1

The proof of Theorem 8.8.1 relies on the following lemma, which allows for removing certain zeros inside the shift-invariant subspace [f ] generated by f . Lemma 8.8.2. Let p ∈ (1, ∞), and suppose that f ∈ pA . If (z − w)g(z) ∈ [f ] for some w ∈ D such that f (w) = 0, then g ∈ [f ].

8.8. BOUNDS FOR EXTRA ZEROS

121

Proof. By hypothesis there are polynomials ϕn , n  1, such that ϕn f → (z −w)g(z) in pA . By continuity of the difference quotient operator Qw (Proposition 7.2.1), we have Qw (ϕn f ) → Qw ((z − w)g) = g. We also have (8.8.3)

f · Qw ϕn + ϕn (w)Qw f = Qw (ϕn f ) → g.

Since convergence in pA implies convergence pointwise in the disk, we have ϕn (w)f (w) → (w − w)g(w) = 0; so, with f (w) = 0, it must be that ϕn (w) → 0. Hence the second term on the left side of (8.8.3) vanishes in the limit. Therefore, f · Qw ϕn → g. The function Qw ϕn is itself a polynomial, and thus we have shown g ∈ [f ].



The above shows that if J has an extra zero w, then J(z) ∈ [f ]. z−w This is utilized below. Proof of Theorem 8.8.1. First let us note that f (0) = 1, which forces J(0) = 1. Thus we may assume that w = 0. By Lemma 8.8.2, J(z) ∈ [f ]. 1 − z/w We can split this function into two terms z z J(z) z J(z)  J(z) + = J(z) + . z = z 1− 1− w 1− w w w w 1 − wz Notice that on the right side the first term is ⊥p to the second term, which belongs to [Sf ]. Therefore, the Pythagorean inequality of Corollary 4.6.12 applies, with the result    r  J(z) r      J(z)rp + K  zJ(z)  , z  1 − z   r |w| 1 − w p w p where r and K are the Pythagorean parameters appropriate to p. In the last term, multiplying by z does not alter the norm. So we can transpose this term to get r   J(z)  K    J(z)r .  (8.8.4) 1− p |w|r  1 − z  w

p

Our last step is to use            1  1  J(z)  =  J(z) − 0  =  J(z) − J(w)   J(z)p ,  |w|  1 − wz p  z − w p  z−w 1 − |w| p where we have used the norm of the difference quotient operator from Proposition 7.2.1. Combining the last two estimates gives us    1 − |w| r  J(z) r K     J(z) r 1−     ,  z |w|r 1 − w p |w| 1 − wz p

122

8. EXTREMAL FUNCTIONS

which simplifies to 1−

 1 − |w| r K  . |w|r |w|

Hence |w|r − (1 − |w|)r  K. When p ∈ (1, 2], the Pythagorean parameters are r = 2 and K = p − 1 and the condition on |w| then simplifies to |w|  p2 . When p ∈ [2, ∞), the Pythagorean parameters are r = p and K = 1/(2p−1 − 1) and the condition becomes 1 .  |w|p − (1 − |w|)p  p−1 2 −1 8.9. Notes Extremal functions in functional analysis have a long and storied history [54, 56, 59, 68, 102, 118, 119, 138, 162]. There are some general results which relate an extremal problem in a Banach space to its associated dual extremal problem [54, 68, 102, 139]. The discussion used to solve the Hardy space extremal problem in (8.2.6) follows [53]. Results about the nature of solutions for extremal problems for Bergman spaces can be found in [53, 81, 82]. 4/3 The construction of an extra zero for A follows from Cheng and Dragas [30], as does the bound for potential extra zeros (Theorem 8.8.1). It turns out that this bound can be sharpened when the given zero set is finite. From Chapter 5, every f in the Hardy space H p can be factored as as f = F G, where F is H p -inner, i.e., f ⊥H p S n f for all n  1, equivalently n |f (ξ)|p ξ dm(ξ) = 0, n  1, T

and G is H p -outer, equivalently,  {S n G : n  0} = H p . There is an analogous result for the Bergman space Ap . With our current definition of p-inner, there is no such result for pA . This stems from the fact that the p-inner function associated with f can have zeros besides those of f . However, there is a weaker factorization theorem for pA functions from a paper of Cheng and Ross [27]. A classical bound for polynomial roots is that if w is a root of the polynomial f (z) = a0 + a1 z + a2 z 2 + · · · + ad z d , with ad = 0, and p ∈ (1, ∞), then

   

a   a p a p

0

1

d−1 p p /p 1/p |w|  1 + + + · · · +

ad ad ad [121]. The orthogonality properties of the one-point extremal function Bw,p from (8.5.4), together with the Pythagorean Inequalities for pA , were used by Cheng, Mashreghi, and Ross to obtain improvements and extensions to such bounds, along with bounds and separation theorems for the zeros of analytic functions [33]. In the case of H p and Ap we do not need a concept of a pre-zero set since for these two spaces any subset of a zero set is also a zero set. In addition, for these two spaces the associated extremal functions have no extra zeros. For pA we do not yet know if a subset of a zero set is also a zero set. But we do know that the associated extremal function can have extra zeros.

10.1090/ulect/075/09

CHAPTER 9

Zeros of pA Functions In the previous chapter we laid the groundwork for our discussion of the pA zero sets by establishing their connections to some associated extremal functions. We now bring this discussion to fruition with a partial characterization of the pA zero sets via p-inner functions. To be more precise, we will give an exact characterization for a sequence in D to be a pre-zero set on which some nontrivial function in pA vanishes (counting multiplicities). This characterization is then used to construct key examples, including a zero set that fails to satisfy the Newman condition, an pA zero set that fails to be a Blaschke sequence when p ∈ (2, ∞), and a Blaschke sequence that fails to be an pA zero set when p ∈ (1, 2). We begin with some historical context. 9.1. The Blaschke Condition pA

is close to the Hardy spaces H p for some p and far away for other p, let Since us review the zero sets for H p . We state a few known results and refer the reader to the notes at the end of this chapter for references and other related topics. Theorem 9.1.1. For a sequence {zk }k1 ⊆ D\{0}, the following are equivalent: % 1 < ∞, (i) |zk | k1  (ii) (1 − |zk |) < ∞, k1

(iii) there exists an f ∈ H ∞ that vanishes exactly on {zk }k1 , including multiplicities. Recall that H ∞ ⊆ p>0 H p (Proposition 5.1.4). Combining this with the factorization theorem for H p (Theorem 5.1.9) we see that the equivalent conditions in (i) and (ii) (the Blaschke condition) characterize where H p functions vanish. We defer the proof of the above well-known and classical result to Remark 9.2.4. When  p ∈ (1, 2], pA ⊆ H p (Theorem 6.3.2). It follows that if f ∈ pA , p ∈ (1, 2], vanishes exactly on {zk }k1 , then this sequence must be a Blaschke sequence. We will see in Theorem 9.6.1 below that the converse is not true. When p ∈ (2, ∞) we know  that H p ⊆ pA (Theorem 6.3.2). From this, one might suspect that pA is a large space that contains functions vanishing on sequences that are non-Blaschke. This is indeed the case. Theorem 9.1.2 (Vinogradov). There exists a function f with & p f∈ A \ {0} p>2

and which vanishes on a non-Blaschke sequence. 123

9. ZEROS OF p A FUNCTIONS

124

We will not give a proof of this theorem since we will prove a related version of this in Theorem 9.5.1 below involving p-inner functions. Despite the fact that the classical Blaschke condition fails to characterize where the functions in pA vanish, the situation becomes nicer when the geometry is just right. Theorem 9.1.3. Let p ∈ (1, ∞). If the sequence {wk }k1 ⊆ D lies on a single radial segment of D and some f ∈ pA vanishes exactly on {wk }k1 , then this sequence must be a Blaschke sequence. Proof. Without loss of generality we may assume that wk = rk ∈ (0, 1). Let D = {z : |z − 12 | < 12 }. For each r ∈ (0, 12 ), let Cr = {z : |z − 12 | = r}. In a moment will show that any function f ∈ pA satisfies   (9.1.4) sup |f (z)|s |dz| : r ∈ (0, 12 ) < ∞, Cr



where s = p /4. In other words, f |D belongs to the Hardy space H s (D). From here it is well known [54] that a zero sequence {rk }k1 for such functions must satisfy the Blaschke-type condition  dist(rk , ∂D) < ∞. k1

But since  rk ∈ [0, 1), the above summability criterion is equivalent to the Blaschke condition k1 (1 − rk ) < ∞. We will finish the proof by showing that any f ∈ pA must satisfy (9.1.4). If z = 12 + reit ∈ Cr then for r ∈ ( 14 , 12 ) and t ∈ [−π, π], 1 − |z|2 = =  Since any f ∈

pA

2 3 4 − r − r cos t ( 32 + r)( 21 − r) + r(1 2 1 4 (1 − cos t)  ct .

satisfies the pointwise estimate  1  1 p , |f (z)|  C 1 − |z|

from (6.1.3), we have

− cos t)

z ∈ D,



1 |dz| 2 )s/p (1 − |z| Cr 1 =A |dz| 2 )1/4 (1 − |z| Cr π 1 dt A 2 )1/4 (|t| −π π 1 ! dt. A |t| −π

|f (z)| |dz|  A s

Cr

This last integral is finite and independent of r ∈ ( 41 , 12 ). Thus f |D ∈ H s (D) and the proof is complete.  This next result says, however, that although the sets on which the functions in pA vanish need not always satisfy the Blaschke condition (when p > 2), they do satisfy a slightly weaker condition.

9.1. THE BLASCHKE CONDITION

125

Theorem 9.1.5. If f vanishes exactly on {zk }k1 ⊆ D for some f ∈ pA , p ∈ (0, ∞), then 

 (1 − |zk |) log

k1

1 −1− 0. Proof. For simplicity in our exposition we can assume that 0 < |z1 |  |z2 |  |z3 |  · · · . Suppose f ∈ pA whose zeros are exactly {zk }k1 . Multiplying by a suitable constant we can assume |f (0)| = 1. By Jensen’s Formula, with r ∈ (0, 1), T

log |f (rξ)| dm(ξ) = log |f (0)| +

n 

log

k=1

=

n  k=1

r = log |zk |



r

0

r |zk |

n(t) dt, t

where z1 , z2 , . . . , zn are the zeros of f , repeated according to multiplicity, that are contained in {z : |z| < r} and, for t ∈ (0, 1), n(t) denotes the number of zeros of f (counting multiplicity) that lie in {z : |z| < t}. Since |f (rξ)|  C

1 , (1 − r)1/p

ξ ∈ T,

by (6.1.3), we conclude that

r

(9.1.6) 0

n(t) 1 dt  A log , t 1−r

r ∈ ( 21 , 1).

Now use the fact that n(t) is an increasing function to get 1 (r − r 2 )n(r 2 )  r



r

r2

n(t) 1 dt  A log , t 1−r

which implies that (9.1.7)

n(r)  A

1 1 log 1−r 1−r

(an interesting fact in itself). By Riemann-Stieltjes integration, integration by parts (twice), the estimate for n(r) from (9.1.7), along with (9.1.6), we have the following

9. ZEROS OF p A FUNCTIONS

126

chain of inequalities   (1 − |zk |) log k1



1 −1− 1 − |zk |

1 −1− dn(r) 1−r 0 1   1 −1  n(r) 1 −1−  dr 1 + (1 + ) log r log = 1−r 1−r r 0 1 1 −1− n(r) dr A log 1−r r 0 1 1  1 −2−  r n(t)  log dt dr = A(1 + ) 1−r t 0 1−r 0 1 1  1 −1− log A dr. 1−r 0 1−r =

1

 (1 − r) log

Since this last integral above is finite, the proof is complete.



Unfortunately, when p > 2, even this weaker condition does not characterize the sequences on which pA functions vanish. Theorem 9.1.8. For each p ∈ (2, ∞), there exists a function in pA that vanishes exactly on a sequence {zk }k1 ⊆ D which satisfies   1 −1 (1 − |zk |) log = ∞. 1 − |zk | k1

We will not include the proof of this since we will use extremal function techniques to prove a related result in Example 9.5.2 below. Finally, we remind the reader of our earlier results of Newman and Shapiro (see Theorem 6.2.8 and Remark 6.2.14) which says that for any sequence {zn }n1 ⊆ D satisfying  1 − |z  k+1 | (9.1.9) sup :k1 0. From Lemma 1.8.8, we get 



(Jn pp − 1)p −1 = (Jn pp − 1)p −1 .

9. ZEROS OF p A FUNCTIONS

128

From here we have Φn pp = 1 − Gn (0)Jn pp = |1 − Gn (0)|p + |Gn (0)|p (Jn pp − 1)

p

1



= 1 −

 p p −1 1 + (Jn p − 1) 1 + (Jn pp − 1) (1 + (Jn pp − 1)p −1 )p Jn pp − 1 . = (1 + [Jn pp − 1]p −1 )p−1 Take p − 1 powers of both sides of the above equation to get 

 Φn pp

(Jn pp − 1)p −1 = . 1 + (Jn pp − 1)p −1



Now solve for (Jn pp − 1)p −1 and then for Jn pp to obtain Jn pp = 1 +

Φn p 

(1 − Φn pp −1 )p−1

.

Since Φn p is a nondecreasing sequence and Φn p < 1, we see that Jn p must be monotone nondecreasing with n, which proves (ii). Moreover, W is a pre-zero set for pA if and only if Jn p is bounded. This proves (iii). If W is a pre-zero sequence for pA , the limiting function J (which exists by (9.2.3)) for the sequence Jn is p-inner. To verify this, note that J ∈ pA \ {0} since Jn (0) = 1 for all n. Moreover, Jn ⊥p S N Jn for all N  1. Using the bilinear  pairing ·, · (see (6.4.1)) between pA and its dual space pA , along with Theorem 3.4.7, we can rewrite this orthogonality condition as

Jnp−1 , S N Jn  = 0,

N  1. p−1 Jn



We now claim that Jn → J in if and only if → J p−1 in pA . To see this, (n) let ak and ak be the kth coefficients of J and Jn , respectively. The hypothesis (n) that Jn → J implies that ak → ak for each k; that is, viewed as functions of the index k, the coefficient sequence Jk converges to J “pointwise.” Furthermore, the elementary bound

p

p

 

(n) p−1

p−1 p p−1

p (n) p−1

[a [a ] − a  2 ] + a

k





k

k k      (n) = 2p |ak |(p−1)p + |ak |(p−1)p    (n) = 2p |ak |p + |ak |p pA

holds for all k and n. The right hand side is summable in k for each n. Thus, it furnishes a suitable dominating sequence of functions of k, with the sequence being indexed by n, for the Dominated Convergence Theorem to apply. Counting p−1 measure in k is the underlying measure. The conclusion is that Jn → J p−1  p in A . The converse holds since the argument is symmetric in p and p . As a consequence of the claim, we see that

J p−1 , S N J = 0,

N  1.

9.3. GEOMETRIC CONVERGENCE TO THE BOUNDARY

129

In other words, via Theorem 3.4.7, J ⊥p S N J for all N . This shows that J is p-inner and thus completes our proof.  Remark 9.2.4. When p = 2, the 2-inner functions Jn corresponding to fn from (9.2.2) turn out to be n n % 1  % wk − z Jn (z) = , wk 1 − wk z k=1

k=1

which, apart from a multiplicative constant, are finite Blaschke products. Furthermore, Parseval’s theorem, and the fact that n

% wk − eiθ



= 1, θ ∈ [0, 2π],

1 − wk eiθ k=1

show that Jn 2 =

n % k=1

1 . |wk |

Thus when p = 2, the necessary and sufficient condition for W = (wk )k1 ⊆ D \ {0} to be a zero set for 2A = H 2 is sup Jn 2 = sup n1

n1

n % k=1

1 < ∞, |wk |

which is equivalent to the Blaschke condition  (1 − |wk |) < ∞. k1

Remark 9.2.5. Theorem 9.2.1 is somewhat limited as a description of where the functions of pA vanish. Given a sequence W in D, it can tell us if there exists a nontrivial function f ∈ pA that vanishes on W . However, it cannot determine whether such an f vanishes at additional points (reflecting multiplicities, as usual). Even further, the characterization is challenging to apply, since there are few general methods for explicitly calculating the p-inner functions Jn . Finally, unlike the Blaschke condition for the zero sets of the Hardy space H p , a complete characterization of the pre-zero sets of pA cannot depend solely on the moduli of the sequence. These challenges remain open questions for further investigation. 9.3. Geometric Convergence to the Boundary In this section we use Theorem 9.2.1 to construct further examples of pre-zero sequences for pA . To do this, we will find a bound on the norms of the co-projection functions Jn for each finite sequence w1 , w2 , . . . , wn . Then, by applying Theorem 9.2.1, we obtain a limiting function with the prescribed zeros. The bound is made possible by constructing a polynomial that is suitably close to each of the associated metric co-projections. Theorem 9.3.1. Fix p ∈ (1, ∞) and let W = {wk }k1 ⊆ D \ {0}. Choose numbers r1 , r2 , r3 , . . . with rk > 1 and  1 1 1− < . (9.3.2) rk p k1

9. ZEROS OF p A FUNCTIONS

130

If the sequence W satisfies 



(1 − |wk |rk )rk −1 < ∞,

k1

then W is a pre-zero set for pA . Proof. As in Theorem 9.2.1, define, for each positive integer n,  z   z  z  1− ··· 1 − , fn (z) := 1 − w1 w2 wn and the p-inner function Jn = fn − fn . By the convergence of the series in (9.3.2) we see that rk → 1. Let

:=

1 1  1− > 0. −  p rk k1

The next steps involve the p-inner function with a single zero Bw,p =

1 − z/w 1 − wp −1 z

in (8.5.4). For each n  1, the function Fn := Bw1 ,r1 Bw2 ,r2 · · · Bwn ,rn belongs to pA and satisfies the conditions Fn (0) = 1,

1  k  n.

Fn (wk ) = 0,

Thus, by the minimality property of the co-projection Jn (Proposition 8.3.8), we have Jn p  Fn p ,

n  1.

The goal now is to obtain an upper bound for Fn p that is independent of n. We next define p1 , p2 , p3 , . . . by first defining p1 via (9.3.3)

1 1 1 + = +1 p1 r1 p

and then pk by (9.3.4)

1 1 1 + = + 1, pk rk pk−1

k  2.

Thus n 1 1  1 1− , = + pn p rk k=1

the sequence {pn }n1 is decreasing, and p∗ := lim pn = n→∞

1 p

+

−1 1 −

> 1. p

9.3. GEOMETRIC CONVERGENCE TO THE BOUNDARY

131

By virtue of the conditions (9.3.3) and (9.3.4), we can apply Young’s Convolution Inequality (see Theorem 12.1.2 below) repeatedly to obtain (9.3.5) Jn p  Fn p  Bw1 ,r1 Bw2 ,r2 · · · Bwn ,rn p  Bw1 ,r1 r1 Bw2 ,r2 · · · Bwn ,rn p1  Bw1 ,r1 r1 Bw2 ,r2 r2 Bw3 ,r3 · · · Bwn ,rn p2 ···  Bw1 ,r1 r1 Bw2 ,r2 r2 Bw3 ,r3 r3 · · · Bwn−1 ,rn−1 rn−1 Bwn ,rn pn−1  Bw1 ,r1 r1 Bw2 ,r2 r2 Bw3 ,r3 r3 · · · Bwn−1 ,rn−1 rn−1 Bwn ,rn p∗ . We will be done if we can find a uniform bound for the final factor, Bwn ,rn p∗ , as well as the product of the remaining factors. By direction calculation we have 

Bw,r tt = 1 +

(1 − |w|r )t , t |w| (1 − |w|(r −1)t )



Bw,r rr = 1 +

(1 − |w|r )r−1 . |w|r

Let us prove the second identity since the proof of the first identity is similar. Indeed,   

 1 

r Bw,r rr = 1 +

wr −1(j−1) wr −1 − w j1

 1

r  r(r −1)(j−1)

= 1 + wr −1 −

|w| w j1



=1+

1 (1 − |w|r )p r |w| 1 − |w|r

=1+

(1 − |w|r )r−1 . |w|p



When t ∈ (1, ∞), r ∈ (1, ∞), and |w| increases to 1, the quantity Bw,r tt tends to the value 1, since 

(9.3.6)

rk





(1 − |w|r )t t(1 − |w|r )t−1 r  |w|r −1 ≈ → 0.  (1 − |w|(r −1)t ) t(r  − 1)|w|t(r −1)−1

Next, we recall that p∗ > 1 and rk decreases to 1 as k tends to infinity. Thus increases to infinity, and consequently for k sufficiently large we have p∗  rk /(rk − 1), (rk − 1)p∗  rk , 

∗



1 − |wk |(rk −1)p  1 − |wk |rk , 

∗

 (1 − |wk |rk )p . This implies that for sufficiently large k, the last factor Bwn ,rn p∗ of (9.3.5) is no greater than 2. Finally, we see that (9.3.5) is uniformly bounded as n increases

9. ZEROS OF p A FUNCTIONS

132

provided that the points w1 , w2 , w3 , . . . satisfy   (1 − |wk |rk )rk −1 < ∞. (9.3.7) k1

This completes the proof.



Any W satisfying (9.3.7) must be a Blaschke sequence. However, by replacing each factor Bwk ,rk (z) in the above construction with Bwk ,rk (z k ), we obtain a pre0 consisting of the complex kth roots of wk for each k. Because zero set W Bwk ,rk (z)rk = Bwk ,rk (z k )rk 0 is a pre-zero set holds, the same estimate for (9.3.5) applies, telling us that W p 0 for A . One can check that W cannot be the union of finitely many sequences tending toward the boundary at exponential rates. Furthermore, the pre-zero set 0 accumulates everywhere on the boundary of D. Therefore, this produces an W example going beyond those resulting from Theorem 6.2.8. 9.4. Slower Than Geometric Convergence to the Boundary We now use Theorem 9.2.1 to produce an example of a pre-zero sequence that does not satisfy the Newman condition from (9.1.9). First, let us observe that if rk is given by rk := e−1/k , k  1, then rk fails to converge to 1 at an exponential rate as k → ∞. In fact,  k 2 1 − rk+1 1 − e−1/(k+1) = lim = lim e1/k(k+1) = 1. lim k→∞ 1 − rk k→∞ k→∞ k + 1 1 − e−1/k The conclusion remains valid if e is replaced by some other base exceeding one, or if the 1/k in the exponent is replaced by 1/kd for any positive integer d. Furthermore, it holds all the more if, instead of being constant, the base increases with k. These issues come into play at the end of this construction. The overall strategy is to identify an increasing sequence of nested finite zero sets and define fk to to be the polynomial with precisely the zeros of the kth set. We will obtain a corresponding sequence of polynomials Fk that carry these zero sets and possibly other zeros as well. Each Fk will furnish a norm estimate of the associated p-inner co-projection function Jk = fk − fk . By showing that these Fk are uniformly bounded in norm, and using the extremal property of Jk , i.e., Jk p  Fk p , we may conclude that the Jk are also bounded, and hence the constructed sequence is contained in the zero set of a nontrivial function in pA . Let p ∈ (1, ∞) and for each k  1 consider polynomials Fk given by  1  z2 1  z8 z  z 4  z 16 z 32 z 64  1− 1− + 4 + 16 + 32 + 64 Fk (z) := 1 − 2 8 r1 2 r2 r2 4 r3 r3 r3 r3   Nk Nk2  1 z z × · · · × 1 − k−1 Nk + · · · + N 2 , (9.4.1) 2 rk rk k where Nk = 2(2 −1) , and r1 , r2 , . . . , rk belong to (0, 1) yet to be determined. Let us make some observations about these Fk . Each factor is a polynomial with a number of roots; among them are a specific set of roots that we will call the targeted roots. The targeted roots are determined in the following way. For each k−1

9.4. SLOWER THAN GEOMETRIC CONVERGENCE TO THE BOUNDARY

133

j = 1, 2, 3, . . . , k, fix some modulus rj with rj ∈ (0, 1) and notice that the jth factor vanishes, as does Fk itself, consequently, at the points rj , rj e2πi/Nj , rj e2·2πi/Nj , . . . , rj e(Nj −1)·2πi/Nj . For each j, the targeted roots are these Nj elements of D, each with modulus rj , uniformly distributed in argument around the disk. The jth factor thus contributes a huge number of targeted roots, and this serves to slow down the rate of convergence to the boundary of the constructed zero set. Any roots other than the targeted roots will have no bearing whatsoever on the argument. Define fk to be the polynomial with fk (0) = 1 and whose roots are precisely the targeted roots of Fk . Next, consider the effect of multiplying out the factors Fk , with intention of calculating its norm in pA . The fact that all occurrences of z in the defining formula for Fk are all powers of 2 implies that in the expansion each z m can occur only once for each m (namely, the combination of factors corresponding to the binary representation of m). Put differently, when you multiply out the defining formula for Fk , there is no need to collect like terms – each power of z can only arise in at most one way. A typical term in this expansion looks like ±

1 1 zm, 2j1 −1 2j2 −1 · · · 2js −1 rjm1 1 rjm2 2 · · · rjms s

where m1 , m2 , . . . , ms are certain powers of 2 adding up to m. Its absolute value bounded above crudely by 1 2j1 −1 2j2 −1

1 · · · 2js −1

N2 N2 rj1j1 rj2j2

N2

,

· · · rjsjs

and notice that there are 2j1 −1 2j2 −1 · · · 2js −1 terms with the same bound. It follows that Fk pp is bounded above by a sum of terms of the form 1 1 , (2j1 −1 2j2 −1 · · · 2js −1 )p−1 r Nj21 p r Nj22 p · · · r Nj2s p j1

j2

js

where the parameters j1 , j2 , . . . , js are now distinct. And now working backwards from this sum, we obtain the bound    1 1 1  Fk pp  1 + p 1 + 1+ 4p 64p r1 2p−1 r2 22(p−1) r3   1 . × ···× 1+ 2 N p 2(k−1)(p−1) rk k The infinite product converges if and only if the following sum converges:  1 . N 2p (k−1)(p−1) rk k k1 2 Our next task will be to identify values of rk that are sufficient for this sum to converge. Plainly, this happens if there is some a > 1 such that 1 N 2p 2(k−1)(p−1) rk k

=

1 k(log k)a

9. ZEROS OF p A FUNCTIONS

134

for all k. This can be rewritten as  k(log k)a 1/2(2k −2)p rk = (k−1)(p−1) . 2 Now remember that rk is not the modulus of the kth of the zeros of Fk , but rather, it is the modulus of a large collection of zeros. To be sure, if (ρn )n1 is an enumeration of the set of targeted zeros in order of nondecreasing modulus, then |ρn | = rk whenever N1 + N2 + · · · + Nk−1 < n  N1 + N2 + · · · + Nk−1 + Nk . Let us estimate the rate at which the targeted roots of Fk tend toward the boundary. Since Nk−1  N1 +N2 +· · ·+Nk−1 and N1 +N2 +· · ·+Nk−1 +Nk  kNk for all k, we have the bounds Nk−1 < n  kNk . With n and k related in this manner, it follows that Nk−1 < n  kNk log2 (1 + log2 n)  k < 2 + log2 (1 + log2 n). Consequently  [log (1 + log n)]{log (1 + log n)}a (2/n)p 2 2 2 2 {2[1 + log2 n]}p−1  [2 + log (1 + log n)]{2 + log (1 + log n)}a 1/(4n4 )p 2 2 2 2 |ρn |  . {(1/2)[1 + log2 n]}p−1

|ρn | 

By the observations made at the beginning of this section, the pre-zero sequence (ρn )n1 fails to approach the boundary at a geometric rate. This completes the construction. Again, this produces an example of a prezero set that fails to satisfy the Newman condition; that is, it cannot be expressed as the union of finitely many sequences tending toward the boundary of D at an exponential rate. 9.5. A Non-Blaschke Zero Set for p > 2 Vinogradov proved that when p > 2 the zero sets for pA need not be Blaschke sequences (Theorem 9.1.2). Here is another demonstration of this fact using p-inner functions. Theorem 9.5.1. For each p ∈ (2, ∞), there exists a non-Blaschke sequence {wk }k1 ⊆ D, in other words,  (1 − |wk |) = ∞, k1

that is a zero set for pA . Proof. It suffices to construct a non-Blaschke pre-zero set. For then, any zero set containing it must also be non-Blaschke. As in the previous constructions, one of the difficulties is to think of polynomials for which we can control both the zero set and the norm in pA . That concern motivates the following definition. For each

9.5. A NON-BLASCHKE ZERO SET FOR p > 2

135

k  1, let the polynomial Fk be given by  1  z 2! 1  z 3! z  z 2·2!  z 2·3! z 3·3!  1− 1 − + + + Fk (z) := 1 − r1 2 r22! 3 r33! r22·2! r32·3! r33·3!  1  z k! z 2·k! z 3·k! z k·k!  × ··· × 1 − + 2·k! + 3·k! + · · · + k·k! , k! k rk rk rk rk where r1 , r2 , r3 , . . . are moduli in (0, 1). We readily see that among the roots of this polynomial are certain targeted roots, consisting of rj , rj e2πi/j! , rj e2πi·2/j! , . . . , rj e2πi·(j!−1)/j! ,

1  j  k.

Observe how the jth factor in Fk contributes j! of such roots. Again, as with the example in the previous section, let fk be the polynomial whose roots are precisely the targeted roots of Fk and such that fk (0) = 1. It is easily proved by induction that 1 + 2 · 2! + 3 · 3! + · · · + j · j! = (j + 1)! − 1 for each j. As a consequence, when the formula for Fk is multiplied out, each resulting power of z can only arise from one combination of factors, and there is no need to collect like terms. (It helps to notice that if we take the largest power of z from each of the first (j − 1) factors and multiply them, then the resulting power is, by design, one less than the smallest power of the jth factor.) The above feature greatly simplifies the estimate of Fk p . Indeed, a typical term in the expansion looks like 1 1 ± z (n1 k1 !+n2 k2 !+nm km !) , k1 k2 · · · km rkn1 k1 ! rkn2 k2 ! · · · rknm km ! 1 2 m where k1 , k2 ,. . . , km are distinct indices between 1 and k, and for 1  l  m, we have 1  nl  kl . The coefficient can be bounded absolutely above by 1 1 , k k ! k k 1 1 2 2 m km ! k1 k2 · · · km rk rk2 ! · · · rkkm 1 where we have simply replaced powers in the denominator by something possibly larger, increasing the fraction overall. There are exactly k1 k2 · · · km terms in the expansion with the same bound. Accordingly we obtain the estimate  1 1 Fk pp  1 + , (k1 k2 · · · km )p−1 rkpk1 k1 ! rkpk2 k2 ! · · · rkpkm km ! 1 2 m where the sum ranges over all selections k1 , k2 ,. . . , km of distinct indices between 1 and k. The right hand side can be expressed as  1+

1 1p−1 r1p

 1+

1 2p−1 r22p·2!



 ··· 1+

which converges as k tends to infinity precisely when  1 < ∞. p−1 r kp·k! k k k1 Convergence is assured if we take, for example,  1 1/kp·k! rk = , kp−2−α

1 kp−1 rkkp·k!

 ,

9. ZEROS OF p A FUNCTIONS

136

where α > 0. This can only make sense when p > 2, and we choose 0 < α < p − 2. The defined sequence of polynomials therefore satisfies sup Fk p < ∞. k1

Since each Fk (0) = 1, we have Jk p  Fk p for all k, where Jk is the p-inner function corresponding to fk , the polynomial with precisely the targeted roots up to the kth index. Now invoke Theorem 9.2.1 to see that the set W of all targeted zeros is contained in the zero set of a nontrivial function from pA . In this case, the corresponding Blaschke criterion becomes 

k!(1 − rk ) =

k1

 k1

=



k!(1 − exp log rk )   k! 1 − exp

k1





k1

−

k!

1 (log k)(p − 2 − α) kp · k!

 k1

=

k!

 log k k1

 1  (log{1/k})(p − 2 − α) kp · k!

kp

1 (log k)2 (p − 2 − α)2 2(kp · k!)2

(p − 2 − α) −

 k1

1 (log k)2 (p − 2 − α)2 , 2(kp)2 k!

(recalling that there are k! roots with modulus rk ) which diverges to infinity. In the above we used 1 − e−x  x −

x2 . 2 

for sufficiently small x.

Example 9.5.2. In the proof of Theorem 9.5.1, we have a pre-zero set W for pA consisting of layers of concentric points. There are k! points in the kth layer, and their common modulus is  rk =

1

1/[kp·k!]

kp−2−α

,

where p − 2 − α > 0. (In what follows, we may omit the index k = 1, since the corresponding root is r1 = 1.) It was shown that for this zero set, the Blaschke condition fails: 

(1 − |wn |) = ∞.

n1

9.5. A NON-BLASCHKE ZERO SET FOR p > 2

137

However, a bit more can be said. In this situation, 1 − rk = 1 − exp log rk  (p − 2 − α) log k  = 1 − exp − kp · k!  (p − 2 − α) log k 2 (p − 2 − α) log k = +O kp · k! kp · k! (p − 2 − α) log k ;  kp · k! 1 log  k log k − k + log k + log p − log log k − log(p − 2 − α); 1 − rk 1  log k + log log k. log log 1 − rk With these estimates, we have   (9.5.3) (1 − |w|) log log 

w∈W ∞ 

 (p − 2 − α) log k 

k!

k=2

1 −a 1 − |w|

kp · k!

a 1 log k + log log k

∞



1 p−2−α  . p k(log k)a−1 k=2

This is finite if a > 2. Otherwise, it diverges to ∞. The test quantity in (9.5.3) provides us with a more precise sense of how far W departs from being a Blaschke sequence. With only a single logarithm, however, the test quantity converges for all a > 0:  w∈W

 (1 − |w|) log

a 1 1 −a   (p − 2 − α) log k   k! 1 − |w| kp · k! k log k − k ∞

k=2

∞



1 p−2−α  . p k1+a (log k)a−1 k=2

There are yet other ways to quantify how badly the Blaschke condition fails for W . With W = {wk }k1 define ΩN :=

N % k=1

1 . |wk |

For a Blaschke sequence, ΩN would be bounded in N . Recalling again that there are k! zeros with the radius rk , we have for this construction ΩN =

k % 1 , n! r n=2 n

where N = N (k) := 2! + 3! + · · · + k!.

138

9. ZEROS OF p A FUNCTIONS

Then log ΩN =

k  n=2

=

n!

p−2−α log n n! · np

k p − 2 − α  log n p n n=2

Ω(2!+3!+···+k!) = (21/2 31/3 · · · k1/k )(p−2−α)/p . This furnishes information on the rate at which ΩN diverges to infinity, and how that rate depends on p > 2. 9.6. Blaschke, But Not a Zero Set As noted earlier in the Hausdorff-Young Theorem (Theorem 6.3.2), for p ∈ (1, 2)  the space pA is a subset of H p . It follows that when p ∈ (1, 2) all nontrivial zero p sets for A must be Blaschke sequences. Our present aim is to show by construction that the containment is proper. Theorem 9.6.1. For each fixed p ∈ (1, 2) there exists a Blaschke sequence W such that any function f ∈ pA vanishing on W must vanish identically. Proof. To prove the result it will suffice to exhibit a Blaschke sequence that fails to satisfy the test from Theorem 9.2.1. This sequence will contain the 2n th roots of unity, multiplied by some common radius rn , for each n  1. We can choose each rn so that the resulting sequence of points satisfies the Blaschke condition. With this selection of points, a lower bound for the norms of associated sequence of p-inner functions can be computed, by use of Theorem 8.3.12. The abundant symmetries in the selection of roots will aid in the resulting computations. It is then shown that these lower bounds diverge to infinity. The proof is then completed by invoking Theorem 9.2.1. We begin with an observation. Let n be an integer greater than 1, and consider the nth roots of unity e2πi·0/n , e2πi·1/n , e2πi·2/n , . . . , e2πi·(n−1)/n . They form a group G under multiplication. For any nonnegative integer j, their respective powers e2πij·0/n , e2πij·1/n , e2πij·2/n , . . . , e2πij·(n−1)/n . constitute a subgroup of G . It is the trivial subgroup precisely if GCD(j, n), the greatest common divisor of j and n, is equal to n. In this case, the sum of these powers is just n. Let us denote the sum of these powers by Θ(n, j): Θ(n, j) := e2πij·0/n + e2πij·1/n + e2πij·2/n + · · · + e2πij·(n−1)/n . When GCD(j, n) < n, Θ(n, j) is equal to zero, due to the symmetric placement of terms around the origin. Moreover, if the subgroup contains r members, where r divides n, then exactly n/r elements of G map to each element of the subgroup. These properties will make the subsequent norm estimates tractable. Next, suppose that 0 < r 1 < r2 < r3 < · · · < 1

9.6. BLASCHKE, BUT NOT A ZERO SET

139

and consider the finite collection SN of points in D comprising 1

1

2

2

r1 e2πi·0/2 , r1 e2πi·1/2 , 2

2

r2 e2πi·0/2 , r2 e2πi·1/2 , r2 e2πi·2/2 , r2 e2πi·3/2 , ... N

N

N

N

rN e2πi·0/2 , rN e2πi·1/2 , rN e2πi·2/2 , . . . , rN e2πi·(2

−1)/2N

.

By choice of the radii rn , we want the union W of these sets SN to serve as the Blaschke sequence that fails to be a zero set. Toward the goal of applying Theorem 9.2.1, we define JN to be the p-inner function with unit constant term that vanishes at these points. (It may happen that JN vanishes at other points as well—this would have no effect on the construction). That is, JN := fN − f/ N , where fN (z) :=

%  z . 1− w

w∈SN

We would like to estimate the norm in pA of JN , using the formula from Theorem 8.3.12, so we enumerate the points of SN , and consider (9.6.2)

−1  , JN p = inf 1 + α1 k1 + α2 k2 + · · · + αM kM p

where the infimum is over the complex parameters α1 , α2 , . . . , αM , and kj is the point evaluation functional at the jth of the M points of SN (it would have to be that M = 21 + 22 + · · · + 2N ). Let us assume that the points of SN with the same radius rj share a common coefficient bj in (9.6.2). This simplifying assumption has the effect of giving us, at worst, an underestimate for JN , consistent with our strategy. Then 1 + α1 k1 + α2 k2 + · · · + αM kM = (1 + 21 b1 + 22 b2 + · · · + 2N bN )   + b1 r1 Θ(21 , 1) + b2 r2 Θ(22 , 1) + · · · + bN rN Θ(2N , 1) z 1   2 + b1 r12 Θ(21 , 2) + b2 r22 Θ(22 , 2) + · · · + bN rN Θ(2N , 2) z 2 + ···   j + b1 r1j Θ(21 , j) + b2 r2j Θ(22 , j) + · · · + bN rN Θ(2N , j) z j + ··· . Writing 

ΔN := inf 1 + α1 k1 + α2 k2 + · · · + αM kM pp ,

9. ZEROS OF p A FUNCTIONS

140

we now have the bound p  ΔN  1 + α1 k1 + α2 k2 + · · · + αm km   p

p

= 1 + 21 b1 + 22 b2 + · · · + 2N bN



p + b1 r1 Θ(21 , 1) + b2 r2 Θ(22 , 1) + · · · + bN rN Θ(2N , 1)



p 2 + b1 r12 Θ(21 , 2) + b2 r22 Θ(22 , 2) + · · · + bN rN Θ(2N , 2)

+ ···

p j + b1 r1j Θ(21 , j) + b2 r2j Θ(22 , j) + · · · + bN rN Θ(2N , j)

+ ··· .

Next, use the fact that if j is a multiple of 2n , then it must be a multiple all of 2, 22 ,. . . , 2n−1 as well. In this situation Θ(2n , j) = 2n for all n, 1  n  j. Therefore, the above sum over j could be grouped into separate sums over odd multiples of 2, odd multiples of 22 , odd multiples of 23 , and so on. When we reach the last layer 2N roots, we will have to sum over all the multiples (not merely the odd multiples), so as to account for all j. We are discarding the terms with odd values of j, since all of the corresponding Θ(2n , j) are zero. Writing O for the set of odd positive integers, and substituting the numerical values of each Θ(2n , j), we obtain

p

ΔN  1 + 21 b1 + 22 b2 + · · · + 2N bN

 j



b1 r Θ(21 , j) + b2 r j Θ(22 , j) + · · · + bN r j Θ(2N , j) p + 1

2

N

j∈2·O

+

 j



b1 r Θ(21 , j) + b2 r j Θ(22 , j) + · · · + bN r j Θ(2N , j) p 1 2 N

j∈22 ·O

+

 j



b1 r Θ(21 , j) + b2 r j Θ(22 , j) + · · · + bN r j Θ(2N , j) p 1 2 N

j∈23 ·O

+ ···  +

j



b1 r Θ(21 , j) + b2 r j Θ(22 , j) + · · · + bN r j Θ(2N , j) p 1 2 N

j∈2N −1 ·O

+

 j



b1 r Θ(21 , j) + b2 r j Θ(22 , j) + · · · + bN r j Θ(2N , j) p 1 2 N

j∈2N ·N

p

= 1 + 21 b1 + 22 b2 + · · · + 2N bN

 j



b1 r · 21 p + 1 j∈2·O

+

 j



b1 r · 21 + b2 r j · 22 p 1 2

j∈22 ·O

+

 j



b1 r · 21 + b2 r j · 22 + · · · + b3 r j · 23 p 1 2 3

j∈23 ·O

+ ···

9.6. BLASCHKE, BUT NOT A ZERO SET

+



141

j 1

 N −1 p

b1 r · 2 + b2 r j · 22 + · · · + bN −1 r j 1 2 N −1 · 2

j∈2N −1 ·O

+

 j



b1 r · 21 + b2 r j · 22 + · · · + bN r j · 2N p . 1 2 N

j∈2N ·N

Each bn always occurs with the factor 2n , so without loss of generality, we might as well work with the coefficients cn := bn 2n , 1  j  N . Thus we have the bound

p

ΔN  1 + c1 + c2 + · · · + cN

(9.6.3)  j p

c1 r

+ 1

j∈2·O

 j



c 1 r + c2 r j p 1 2

+

j∈22 ·O

 j



c1 r + c2 r j + · · · + c3 r j p 1 2 3

+

j∈23 ·O

+ ···  +

j



c1 r + c2 r j + · · · + cN −1 r j p 1 2 N −1

j∈2N −1 ·O

 j



c1 r + c2 r j + · · · + cN r j p . 1 2 N

+

j∈2N ·N

We will be done if we can find constants c1 , c2 , . . . , cN (which can depend on N ), and a sequence of radii {rn }n1 , such that the expression on the right hand side tends to zero as N −→ ∞, and the resulting sequence W satisfies the Blaschke condition. With this in mind, let us take c1 = c2 = · · · = cN = −1/N , and rn = e−1/2

n

n3−p

,

n  1.

Already the first term in (9.6.3) is zero, so we need only be concerned about the remaining sums. Notice that e−x  1 − x, x  0. Hence we have 

(1 − |w|) =

∞ 

2n (1 − rn )

j=1

w∈W

= 

∞  j=1 ∞  j=1

=

∞  j=1

2n (1 − e−1/2

n

2n 2n n3−p 1 n3−p

n3−p

)

9. ZEROS OF p A FUNCTIONS

142

which is finite. That is, the prescribed collection of points  W = SN ⊆ D N 1

is a Blaschke sequence. Moving on, consider the final sum in (9.6.3).  j



c1 r + c2 r j + · · · + cN r j p (9.6.4) 1 2 N j∈2N ·N

=

=

1 N p

 j



r + r j + · · · + r j p 1 2 N

j∈2N ·N ∞ 

−2N j/21 13−p

1 N p

e

+ e−2

N

j/22 23−p

+ · · · + e−2

N

 j/2N N 3−p p

.

j=1

Among the terms inside the absolute values, (9.6.5)

e−2

N

j/21 13−p

+ e−2

N

j/22 23−p

+ · · · + e−2

N

−N j

there are some exceeding than the quantity e

j/2N N 3−p

,

. For such terms, indexed by s,

−2 j  −N j 2s s3−p 2N  2s s3−p N N

N  s + (3 − p) log2 s + log2 N N − log2 N  s + (3 − p) log2 s N − log2 N  s + (3 − p) log2 N N − (4 − p) log2 N  s. Since we already have s  N , there can be at most (4 − p) log2 N such terms. They 3−p are bounded above by the largest of them, namely e−j/N . Among the remaining terms in (9.6.5), there are no more than N of them, and they are each bounded above by e−N j . Thus, combining these estimates with the elementary bound     |x + y|p  2p −1 (|x|p + |y|p ), we have

−2N j/21 13−p N 2 3−p N N 3−p p

e + e−2 j/2 2 + · · · + e−2 j/2 N















 2p −1 N p e−N jp + 2p −1 (4 − p)p [log2 N ]p e−jp /N

3−p

.



Now sum over j, and divide by N p , to get 

(9.6.6)











2p −1 e−N p 2p −1 (4 − p)p [log2 N ]p e−p /N +  1 − e−N p N p (1 − e−p /N 3−p )

3−p

as an upper bound for the final sum (9.6.4). The first term in the bound (9.6.6) tends to zero as N increases without bound. For N large, the second term behaves like 







2p −1 (4 − p)p [log2 N ]p e−p /N p N p+p −3

3−p

,

9.6. BLASCHKE, BUT NOT A ZERO SET

143

which also tends to zero as N → ∞. Thus we have the expression in (9.6.4) under control. We now lay this aside, and turn to the mth of the sums in (9.6.3), 1  m < N , which is  j

 1 j p

r + r j + · · · + rm .  1 2 p N m j∈2 ·O

By re-indexing, we find that this is equivalent to

 m 1 

j2m j2m p r1 + r2j2 + · · · + rm . (9.6.7)  p N j∈·O

Focusing on the summand of (9.6.7), we have the expression e−2

m

j/21 13−p

+ e−2

m

j/22 23−p

+ e−2

m

j/23 33−p

+ · · · + e−2

m

j/2m m3−p

inside the absolute value signs. Again, there are terms which exceed the quantity e−N j , in which case (with the terms indexed by t) e−2

m

j/2t t3−p

 e−N j

2m j/2t tp−3  N j 2m  2t t3−p N m  t + (3 − p) log2 t + log2 N m  t + (3 − p) log2 m + log2 N m − (3 − p) log2 m − log2 N  t. Since t  m, there can be at most (3 − p) log2 m + log2 N of these terms, which 3−p we can estimate with the largest of them, namely e−j/m . As before, the sum of the remaining terms is bounded above by N e−N j . Thus the expression (9.6.7) is bounded above by 

(9.6.8)











2p −1 N p e−N p 2p −1 [(3 − p) log2 m + log2 N ]p e−p /m +   N p (1 − e−2N p ) N p (1 − e−2p /m3−p )

3−p

.

If we perform the sum of the quantity (9.6.8) from m = 1 to N − 1, and take N → ∞, the contribution from the first term behaves like 



(N − 1) · 2p −1 e−N p → 0. 1 − e−2N p Next, with respect to the second term of (9.6.8), the expression in square brackets is bounded as follows (3 − p) log2 m + log2 N  (4 − p) log2 N for each m, 1  m  N − 1. Also, there exists a positive constant C depending only on p such that 1 C  , 0 < x < 2p . 1 − e−x x Therefore, the contribution from the second term of (9.6.8) is no greater than 

(9.6.9)



2q−1 C(4 − p)p N (log2 N )p . N p+p −3

144

9. ZEROS OF p A FUNCTIONS

It is elementary to check that if p = 2, then p + p > 4. Hence the quantity (9.6.9) tends to zero as N increases without bound. We have shown that ΔN → 0 as N → ∞. This shows that JN p diverges to infinity with increasing N . By Theorem 9.2.1, the Blaschke sequence W fails to be a zero set for pA . This completes the proof.  Theorems 9.5.1 and 9.6.1 together suggest that the zero sets for pA vary depending on p ∈ (1, ∞). This is indeed the case. Theorem 9.6.10. Let 1 < p1 < p2 < ∞. There exists a sequence W of distinct points in D such that a nontrivial function f ∈ pA2 vanishes on W . Moreover, any function in pA1 vanishing on W must be identically zero. The construction of the desired set W , which we omit here and refer the reader to the end notes, uses ideas from the proofs of Theorems 9.5.1 and 9.6.1. 9.7. Zero Sets When 0 < p  1 From Proposition 6.6.9, pA is contained in a class of functions which are smooth up to the boundary and thus there are further restrictions on the zeros beyond the Blaschke condition. Proposition 9.7.1. If p ∈ (0, 1], W ⊆ D, and f ∈ pA vanishes exactly on W , then W is a Blaschke sequence whose closure meets T on a set of measure zero. Proof. Suppose f ∈ pA vanishes on W . Since f must be continuous up to the boundary, then f must also vanish on W ∩ T. But since log |f (ξ)| dm(ξ) > −∞ T

(Proposition 6.5.1), it must be the case that m(W ∩ T) = 0.



When p ∈ (0, 1) one can completely characterize the sets W ⊆ D on which the functions of pA vanish. By Proposition 6.6.9 and Remark 6.6.11 each f ∈ pA must satisfy some Lipschitz condition (depending on p) on D, that is, |f (z) − f (w)|  C|z − w|α ,

z, w ∈ D

for some α ∈ (0, 1). We must have log |f (ξ)|  α log ρ(ξ) + log C, where ρ(ξ) = inf{|ξ − w| : w ∈ W }, But since

T

we conclude that

ξ ∈ T.

log |f (ξ)| dm(ξ) > −∞,



T

log ρ(ξ) dm(ξ) > −∞.

Here is the exact criterion for W to be a zero set for pA for any p ∈ (0, 1). Theorem 9.7.2. Let p ∈ (0, 1). A sequence W ⊆ D, counting multiplicities, is a zero set for pA if and only if W is a Blaschke sequence and log ρ(ξ) dm(ξ) > −∞. T

9.8. A NOTE ABOUT SAMPLING IN p A

145

Of course one would like some specific examples of sequences {wn }n1 in D that satisfy the hypotheses of Theorem 9.7.2. Example 9.7.3. Let {rn }n1 ⊆ (0, 1) be any Blaschke sequence. Then for sufficiently small θ > 0 we have sin θ = inf{|eiθ − x| : x ∈ [0, 1]}  |eiθ − rn |,

n  1,

and so sin θ  inf{|eiθ − rn | : n  1} = ρ(eiθ ).

But since



log sin θ dθ > −∞, 0



it follows that

T

log ρ(ξ) dm(ξ) > −∞.

Thus {rn }n1 is a zero set for pA for every p ∈ (0, 1). Example 9.7.4. Let E be a closed subset of T for which log dist(ξ, E) dm(ξ) > −∞. (9.7.5) T

Choose a countable dense set {ξn }n1 of E and a Blaschke sequence {rn }n1 contained in (0, 1). Using the previous example, one can show that {rn ξn }n1 is a zero set for pA for all p ∈ (0, 1). It is also true that if T \ E comprises the disjoint open arcs {Jn }n1 , then the condition in (9.7.5) is equivalent to the summability condition  m(Jn ) log m(Jn ) > −∞. n1

9.8. A Note About Sampling in pA For the Bergman spaces Ap there is a notion of sampling. A sequence {zn }n1 ⊆ D is called a sampling sequence for Ap if there are positive constants A and B for which 1  2 p p p (9.8.1) Af A  (1 − |zn |) |f (zn )|  Bf Ap . f ∈ Ap . n1

The above definition says that the integral norm on Ap is equivalent to a discrete norm on the sequence {zn }n1 . These sequences have been characterized and have many interesting properties. For example, work of Horowitz [89] produces an example of two zero sets for Ap whose union is no longer a zero set for Ap . Work of Hedenmalm [83] produces two Ap zero sets whose union is a sampling set. As we head towards a possible condition for sampling for pA , let us rephrase the Bergman space sampling condition in terms of kernel functions. For the Bergman space, every element of (Ap )∗ can be represented as := ϕg (f ) f (z)g(z)dα(z), f ∈ Ap , D

p

for some g ∈ A . With this representation, the Bergman kernel function 1 , λ, z ∈ D, Kλ (z) := (1 − λz)2

9. ZEROS OF p A FUNCTIONS

146

satisfies the reproducing property ϕKλ (f ) = f (λ),

f ∈ Ap ,

λ ∈ D.

Furthermore, Kλ Ap 

1 . (1 − |λ|)2(p −1)/p

See [81, p. 7 and p. 18] for details. For a sequence {zn }n1 ⊆ D, one can examine whether the normalized kernel functions  Kzn κzn := ∈ Ap Kzn Ap p form a frame for A . The frame condition in this case would be 1/p  |ϕκzn (f )|p  C2 f Ap , f ∈ Ap . C1 f Ap  n1

Working out the value of |ϕκzn |p we get |ϕκzn |p =

1 |ϕKzn (f )|p Kzn p p A

1 = |f (zn )|p Kzn p p A





 (1 − |zn |)p·2(p −1)/p |f (zn )|p = (1 − |zn |)2 |f (zn )|p . 

This says that {κzn }n1 ⊆ Ap is a frame for Ap when 1/p  (1 − |zn |)2 |f (zn )|p  C2 f Ap , C1 f Ap 

f ∈ Ap ,

n1

which is precisely the sampling condition from (9.8.1). For the Hardy space H p we have seen that every element of (H p )∗ has the form f (ξ)g(ξ) dm(ξ), f ∈ H p , ψg (f ) := T



for some g ∈ H p . Similarly as with the Bergman space, but with the kernel Kλ replaced with the kernel kλ (z) =

1 , 1 − λz

λ, z ∈ D,

we see that ψkλ (f ) = f (λ),

f ∈ H p,

λ ∈ D.

By [81, p. 7] we have 1 . (1 − |λ|)(p −1)/p As we did for the Bergman space, we explore whether the normalized kernels kλ p 

 kzn ∈ Hp kzn H p

9.8. A NOTE ABOUT SAMPLING IN p A

147

form a frame for H p . In this caae the frame condition turns out to be (9.8.2)

Af H p 



(1 − |zn |)|f (zn )|p

1

p

 Bf H p ,

f ∈ H p.

n1

If we were to take this as a definition of sampling in H p , we can show there are no sampling sequences for H p . If there were, then the upper  inequality in (5.1.9) applied to the constant function f ≡ 1 would show that n1 (1 − |zn |) < ∞, which would mean that {zn }n1 is a Blaschke sequence. Since Blaschke sequences are zero sequences for H p , then applying the lower inequality in (5.1.9) to the Blaschke product with these zeros (which has H p norm equal to one) would yield a contradiction. As it turns out, there is no useful version of sampling for pA . Following our analysis for the Bergman and Hardy spaces, we represent every Φ ∈ (pA )∗ as   Φg (f ) = an bn , f (z) = an z n ∈ pA , n0

for some g(z) =



n0 bn z

n

n0



∈ pA . If Lλ (z) =

1 , 1 − λz

λ, z ∈ D,

then ΦLλ (f ) = f (λ),

f ∈ pA ,

λ ∈ D,

and Lλ p 

1 . (1 − |λ|)1/p

As with the Bergman and Hardy spaces, the condition that the normalized reproducing kernels  Lzn ∈ pA Lzn p

form a frame for pA becomes Af p 



p

|f (zn )|p (1 − |zn |) p

1

p

 Bf p ,

f ∈ pA .

n1

As before, we take this as the definition of sampling for pA . Apply the definition of sampling to the monomials f (z) = z N , N  1, to get Ap 



p

|zn |N p (1 − |zn |) p  B p ,

N  1.

n1

An argument with the Dominated Convergence Theorem shows that the middle term in the above inequality goes to zero as N → ∞. This yields a contradiction. Thus there are no sampling sets for pA .

148

9. ZEROS OF p A FUNCTIONS

9.9. Notes Theorem 9.1.2 is from Vinogradov [164]. Theorem 9.1.3 is from Shapiro and Shields [151] while Theorem 9.1.5 appears in the text [53]. The analogous result for Bergman spaces appears in a paper of Horowitz [89]. Both of these theorems work for large classes of analytic functions on D and our proof of Theorem 9.1.3 follows a proof from [53]. Theorem 9.1.8 follows from work of Girela and Pel´aez [69]. The paper of Cheng, Mashreghi, and Ross [36] is the source for the zero set results for pA , p ∈ (1, ∞) (Theorem 9.2.1), along with the zero set constructions in Theorem 9.3.1, Section 9.4, and Theorem 9.5.1. The idea of using certain extremal functions to characterize zero sets extends to broader classes of function spaces (see for instance Cheng, Mashreghi, and Ross [37]). The example of a Blaschke sequence that fails to be a zero set for pA , p ∈ (1, 2), comes from a paper of Cheng and Dragas [29]. Theorem 9.6.10 is from Cheng [28]. The criterion for the zero sets for pA , where p ∈ (0, 1) (Theorem 9.7.2), is actually part of a more general result by Taylor and Williams concerning Lipschitz functions [161]. The alert reader has probably noticed that since functions in pA , p ∈ (0, 1), have a certain smoothness on T, one can talk about zero sets not only on D but also on T (including multiplicity). Shirokov’s book [156] has a wealth of information about this. For a thorough treatment of sampling in various spaces of analytic functions, we point the reader to [53, 81, 148]

10.1090/ulect/075/10

CHAPTER 10

The Shift We have already discussed some of the basic properties of the shift operator (Sf )(z) = zf (z) on pA in Section 7.1, including its norm, adjoint, and spectrum. In this chapter we focus on the invariant subspaces of S. These are the subspaces M ⊆ pA for which SM ⊆ M . For p = 2, these invariant subspaces are completely understood and were characterized by Beurling (Theorem 10.4.1) back in 1949. This seminal result has successfully inspired many important results in operator and function theory. In fact, many invariant subspace results often attempt to model Beurling’s characterization. For all other p, a complete description of the invariant subspaces is still unknown. As we will see, when p > 2, Theorems 10.4.3 and 10.4.5 demonstrate that the invariant subspaces can be extremely complicated. 10.1. Finite Co-Dimensional Invariant Subspaces We first examine the invariant subspaces M ⊆ pA for which the quotient space is finite dimensional. These are called the finite co-dimensional invariant subspaces.

pA /M

Proposition 10.1.1. Let p ∈ (1, ∞) and suppose that q is a polynomial with all of its zeros inside D. Then qpA is an S-invariant subspace of pA for which dim(pA /qpA ) = deg q. Proof. First suppose that the roots of q are distinct points w1 , w2 , . . . wd of D. If q has roots of higher multiplicity, we can apply the argument below with the appropriate number of vanishing derivatives. From the comments following Definition 8.1.4, we note that qpA is a subspace of pA . Next we will show that qpA = {f ∈ pA : f (wj ) = 0, 1  j  d}. The ⊆ containment follows by inspection. For the ⊇ containment, use the difference quotient operator Qw from Proposition 7.2.1 to see that if f ∈ pA with f (wj ) = 0, 1  j  d, then Qw1 Qw2 Qw3 · · · Qwd f ∈ pA . But since c Qw1 Qw2 Qw3 · · · Qwd f = f q for some nonzero constant c, the ⊇ containment follows. To study the co-dimension, take any F ∈ pA and observe that g := Qw1 · · · Qwd F ∈ pA . Direct computation also shows that g=

F − q1 , Cq 149

150

10. THE SHIFT

where C is a nonzero constant and q1 is a polynomial of degree less than d := deg q. Hence F + qpA = q1 + qpA , which means that the cosets z j + qpA ,

0  j  d − 1,

pA /qpA .

span From here it is straightforward to see that these cosets are linearly independent and so dim(pA /qpA ) = d.  The main result of this section is the converse of Proposition 10.1.1. Theorem 10.1.2. Let p ∈ (1, ∞). Suppose that M is an S-invariant subspace of pA of finite co-dimension. Then there exists a polynomial q, with all of its zeros in D, such that M = qpA . The proof requires the following lemma. Lemma 10.1.3. For p ∈ (1, ∞) and ξ ∈ T, the set (z − ξ)pA is dense in pA . Proof. Without loss of generality, assume ξ = 1. For each N  1, define pN (z) =

N −1  j=0

N −j j z N

and perform a routine calculation to obtain the identity 1 − (1 − z)pN (z) =

N 1  j z . N j=1

Thus N p 1  1 1   1 − (1 − z)pN (z)pp =  z j  = p N = p−1 , N j=1 N N p

which goes to zero as N → ∞. This implies that the constant function 1 belongs to the closure of (z − 1)pA . But since (z − 1)pA is an S-invariant subspace of pA which contains 1, then it contains all polynomials. By the density of the polynomials in pA (Proposition 6.1.6), we see that (z − 1)pA = pA , which proves the result.  Proof of Theorem 10.1.2. Define the operator T on pA /M by T (f + M ) = Sf + M . Then T is well defined, for if f1 + M = f2 + M , then f1 − f2 ∈ M , and 

T (f1 + M ) = T f2 + (f1 − f2 ) + M = Sf2 + S(f1 − f2 ) + M . But since f1 − f2 ∈ M , and M is S-invariant, the above equalities continue with = Sf2 + M = T (f2 + M ) which makes T well defined. For any polynomial Q, the S-invariance of M will show that Q(T )(f + M ) = Qf + M .

(10.1.4) Since

dim(pA /M )

= d < ∞, the cosets zj + M ,

0  j  d,

10.2. A QUICK REVIEW OF FREDHOLM THEORY

151

are linearly dependent. Thus there is a polynomial Q0 (not identically equal to zero) such that d  Q0 (z) = cj z j ∈ M . j=0

Hence by (10.1.4) Q0 (T )(f + M ) = Q0 f + M = M for all polynomials f . This implies that Q0 (T ) is zero on pA /M . Factor the polynomial Q0 as Q0 = qh, where q is polynomial whose roots are contained in D and h is a polynomial whose roots are contained in C \ D. If |w| > 1 then the fact that the spectrum of S is contained in D (Proposition 10.1.4) shows that (z − w)pA = pA . From Lemma 10.1.3 we know that (z − ξ)pA is dense in pA for every ξ ∈ T. It follows that hpA is dense in pA and thus qpA ⊆ Q0 pA ⊆ M . Now use Proposition 10.1.1 to get dim(pA /M )  dim(pA /qpA ) = deg q  deg Q0  d = dim(pA /M ). Hence dim(pA /M ) = dim(pA /qpA ). But since

qpA

⊆ M , we see that M = qpA .



10.2. A Quick Review of Fredholm Theory For a Banach space X recall that B(X ) is the algebra of bounded operators on X . Also recall that ker T and Ran T denote the kernel and range of T . Definition 10.2.1. We say T ∈ B(X ) is semi-Fredholm if Ran T is closed and either ker T or X / Ran T is finite dimensional. We say that T is Fredholm if both ker T and X / Ran T are finite dimensional. If ker T and X / Ran T are finite dimensional, then Ran T is closed. Thus every Fredholm operator is a semi-Fredholm. Also worth pointing out is the following theorem of Atkinson which gives an equivalent characterization of Fredholm operators. Let K (X ) denote the norm closed ideal of compact operators on X . Since K (X ) is a closed ideal in B(X ), then the quotient space B(X )/K (X ), called the Calkin algebra, has a well defined addition, scalar multiplication, and multiplication via its cosets T + K (X ),

T ∈ B(X ).

Theorem 10.2.2 (Atkinson). For T ∈ B(X ) the following are equivalent: (i) T is Fredholm. (ii) The coset T + K (X ) is an invertible element of B(X )/K (X ). Definition 10.2.3. For a semi-Fredholm operator T , define the Fredholm index ind(T ) to be ind(T ) := dim(ker T ) − dim(X / Ran T ). The “nullity-rank” theorem from elementary linear algebra says that if T is a linear transformation on a finite dimensional Banach space, then ind(T ) = 0. This result extends to compact operators.

152

10. THE SHIFT

Proposition 10.2.4. Let T , T1 and T2 ∈ B(X ) for a Banach space X . We have the following. (i) If T is a semi-Fredholm operator then ind(T + K) = ind(T ) for every K ∈ K (X ). (ii) If T1 and T2 are two semi-Fredholm (Fredholm) operators then T1 T2 is also a semi-Fredholm (Fredholm) operator and ind(T1 T2 ) = ind(T1 ) + ind(T2 ). Let us apply this to X = pA and T = S (the unilateral shift on pA ), p ∈ (1, ∞). Proposition 10.2.5. The shift S on pA is Fredholm with ind(S) = −1. Proof. Notice that ker S = {0} and Proposition 10.1.1 shows that the range of S is closed with dim(pA /SpA ) = 1.  As it turns out, the index function is continuous in the following sense. Proposition 10.2.6. The semi-Fredholm operators are an open subset of B(X ) and the Fredholm operators are an open subset of the semi-Fredholm operators. Furthermore, the Fredholm index function ind is a continuous function from the semiFredholm operators to Z ∪ {−∞, ∞} and a continuous function from the Fredholm operators to Z. Next we discuss an important part of the spectrum of an operator. Definition 10.2.7. For T ∈ B(X ) define the essential spectrum of T , denoted by σe (T ), to be the λ ∈ C such that T − λI is not a Fredholm operator. From Atkinson’s Theorem we see that λ ∈ σe (T ) if and only if the coset T − λI + K (X ) is not invertible in the Calkin algebra B(X )/K (X ). From this we also see that σe (T ) = σe (T + K) for any K ∈ K (X ). Note that σe (T ) ⊆ σ(T ) (the spectrum of T ). Proposition 10.2.8. The essential spectrum of S is equal to T. Proof. For λ ∈ D we have ker(S − λI) = {0}. Moreover, Proposition 10.1.1 shows that the range of S − λI is closed and dim(pA /(S − λI)pA ) = 1. Thus S − λI is a Fredholm operator for every λ ∈ D and so, since σe (S) ⊆ σ(S) = D, we have σe (S) ⊆ T. For any ξ ∈ T, we see from Lemma 10.1.3 that the range of S − ξI is dense in pA . However, (S − ξI)pA is not equal to pA since that would imply that 1 = (S −ξI)g for some g ∈ pA . But then this would mean that g = 1/(z −ξ) belongs to pA , which is false (the Taylor coefficients of g do not form an p sequence). In summary, the range of S − ξI is not closed for any ξ ∈ T and so ξ ∈ σe (S). Thus σe (S) = T.  The above proof also shows the following. Proposition 10.2.9. For any λ ∈ D we have ind(S − λI) = −1. 10.3. The Division Property In this section we discuss the complexity of the S-invariant subspaces of pA . For any λ ∈ D note that the continuity of the difference quotient operator Qλ on pA shows that (z − λ)f p  cλ f p , f ∈ pA ,

10.3. THE DIVISION PROPERTY

153

where cλ = 1/Qλ . This implies, as we have seen previously, that S − λI is injective with closed range and so it is a semi-Fredholm operator. The same is true for (S − λI)|M , where M is an S-invariant subspace. The Fredholm index of (S − λI)|M is equal to the dimension of the kernel of (S − λI)|M (which is equal to zero) minus the dimension of the co-kernel, i.e., dim M /(S − λI)M . Furthermore, by Proposition 10.2.6, we have the following. Proposition 10.3.1. For any S-invariant subspace M of pA the Fredholm index of (S − λI)|M is the same for all λ ∈ D. As we will see later on in this chapter, the negative of the Fredholm index of (S − λI)|M can be any number in N ∪ {∞}. We will now give an equivalent formulation of when the Fredholm index is equal to minus one. For an S-invariant subspace M of pA define Z(M ) := {λ ∈ D : f (λ) = 0 for all f ∈ M } to be the “set of common zeros” of M . Here multiplicities are not taken into account. Definition 10.3.2. We say that an S-invariant subspace M of pA has the division property if Qλ f ∈ M whenever f ∈ M and λ ∈ D \ Z(M ) with f (λ) = 0. Here are some examples of invariant subspaces of pA with the division property. Proposition 10.3.3. Let p ∈ (1, ∞). If W = {wk }k1 is a sequence of distinct points of D, and {mk }k1 is a sequence of positive integers, then the subspace   RW := f ∈ pA : f (j) (wk ) = 0, 0  j  mk − 1, k  1 of pA from Definition 8.1.4 has the division property. Proof. Here W ⊆ Z(RW ) and so if λ ∈ Z(RW ) and f (λ) = 0, then f (z) z−λ belongs to pA and vanishes (with the required multiplicity) whenever z ∈ W . Thus Qλ f ∈ RW .  (Qλ f )(z) =

By Lemma 8.8.2 we have the following. Proposition 10.3.4. Let p ∈ (1, ∞). If f ∈ pA \ {0} then [f ], the S-invariant subspace generated by f , has the division property. We now relate the division property for M with the integer dim(M /(z − λ)M ). Proposition 10.3.5. Suppose that p ∈ (1, ∞). Let M be a nonzero S-invariant subspace of pA and Ω = D \ Z(M ). Then the following statements are equivalent: (i) for all λ ∈ Ω, and f ∈ M with f (λ) = 0, we have Qλ f ∈ M ; (ii) there is a λ0 ∈ Ω such that for every f ∈ M with f (λ0 ) = 0 we have Qλ0 f ∈ M ; (iii) dim(M /(S − λI)M ) = 1 for all λ ∈ Ω; (iv) there is a λ0 ∈ Ω such that dim(M /(S − λ0 I)M ) = 1.

154

10. THE SHIFT

Proof. Clearly (i) =⇒ (ii) and (iii) =⇒ (iv). The implication (iv) =⇒ (iii) follows from Proposition 10.3.1. Now fix λ0 ∈ Ω and observe that (ii) implies that (S − λ0 I)M = {f ∈ M : f (λ0 ) = 0}. We now argue that dim(M /(S − λ0 I)M ) = 1. Indeed, since λ0 ∈ Z(M ), there is an f ∈ M with f (λ0 ) = 0. Thus the coset f + (S − λ0 I)M is nonzero. So dim M /(S − λ0 I)M  1. If g ∈ M and cf + dg + (S − λ0 I)M is the zero coset for some constants c and d, then cf (λ0 ) + dg(λ0 ) = 0 and so g(λ0 ) = kf (λ0 ). The above shows that the coset g + (S − λ0 I)M is a constant multiple of f + (S − λ0 I)M . Thus we have shown (ii) =⇒ (iv). Now suppose that λ0 ∈ Ω with dim(M /(S − λ0 I)M ) = 1. We know that (S − λ0 I)M ⊆ {f ∈ M : f (λ0 ) = 0} and (S − λ0 )M is closed. But since dim M /(S − λ0 I)M = 1 and dim(M /{f ∈ M : f (λ0 ) = 0}) = 1, it follows that (S − λ0 I)M = {f ∈ M : f (λ0 ) = 0}. This proves (iv) =⇒ (ii). The same argument proves (iii) =⇒ (i) and thus completes the proof.  Definition 10.3.6. For an S-invariant subspace M of pA we define the index of M to be dim(M /(SM )). Corollary 10.3.7. For a non-zero S-invariant subspace M of pA the following are equivalent: (i) M has the division property; (ii) dim(M /(SM )) = 1. 10.4. Beurling’s Theorem Recall from Proposition 6.3.1 that 2A is equal to the Hardy space H 2 , which is a Hilbert space with inner product (f, g) = f (ξ)g(ξ) dm(ξ). T

With this inner product we see that if u is an inner function, i.e., a bounded analytic function on D for which |u(ξ)| = 1 for almost every ξ ∈ T, then uH 2 is a subspace of H 2 (since multiplication by u is an isometry) and is S-invariant. The following celebrated result of Beurling is often the launching point for many analogous investigations of other spaces of analytic functions. Theorem 10.4.1 (Beurling). If M is a nonzero subspace of H 2 that is invariant under S, then there is an inner function u such that M = uH 2 .

10.4. BEURLING’S THEOREM

155

Proof. First notice that SM  M . If this were not the case, then SM = M and so f (z)/z ∈ M whenever f ∈ M . Applying this k times, we conclude that f (z)/z k ∈ H 2 for all k  1. In particular, this means that f (z)/z k is an analytic function on D for all k  0. By a power series argument, this can only happen when f ≡ 0, which contradicts the assumption that M = {0}. Second, since SM  M , one observes that M ∩ (SM )⊥ = {0}

(10.4.2)

and so M ∩ (SM )⊥ contains a function u that is not identically zero. We now argue that there exists a c > 0 such that |u| = c almost everywhere. To be sure, n |u(ζ)|2 ζ dm(ζ) = u, S n u = 0, n  1. T

Taking complex conjugates of both sides of the preceding equation, we also see that |u(ζ)|2 ζ n dm(ζ) = 0, n  1. T

This means that the Fourier coefficients of |u|2 vanish for all n ∈ Z \ {0} and so |u|2 = c almost everywhere on T for some c > 0. Without loss of generality, we can assume that c = 1. Thus u is an H 2 function with unimodular boundary values on T and so by Smirnov’s Theorem (Theorem 5.1.21) we conclude that u ∈ H ∞ . Hence u is an inner function. Third, we now prove that [u] = uH 2 . To see this, observe that [u] ⊆ uH 2 (we have used the fact that uH 2 is closed since u is inner). For the other containment, let g = uG ∈ uH 2 and let GN be the N th partial sum of the Taylor series of G. Notice that uGN ∈ [u] since GN is a polynomial. Also notice that GN → G in H 2 and so, since |u| = 1 almost everywhere, 2 uGN − uG = |uGN − uG|2 dm T |GN − G|2 dm = T

= GN − G2 → 0. This means that uG ∈ [u]. Finally, we check that [u] = M . Since u ∈ M and M is S-invariant, we obtain [u] ⊆ M . To show the reverse inclusion, suppose that f ∈ M and f ⊥ [u]. In order to show that f ≡ 0, we proceed as follows. Since f ⊥ [u], f (ζ)u(ζ)ζ n dm(ζ) = f, S n u = 0, n  0. T

However, since u ⊥ SM by (10.4.2), we also know that f (ζ)u(ζ)ζ n dm(ζ) = S n f, u = 0, T

n  1.

The previous two equations imply that all of the Fourier coefficients of f u vanish. Thus f u = 0 almost everywhere on T. But we have already shown that |u| = 1

156

10. THE SHIFT

almost everywhere on T and so f = 0 almost everywhere on T. This proves that f ≡ 0 and thus [u] = M .  Though the above proof of Beurling’s Theorem is very much a Hilbert space proof designed for H 2 , there is an adaptation of the proof that works for the Hardy spaces H p , p ∈ (1, ∞), and shows that every nontrivial invariant subspace takes the form uH p for some inner u. In fact, the result can be extended to all p ∈ (0, 1) (see the end notes of Chapter 5). However, for the pA spaces, any attempted version of Beurling’s theorem for the p = 2 case encounters difficulties. Indeed, if one looks at the proof of Beurling’s Theorem, one sees that if f ∈ H 2 and f is the metric projection of f onto [Sf ] (which in this case will be the orthogonal projection) then the co-projection J = f − f turns out to be the inner factor of f and [J] = [f ]. When p = 2, the corresponding result does not always hold (Corollary 8.7.7) since there are some circumstances where J has zeros besides those of f , making the above equality impossible. Furthermore, Beurling’s Theorem also shows that every S-invariant subspace M of 2A is cyclic, in that M = [u] for an inner function u. From our earlier examination of the division property and the index of an invariant subspace, notice that cyclic invariant subspaces have the division property, or equivalently, have index equal to one. Here is a dramatic result which shows that when p > 2, the invariant subspaces of pA can be very far from cyclic. Theorem 10.4.3 (Abakumov-Borichev). Let p ∈ (2, ∞) and n ∈ N ∪ {∞}. Then there is an S-invariant subspace M of pA for which dim(M /SM ) = n. Remark 10.4.4. When p ∈ (1, 2) it is unknown whether there are S-invariant subspaces M of pA with dim(M /(SM )) > 1. For the Hardy space H 2 any two (nonzero) S-invariant subspaces have a nonzero intersection. In fact, if u and v are inner functions then (uH 2 ) ∩ (vH 2 ) = uvH 2 . For pA this is far from true. Theorem 10.4.5 (Abakumov). For each p ∈ (2, ∞) there is a family {Mλ : λ ∈ C} of nonzero S-invariant subspaces of pA for which Mλ ∩ Mλ = {0},

λ = λ .

The proofs of the results above are quite long and so we point the reader to the end notes for the original references. 10.5. Notes We mentioned some general theorems which say that the shift, or its compressions, model various types of Hilbert space operators. Results in this regard are found in the texts [61, 62, 130–132]. The proof of Theorem 10.1.2 uses very few specific properties of pA and so the result is true for other Banach spaces of analytic functions (see Aleman [8] and Axler and Bourdon [14]). Our proof of Theorem 10.1.2 follows an argument from [14]. Proposition 10.3.5 is from a paper of Richter [136]. The proofs in our survey of the Fredholm theory follow the ones from the texts [3, 44]. For more on the division property see the papers of Aleman

10.5. NOTES

157

and Richter [9] and Richter [136]. See [19] for the original proof of Beurling’s Theorem (Theorem 10.4.1). The proof presented here is from [66]. Theorem 10.4.3 is one of Abakumov and Borichev [1]. Results similar to this (large index S-invariant subspaces) hold for the Bergman spaces [70, 83]. Theorem 10.4.5 is due to Abakumov [2].

10.1090/ulect/075/11

CHAPTER 11

The Backward Shift In this chapter we will continue the investigation begun in Chapter 7 of the backward shift operator f (z) − f (0) (S ∗ f )(z) = z on pA . We have already seen that S ∗ on pA , p ∈ (1, ∞), is the adjoint of the shift  S on pA via the sesquilinear pairing     (f, g) = an bn , f (z) = an z n ∈ pA , g(z) = bn z n ∈ pA , n0

n0

n0

(Proposition 7.1.2 ). In previous chapters we used the bi-linear pairing f, g =  n0 an bn (notice the difference in brackets and the lack of complex conjugation 

in the second slot) to represent the dual pairing between pA and pA . However, the early work on S ∗ comes from the Hilbert space case H 2 . Thus we will follow historical precedent and use the sesquilinear pairing in this chapter. Here we will cover the S ∗ -invariant subspaces of pA with a special focus on the vectors f ∈ pA for which  [f ]∗ := {S ∗n f : n  0} = pA . 

These are called the cyclic vectors for S ∗ . If M is an S-invariant subspace of pA , then M ⊥ = {g ∈ pA : (f, g) = 0, f ∈ M }, the annihilator of M , is an S ∗ -invariant subspace of pA and vice versa. We will make great use of this annihilator concept to obtain information about the S ∗ -invariant subspaces, especially when p = 2. From the previous chapter, we know that when p = 2 the S-invariant subspaces do not have a complete description and similarly the same will be true for the S ∗ -invariant subspaces. However, as we will see in this chapter, there are some partial results that make connections to several areas of analysis and number theory. 11.1. Pseudocontinuations Let us start with the p = 2 case, where a lot can be said. We spend some time on this specific case since much of what we can say about the backward shift on pA is motivated by what happens in the 2A situation. Here, we remind the reader of our discussion from Chapter 5 that 2A = H 2 , the Hardy space, is a Hilbert space with inner product (f, g) =

f (ξ)g(ξ) dm(ξ), T

where the functions in the integrand are the radial boundary functions of f, g ∈ H 2 on T and m is normalized Lebesgue measure on T (see Theorem 5.1.5). If N is an S ∗ -invariant subspace of H 2 , then its annihilator N ⊥ is an S-invariant subspace 159

160

11. THE BACKWARD SHIFT

which can be described using Beurling’s Theorem (Theorem 10.4.1) as N for some inner function u. Thus N = (uH 2 )⊥ .

⊥

= uH 2

Remark 11.1.1. In this chapter when we use the term “inner function” we mean in the classical sense, that is to say, u ∈ H ∞ and |u(ξ)| = 1 for almost all ξ ∈ T. The description of (uH 2 )⊥ involves the concept of a “pseudocontinuation.” Let us develop the main theorem here and make all the connections later. Definition 11.1.2. Let De = {z : |z| > 1} ∪ {∞} be the extended exterior disk and let    an  2 H 2 (De ) := F (z) = ∈ Hol(D ), |a | < ∞ e n zn n0

n0

be the Hardy space of the extended exterior disk. Note that F ∈ H 2 (De ) if and only if F (1/z) ∈ H 2 (D). In the result below, we adhere to the following notation: If f ∈ H 2 (D) and G ∈ H 2 (De ) then f (ξ) := lim− f (rξ) and G(ξ) := lim+ G(rξ) r→1

r→1

exist for almost every ξ ∈ T. Theorem 11.1.3 (Douglas-Shapiro-Shields). Fix an inner function u. For f ∈ H 2 = H 2 (D) the following are equivalent: (i) f ∈ (uH 2 )⊥ ; (ii) there is a g ∈ H 2 for which f (ξ) = ξg(ξ)u(ξ) for almost every ξ ∈ T; (iii) there is a G ∈ H 2 (De ) with G(∞) = 0 and such that f (ξ)u(ξ) = G(ξ) for almost every ξ ∈ T. Proof. Let us first establish (ii) =⇒ (iii). If g ∈ H 2 is such that f (ξ) = ξg(ξ)u(ξ) for almost every ξ ∈ T, then 1 G(z) = g(1/z) z belongs to H 2 (De ), vanishes at infinity, and lim

r→1−

f (rξ) f (ξ) = = ξg(ξ) = lim G(rξ) u(rξ) u(ξ) r→1+

for almost every ξ ∈ T. Thus (ii) =⇒ (iii). To show (iii) =⇒ (ii), note that if G ∈ H 2 (De ) with G(∞) = 0, then g(z) = G(1/z) belongs to H 2 (D) and vanishes at the origin. This implies that g(z) = zg1 (z) for some g1 ∈ H 2 (D). But since G(ξ) = f (ξ)u(ξ) for almost every ξ ∈ T, we have f (ξ)u(ξ) = G(ξ) = G(1/ξ) = g(ξ) = ξg1 (ξ). Hence f (ξ) = u(ξ)ξg1 (ξ), which proves (ii). We will now show that (i) ⇐⇒ (ii). If f ∈ H 2 then n f (ξ)u(ξ) ξ dm(ξ) = 0, f ∈ (uH 2 )⊥ ⇐⇒ T

n  0,

⇐⇒ uf ∈ ξH 2 ⇐⇒ f ∈ uξH 2 , which completes the proof.



11.1. PSEUDOCONTINUATIONS

161

The third condition of Theorem 11.1.3 naturally brings us to the concept of a “pseudocontinuation.” Recall (5.1.19) that f  N(D) = : f ∈ H ∞ (D), g ∈ H ∞ (D) \ {0} g is the class of meromorphic functions of bounded type. From Theorem 5.1.7 we know that for each f ∈ H ∞ \ {0}, the radial limit f (ξ) = lim f (rξ) − r→1

exists and is nonzero for almost every ξ ∈ T. This means that any h = f /g ∈ N(D) has a well-defined radial limit almost everywhere on T. The same can be said, but with radial limits coming from De instead of D, about F  : F, G ∈ H ∞ (De ) , N(De ) = G the meromorphic functions of bounded type on De . From (5.1.20) we have H 2 (D) ⊆ N(D) and H 2 (De ) ⊆ N(De ). In condition (iii) of Theorem 11.1.3 we see that the radial boundary values of f /u ∈ N(D) are almost everywhere the same as those for G ∈ N(De ). So, in a way, we can think of G has a “continuation” of f /u to the exterior disk even though G might not be an analytic continuation of f /u (see below). With this as our motivation, we make the following definition. Definition 11.1.4. If f ∈ N(D) and F ∈ N(De ), we say that F and f are pseudocontinuations of each other if lim f (rξ) = lim F (rξ) +

r→1−

r→1

for almost every ξ ∈ T. Thus condition (iii) in Theorem 11.1.3 can be replaced by the following equivalent condition: The meromorphic function f /u has a pseudocontinuation F ∈ H 2 (De ) that vanishes at infinity. As it turns out, the concept of a pseudocontinuation is used to describe the cyclic vectors for S ∗ , i.e., those f ∈ H 2 for which  [f ]∗ = {S ∗n f : n  0} = H 2 . If u is an inner function then u has the pseudocontinuation 1 , z ∈ De . (11.1.5) U (z) = u(1/z) This is a good place to note that though U is a pseudocontinuation of u, it may not be an analytic continuation. For example, if u is an infinite Blaschke product whose zeros accumulate everywhere on T, then u does not have an analytic continuation across any point of T, even though it has a pseudocontinuation U . Theorem 11.1.6 (Douglas-Shapiro-Shields). A function f ∈ H 2 is not cyclic for S ∗ if and only if it has a pseudocontinuation to a function of bounded type on De . Proof. If f is non-cyclic for S ∗ , then f ∈ (uH 2 )⊥ for some inner function u. By Theorem 11.1.3, f /u has a pseudocontinuation G ∈ H 2 (De ). But since u has a pseudocontinuation given by (11.1.5), f has a pseudocontinuation U G that is of bounded type.

162

11. THE BACKWARD SHIFT

For the other direction, suppose there exists an F ∈ N(De ) for which F (ξ) = f (ξ) almost everywhere on T. The definition of N(De ) says that F (z) =

u1 (1/z)g1 (1/z) , u2 (1/z)g2 (1/z)

where u1 , u2 are inner functions on D and g1 , g2 are bounded outer functions on D. But f = F almost everywhere on T and so it follows that for almost every ζ ∈ T,

 g1 (ζ)  g1 (ζ)  f (ζ) = u2 (ζ)u1 (ζ) = ζu2 (ζ) · ζu1 (ζ) . g2 (ζ) g2 (ζ) The first factor on the far right hand side of the previous line is the boundary function for an inner function u, while the second factor is the complex conjugate of a boundary function for a g ∈ N + (the Smirnov class from (5.1.18)) which vanishes at the origin. Since g also has L2 boundary values, we see from Smirnov’s theorem (Theorem 5.1.21) that g ∈ H 2 . From Theorem 11.1.3, f ∈ (uH 2 )⊥ which means that f is non-cyclic.  Some examples of cyclic vectors are in order. Certainly any inner function u is not a cyclic vector since it has a pseudocontinuation of bounded type given by (11.1.5). One can also see this directly from [u]∗ = (uzH 2 )⊥ . We can extend our class of examples with the following compatibility result. Proposition 11.1.7. Suppose that f ∈ H 2 has a pseudocontinuation F ∈ N(De ) and f also has an analytic continuation across an arc I ⊆ T. Then this analytic continuation must be equal to F near I. Proof. The function f1 , defined to be the analytic continuation of f on Ω = {rξ : 1 < r < r1 , ξ ∈ I} belongs to N(Ω) as does F |Ω . Both of these functions have the same radial boundary values almost everywhere on I. Therefore f1 = F |Ω (Theorem 5.1.7).  Proposition 11.1.7 can be used to create various examples of cyclic vectors for the backward shift such as f (z) = log(1 − z). This function has a branch cut at z = 1, and so it cannot possibly have a pseudocontinuation to an F ∈ N(De ), or else F would inherit the branch cut of f (see [66] for more details). For another example, consider the function  2−n f (z) = . z − (1 + n1 ) n1 Notice how the poles of f are not a Blaschke sequence in De . One can show that f ∈ H 2 is cyclic since it cannot possibly have a pseudocontinuation of bounded type, or else this pseudocontinuation would have too many poles. There are many other results, too numerous to get into here, and we refer the reader to the end notes of this chapter for further references. 11.2. Other Types of Continuations In this section we develop a replacement for the concept for a pseudocontinuation that is appropriate for the pA spaces. We are guided by the following observation: If f ∈ H 2 = 2A is a non-cyclic vector for S ∗ then Theorem 11.1.6 says

11.2. OTHER TYPES OF CONTINUATIONS

163

that f has a pseudocontinuation of bounded type in that there are two functions F, G ∈ H ∞ (De ) for which F (ξ) f (ξ) = G(ξ) for almost every ξ ∈ T. In the above equation we mean that the radial limits from D on the left hand side and the radial limits from De on the right hand side are equal almost everywhere. We can also think about this in terms of Fourier series in the following way: If   Ak  Bk f (z) = ak z k , F (z) = , G(z) = , k z zk k0

k0

k0

then the identity f (ξ) = F (ξ)/G(ξ), or equivalently f (ξ)G(ξ) = F (ξ), for almost every ξ ∈ T, can be written in terms of Fourier series as 2

2

(a0 + a1 ξ + a2 ξ 2 + · · · )(B0 + B1 ξ + B2 ξ + · · · ) = A0 + A1 ξ + A2 ξ + · · · . Notice how we are formally multiplying out the two Fourier series on the left and gathering up the terms corresponding to like powers (positive and negative) of ξ to obtain the Fourier series on the right. In other words, we are showing that the Taylor coefficients of the functions f , F , and G satisfy the identities   An = ak Bk+n , n  0 and ak+n Bk = 0, n  1. k0

k0

We will now develop a language to do this in the pA setting where we do not have Fourier expansions due to the possible lack of radial boundary values (Corollary 6.5.4). We will also develop the idea that to each non-cyclic vector f there is a meromorphic function defined on De which can be thought of as a “continuation” of f . To this end, let p ∈ (1, ∞) and  f (z) = ak z k ∈ pA . k0

For {Bk }k0 ∈ 

p

define G(z) =

 Bk k0

zk

.

A computation with H¨ older’s inequality shows that the series above converges uniformly on compact subsets of De and thus G defines an analytic function on De .  Furthermore, owing to the fact that the sequence {Bk }k0 belongs to p , we can  think of the function G as an element of pA (De ), the analytic functions on De with p -summable Taylor coefficients. 0n }n0 defined by H¨older’s Inequality shows that the sequences {An }n1 and {A   0n := (11.2.1) An := ak Bk+n , A ak+n Bk k0

k0

are bounded sequences. From here one can define the formal Laurent series  An  0n z n + . f #G := A zn n0

n1

The “product” f #G is merely the formal product of the Laurent series f and G where we gather up coefficients corresponding to like powers (positive and negative)

164

11. THE BACKWARD SHIFT

of z. At this point, f #G is only a formal Laurent series as we have not established any annulus of convergence. We now relate this vectors for S ∗ .  formal kseriesp multiplication to non-cyclic ∗ Suppose that f (z) = k0 ak z ∈ A is non-cyclic for S , i.e., [f ]∗ = pA . By the  p k Hahn-Banach Separation Theorem, there is a g(z) = k0 bk z ∈ A \ {0} for which  (S ∗n f, g) = an+k bk = 0, n  0. k0

If G(z) =

 bk , zk

k0

then G ∈ (11.2.2)

 pA (De )

and by (11.2.1) we have  Ak f #G = . zk k1

Moreover, since the coefficients An are bounded, the formal Laurent series f #G is a Taylor series (converging in De ) whose coefficients belong to ∞ . So f #G ∈ ∞ A (De ) and vanishes at infinity. This argument can also be reversed to obtain the following. Proposition 11.2.3. Suppose p ∈ (1, ∞) and f ∈ pA . Then f is a non-cyclic  vector for S ∗ if and only if there is a G ∈ pA (De ) and an F ∈ ∞ A (De ) which vanishes at infinity such that f #G = F . By the equality f #G = F , we mean that, as in (11.2.2), f #G is a Taylor series in De that is equal to the Taylor series of F . At this point the skeptical reader might view all of this as merely dressing up the definition of non-cyclicity of f in terms of a formal, and perhaps not so enlightening, language of multiplication of Laurent series. However, by considering the identity f #G = F to mean that f (on D) is regarded as being “continued” to the meromorphic function F/G (on De ), the next result says that this type of continuation is compatible with analytic continuation. 

Theorem 11.2.4. Let p ∈ (1, ∞). Suppose that f ∈ pA , G ∈ pA (De ), and F ∈ ∞ A (De ) with f #G = F . If f has an analytic continuation across a fixed point ξ ∈ T to a bounded open neighborhood U of ξ, then f = F/G on U . Before getting to the proof of this theorem, let us show that functions like f (z) = log(1 − z) (which belong to pA by the proof of Proposition 6.2.1), that have a branch point, are cyclic vectors. Corollary 11.2.5. Suppose that p ∈ (1, ∞). If f ∈ pA has a branch point in T, then f is a cyclic vector for S ∗ . 

Proof. If f were non-cyclic, there exist a G ∈ pA (De ) and an F ∈ ∞ A (De ) vanishing at infinity such that f #G = F . However, by Theorem 11.2.4, this would force the meromorphic function F/G to have a branch cut in its domain De , which is a contradiction.  The proof of Theorem 11.2.4 requires the following result about normal families. Recall that a family F of analytic functions on a domain Ω forms a normal family if each sequence in F has a subsequence that converges uniformly on compact subsets

11.2. OTHER TYPES OF CONTINUATIONS

165

of Ω to a function in Hol(Ω). Also recall Montel’s Theorem [43, p. 153] which says that F is a normal family if and only if for each compact subset K ⊆ Ω, there is an MK such that |f (z)|  MK for all z ∈ K and f ∈ F . In the statement below, area measure on the complex plane is denoted by dA. Lemma 11.2.6. Let U be an open set in C and let F be a family of functions from Hol(U ). If there is a positive function ρ ∈ L1 (U, dA) such that log+ |f (z)|  ρ(z),

f ∈ F,

z ∈ U,

then F is a normal family. Proof. By Montel’s Theorem it suffices to show that given any compact set K ⊆ U, sup{|f (z)| : z ∈ K, f ∈ F } < ∞.

(11.2.7)

To this end, fix a compact set K ⊆ U and pick a δ > 0 such that Kδ := {z ∈ C : dist(z, K)  δ} ⊆ U. Then for f ∈ F and z ∈ K, the subharmonicity of log+ |f | implies that 1 + log |f (z)|  2 log+ |f (w)| dA(w) πδ |w−z|δ 1 ρ(w) dA(w).  2 πδ Kδ   1 ρ(w) dA(w) 2 πδ Kδ for all f ∈ F and z ∈ K. This implies (11.2.7) and completes the proof.

Thus

|f (z)|  exp

Proof of Theorem 11.2.4. Let   Bk  f (z) = ak z k ∈ pA , G(z) = ∈ pA (De ), k z k0

k0

where Ak =



aj+k Bj ,

k  0.

N  Bn , zn n=0

z = 0,

j0

For each N  0 define the function GN (z) := and the complex numbers CN,s :=

N 

aj+s Bj ,

s  0,

j=0

CN,−t :=

N −t 

aj Bj+t ,

0  t  N.

j=0

For fixed N and s, H¨older’s Inequality shows that (11.2.8)

|CN,s |  f p Gp

F (z) =

 Ak k0

zk



,

166

11. THE BACKWARD SHIFT

and thus the numbers CN,s are uniformly bounded in N and s. From here it follows that the function HN defined on D by  HN (z) := CN,s z s s1

is analytic. If the function IN is defined by IN (z) :=

N  CN,−t t=0

zt

z = 0,

,

a direct computation shows that (11.2.9)

HN (z) = f (z)GN (z) − IN (z),

z ∈ D \ {0},

N  0.

If we assume that f has an analytic continuation (also denoted by f ) to a bounded open neighborhood U of a fixed point ξ ∈ T, we use (11.2.9) to conclude the same is true for the function HN . Note that GN and IN are rational functions with a pole at z = 0. To finish the proof, we will produce a subsequence {Nk }k1 such that HNk → 0

(11.2.10) and

INk → F

(11.2.11)

uniformly on compact subsets of U ∩ De . Thus, equation (11.2.9), and the fact that GNk → G uniformly on compact subsets of U ∩ De , yield f = F/G on U ∩ De . In order to prove (11.2.10) and (11.2.11), we need to make some estimates. For z ∈ D, we can use (11.2.8) to obtain   1 |HN (z)|  . |CN,s ||z|s  f p Gp |z|s = f p Gp 1 − |z| s1

s0

For z ∈ De , H¨older’s inequality yields |GN (z)| 

N 

|Bj ||z|−j 

j=0

 Gp 





|Bj ||z|−j

j0

1 1 − |z|1p

1/p





1 1/p 1 1 − |z|

1 1 . 1 − |z|

We also have, for z ∈ De , (11.2.12)

|IN (z)| 

N 

|CN,−t ||z|−t

t=1



 t0

|aj ||Bj+t |

j0

 f p Gp

1 1 1 − |z|



|z|−t

11.2. OTHER TYPES OF CONTINUATIONS

167

and so (from (11.2.9)) for z ∈ U ∩ De we have C . |z| − 1

|HN (z)|  |f (z)||GN (z)| + |IN (z)|  In summary, C

, |HN (z)| 

1 − |z|

z ∈ U.

This means that 1

, log+ |HN (z)|  C log+

1 − |z|

But since

z ∈ U.



1

dA(z) < ∞, log+

1 − |z|

U

we can apply Lemma 11.2.6 to see that {HN }N 1 forms a normal family on U . From (11.2.12), {IN }N 1 also forms a normal family on De . From here, one can find a subsequence {Nk }k1 so that {HNk }k1 and {INk }k1 converge uniformly on compact subsets of U (respectively De ). Since {INk }k1 converges uniformly on compact subsets of De to some I ∈ Hol(De ), which we denote by  C−t , I(z) := zt t1

then (by Cauchy’s theorem) for each t, the numbers CNk ,−t converge to C−t as Nk → ∞. Recalling that  At = aj Bj+t , t  1, j0

are the Taylor coefficients (about z = ∞) of F , we observe that ∞ k −t

N 



|CNk ,−t − At | =

aj Bj+t − aj Bj+t



=

j=0

j=0



aj Bj+t



jNk −t+1





|aj Bj+t |,

jNk −t+1

which is the tail end of an absolutely convergent series and so CNk ,−t → At . Thus  At = F (z) INk (z) → zt t1

uniformly on compact subsets of De , which proves (11.2.11). To prove (11.2.10), recall that HNk converges uniformly on compact subsets of U and so it suffices to prove HNk (z) → 0 for each z ∈ U ∩ D. To this end, notice that for each z ∈ U ∩ De ,  |HNk (z)|  |CNk ,s ||z|s . s1

168

11. THE BACKWARD SHIFT

Furthermore, by the hypothesis f #G = F , we have   0s := aj+s Bj = CNk ,s + aj+s Bj , 0=A j0

and so |HNk (z)| 



s  0,

jNk +1

|CNk ,s ||z|s 

s1

Since

 s1





|aj+s ||Bj |



|z|s .

jNk +1

|aj+s ||Bj |

jNk +1

is again the tail end of a convergent series, we can apply the dominated convergence theorem to see that HNk → 0 pointwise on U ∩ D. This proves (11.2.10) and the proof of the theorem is now complete.  11.3. Finite Dimensional Invariant Subspaces Though the S ∗ -invariant subspaces of pA lack a complete description when p = 2, we can at least describe those of finite dimension. Recall from (8.3.11) that (m) the reproducing kernel functions kw satisfy the defining conditions (11.3.1)

(m) (f, kw ) = f (m) (w),

w ∈ D,

m  0,

f ∈ pA ,

and are given by (m) kw (z) =

m!z n , (1 − wz)m+1

m  0,

z, w ∈ D.

Theorem 11.3.2. Let p ∈ (1, ∞). Suppose that N is an S ∗ -invariant subspace of Then N is finite dimensional if and only if there are points w1 , . . . , wn in D and nonnegative integers s1 , . . . sn such that  (m) : 0  m  sj , 1  j  n}. N = {kw j pA .



Proof. The space N ⊥ is an S-invariant subspace of pA . Furthermore, using  the natural isometric isomorphism between pA /N ⊥ and N ∗ , it follows that N is finite dimensional if and only if N ⊥ is finite co-dimensional. Now use Theorem  10.1.2 to write N = (qpA )⊥ , for some polynomial q whose zeros all lie in D, and then use (11.3.1).  11.4. Gap Series Theorems A theorem of Hadamard states that if {nk }k1 is an increasing sequence of positive integers satisfying n  k+1 inf : k  1 > 1, nk and the power series  f (z) = ak z nk k1

has radius of convergence equal to one, then f does not have an analytic continuation across any arc of T. These types of series are called Hadamard gap series. We have seen in Section 11.2 that the cyclic vectors for S ∗ on pA do not have a generalized analytic continuation to De in the sense of Proposition 11.2.3. In this

11.4. GAP SERIES THEOREMS

169

section we discuss the following result of Abakunov which shows that, in certain circumstances, Hadamard gap series do not have a generalized analytic continuation either and thus form cyclic vectors for S ∗ . Theorem 11.4.1 (Abakumov). Suppose p ∈ (1, ∞) and {nk }k1 is an increasing sequence of positive integers with n  k+1 inf : k  1 > 2. nk Assume  ak z nk ∈ pA f (z) = k1

with ak = 0 for all k and set Rk = 

(11.4.2)

|ak |p . p j>k |aj |

Then f is a cyclic vector for S ∗ if and only if 1  (11.4.3) |Rk | p−1 = ∞. k1

To give the reader a flavor of how this theorem works, and to avoid some of the technical details, we will only prove one direction. Before proceeding, we pause for a moment to comment about the “remainders” Rk from (11.4.2). From an exercise in [144, Ch. 3, Prob. 12], one can prove that if {ck }k1 is a summable sequence of positive numbers and ck rk =  j>k cj  are the remainders, then k1 rk = ∞. Applying this to our set up, we see that  k1 Rk = ∞. The divergence criterion from (11.4.3) requires some extra effort on the Rk to form a divergent series. Certainly when ak = 2−k this criterion is satisfied. We  will follow Abakumov’s proof and first verify the following two lemmas. For f (z) = k0 ak z k ∈ pA , let Ω(f ) := {k  0 : ak = 0}. Lemma 11.4.4. Let p ∈ (1, ∞). Suppose that n  k+1 (11.4.5) inf :k1 >2 nk and  ak z nk ∈ pA . f (z) = k1

Then, for any two unequal nonnegative integers d1 and d2 , the set Ω(S ∗d1 f ) ∩ Ω(S ∗d2 f ) contains at most one element. Proof. Suppose that d1 > d2 . Towards a contradiction suppose that there are integers m < n so that m, n ∈ Ω(S ∗d1 f ) ∩ Ω(S ∗d2 f ). Then m + d1 , n + d1 , m + d2 , n + d2 ∈ Ω(f ).

170

11. THE BACKWARD SHIFT

It follows that n + d1 > n + d2 ,

n + d1 > m + d1 .

By (11.4.5) we have 1 2 (n

+ d1 ) > n + d2 ,

1 2 (n

+ d1 ) > m + d1 .

Add and then simplify to get 1 2 (n

+ d1 ) + 12 (n + d1 ) > (n + d2 ) + (m + d1 ), n + d1 > n + d2 + m + d1 , 0 > d2 + m, 

which is clearly a contradiction.

A subset (not necessarily a subspace) F ⊆ pA is S ∗ -invariant if f ∈ F =⇒ S f ∈ F. ∗

Lemma 11.4.6. Let p ∈ (1, ∞). Suppose that F is an S ∗ -invariant subset of pA such that 1 ∈ [f ]∗ for every f ∈ F . Then every element of F is a cyclic vector for S ∗.  Proof. Suppose that f (z) = k0 ak z k ∈ F . By the density of the polynomials in pA (Proposition 6.1.6), it suffices to show that z m ∈ [f ]∗ for every m  0. We already know that 1 ∈ [f ]∗ so we just need to prove the m > 0 case. We will do this by induction. Indeed, suppose that 1, z, z 2 , . . . , z m−1 ∈ [f ]∗ . If Wm denotes the mapping Wm



  bk z k = bk z k ,

k0

km

then Wm is a contractive linear operator on pA and one can verify that Wm = S m S ∗m . By the S ∗ -invariance of F we know that S ∗m f ∈ F and, by hypothesis, 1 ∈ [S ∗m f ]∗ . Thus  z m ∈ {S m S ∗m S ∗k f : k  0} = Wm [f ]∗ . m−1 This shows there is a polynomial q(z) = k=0 ck z k such that q(z) + z m ∈ [f ]∗ . By our induction hypothesis, q ∈ [f ]∗ and so z m ∈ [f ]∗ .  Proof of Theorem 11.4.1. As mentioned earlier, we will only prove one direction and follow Abakumov’s original proof. Suppose (11.4.3) holds. For fixed 1/(p−1)

> 0 let ck = Rk and choose a positive integer N large enough so that N 

1−p ck

< .

k=1

The condition in (11.4.3) makes this possible. Define g=

N  ck ∗nk S f. ak

k=1

11.5. NOTES

171

By Lemma 11.4.4 the sets Ω(S ∗ni f ) ∩ Ω(S ∗nj f ) contain at most one element and hence no  integer k has more than one representation of the form k = nl − nj . Thus if g(z) = k1 gk z k then  

 p |al |p  cj

p |gk |p = al = cj

aj |aj |p 1jN 1jN k1

k1

=

=

{l:nl −nj =k}

N 

cpj

j=1

Rj

N  k=1

The above inequality shows that 1∈

=

N 

l>j

cj

j=1

ck

N

p 1−p  cj



anj < |g(0)|p .

a j j=1



{S ∗nk f : k  1}.

To see this, note that g − g(0)pp < |g(0)|p and so p   dist g(0), {S ∗nk f : k  1} < |g(0)|p . Now apply Lemma 11.4.6 to the set F consisting of the functions  ak z nk ∈ pA k1

satisfying (11.4.3), to see that f is a cyclic vector for S ∗ .



11.5. Notes The subspaces (uH 2 )⊥ , where u is an inner function, are called model spaces and have many applications to operator theory [40, 66, 130–132]. The term model spaces comes from the fact that the compression of the shift operator to a model space can be used to represent a large class of contraction operators on Hilbert spaces. Pseudocontinuations, along with other types of generalized analytic continuations, are discussed in the text [140] and Shapiro’s paper [149]. Theorems 11.1.3 and 11.1.6 are from the paper of Douglas, Shapiro, and Shields [50] and the proofs presented here are from [66]. Most of the results in Section 11.2 are from [140]. Lemma 11.2.6 is found in Sundberg [160]. The gap series result given in Theorem 11.4.1 is by Abakumov [2]. We refer the reader to [50, 140] and to Shapiro’s early work on the subject [150] for further results about Hadamard and Fabry gap series and cyclic vectors. See [73] for Hadamard’s original paper on gap series. We also point out some interesting number theory connections between cyclic vectors and gap series [140].

10.1090/ulect/075/12

CHAPTER 12

Multipliers of pA For any Banach space of analytic functions X on D, one is interested in the pointwise multipliers of X . These are the ϕ ∈ Hol(D) for which ϕX ⊆ X . Equivalently, via the Closed Graph Theorem, these are the ϕ ∈ Hol(D) which define bounded multiplication operators f → ϕf on X . There is a large literature on multipliers for various function spaces that brings in many unexpected tools from analysis. For the Hardy and Bergman spaces, it is not difficult to show that the multipliers are just the bounded analytic functions on D. As we survey the subject in this chapter, we will see that the situation for pA is more complicated. 12.1. Convolutions For two sequences of complex numbers a = {ak }k0 and b = {bk }k0 , define the convolution a ∗ b to be the sequence whose kth term is (12.1.1)

k 

aj bk−j ,

k  0.

j=0

The convolution can be defined for any pair of one-sided sequences a and b, since the sum in (12.1.1) is always finite. However, if we require the convolution to belong to a particular p class, then Young’s Convolution Inequality is useful. Theorem 12.1.2 (Young). Suppose p, q, r ∈ [1, ∞] with 1 1 1 + = + 1. p q r p q r If a ∈  and b ∈  , then a ∗ b ∈  with a ∗ br  ap bq . Convolutions can be defined much more broadly, e.g., for functions on groups endowed with a measure. We will not get into this here but the reader should be aware of the sizable literature on this material and that some of what we will say about multipliers on pA has analogues in the general area of Fourier multipliers. 12.2. The Space of Multipliers Definition 12.2.1. Let p ∈ [1, ∞]. A function ϕ ∈ Hol(D) is called a multiplier of pA if ϕpA ⊆ pA . The set of multipliers of pA will be denoted by Mp . By working out the Cauchy product of two power series, we see that the product of two pA functions a and b corresponds to the convolution a ∗ b of their associated p sequences a and b. To distinguish the multipliers of pA from the Hadamard multipliers from Section 7.3, some authors use the term “pointwise multiplier” for what we are calling a “multiplier.” If ϕ is a multiplier on pA , then the mapping f → ϕf is obviously linear. More can be said. 173

12. MULTIPLIERS OF p A

174

Proposition 12.2.2. Let p ∈ [1, ∞]. If ϕ ∈ Mp then the linear transformation Mϕ : pA → pA ,

Mϕ f = ϕf,

is continuous. Proof. Suppose that fn → f and ϕfn → g in pA norm. Then by (6.1.3), ϕfn → ϕf and ϕfn → g pointwise on D. Hence ϕf is equal to g. This says that the graph of Mϕ is closed. By the Closed Graph Theorem, Mϕ is continuous.  The previous proposition allows us to define the multiplier norm of ϕ ∈ Mp by ϕMp := sup{ϕf p : f ∈ pA , f p  1}.

(12.2.3)

In short, the multiplier norm of ϕ coincides with the operator norm of Mϕ on pA . Proposition 12.2.4. Let p ∈ (1, ∞). If ϕ ∈ Mp then ϕ is bounded and ϕ∞ = sup{|ϕ(z)| : z ∈ D}  ϕMp . Proof. Since ϕ ∈ Mp and the unit constant functions belong to pA , then so does ϕn for all n  1. Thus, for each z ∈ D, we can use Proposition 6.4.5 to see that ϕnMp |ϕ(z)|n = |Λz Mϕn 1|  Λz Mϕ n 1p  1 , (1 − |z|p ) p where Λz f = f (z), f ∈ pA is the point evaluation functional at z. Taking the nth root of both sides of the inequality above and then letting n → ∞ yields the result.  The previous proposition, along with the fact that the constant functions belong to pA , imply the following corollary. Corollary 12.2.5. For p ∈ (1, ∞) we have Mp ⊆ H ∞ ∩ pA . When p = 1, Young’s Theorem (Theorem 12.1.2) ensures that 1A , the Wiener algebra, coincides with the algebra of multipliers on 1A . In other words, M1 = 1A . When p = 2 we know that 2A = H 2 with equal norms. This identification allows us to describe M2 . Proposition 12.2.6. M2 = H ∞ . Proof. Corollary 12.2.5 says that M2 ⊆ H ∞ ∩ H 2 = H ∞ (Proposition 5.1.4). For the reverse containment, let ϕ ∈ H ∞ and f ∈ H 2 . For any r ∈ (0, 1) we have |ϕ(rξ)f (rξ)|2 dm(ξ)  ϕ∞ |f (rξ)|2 dm(ξ) T

T

from which it follows from the definition of H 2 that ϕf ∈ H 2 . Thus ϕ is a multiplier of H 2 .  We will see in Corollary 12.6.3 below that Mp  pA ∩ H ∞ ,

p ∈ (1, ∞) \ {2}.

We can adopt a notion of multipliers for pA when p ∈ (0, 1], that is, ϕ ∈ Hol(D) for which ϕpA ⊆ pA . However, by Theorem 6.6.13, pA is an algebra and so the multipliers turn out to be all of pA . In other words, Mp = pA ,

p ∈ (0, 1].

12.2. THE SPACE OF MULTIPLIERS

175

Since the shift operator (Sf )(z) = zf (z) is continuous on pA , p ∈ [1, ∞), we know that q(S)pA ⊆ pA for any polynomial q. Thus any polynomial is a multiplier. Also note that the multiplication operator Mϕ is injective for all ϕ ∈ Mp \ {0}. There is a natural Banach space structure on the multipliers. Indeed, the multipliers form a vector space that is endowed with the multiplier (operator) norm from (12.2.3). The final piece needed is to show that Mp is a Banach space is completeness. Proposition 12.2.7. If p ∈ [1, ∞), then Mp is closed in operator norm. Proof. Suppose that {ϕk }k1 is a Cauchy sequence in Mp . Since the space B(pA ) of bounded linear operators on pA is complete, there is an A ∈ B(pA ) such that Mϕk → A in operator norm. Furthermore, the constant function 1 belongs to pA , so ϕk 1 converges to ϕ := A1 in pA norm. Now if f ∈ pA , then ϕk f → Af . At the same time, ϕk f → ϕf uniformly on compact subsets of D. This forces Af = ϕf , from which we see that ϕ is a multiplier and A = Mϕ . Thus Mp is closed.  From Proposition 7.2.1, the difference quotient operator (Qw f )(z) =

f (z) − f (w) , z−w

w ∈ D,

is continuous on pA and thus Qw  := Qw B(pA ) < ∞. Proposition 12.2.8. Let p ∈ [1, ∞). If ϕ ∈ Mp , then Qw ϕ ∈ Mp for every w ∈ D. Moreover, (12.2.9)

Qw ϕMp  Qw (ϕMp + |ϕ(w)|).

Proof. For any f ∈ pA , one can quickly verify the identity (Qw ϕ)f = Qw (ϕf ) − ϕ(w)Qw f. The right hand side belongs to pA and hence (Qw ϕ) is a multiplier whose multiplier norm Qw ϕMp satisfies (12.2.9).  Though Mϕ is injective for all ϕ ∈ Mp \ {0}, it does not always have closed range. For example, when p ∈ (1, ∞), the multiplier ϕ(z) := 1 − z on pA has dense range R(ϕ) (Lemma 10.1.3). However, Mϕ fails to be surjective and thus is not closed. To see this, consider the family of functions fk,t (z) =

zk , 1 − tz

k  0,

t ∈ (0, 1).

Clearly fk,t ∈ pA and thus Mϕ fk,t ∈ R(ϕ). A calculation with power series and the pA norm shows that for each k  0 Mϕ fk,t − z k pp =

(1 − t)p , 1 − tp

which goes to zero as t → 1− . Thus z k ∈ R(ϕ) for every k  0 and consequently R(ϕ) = pA . However, the unit constant function does not belong to R(ϕ) or else (1 − z)−1 ∈ pA . But this is impossible since the Taylor coefficients of (1 − z)−1 do not form a sequence belonging to p .

12. MULTIPLIERS OF p A

176

12.3. Mp as the Commutant Clearly ϕ(z) = z is a multiplier of pA . In fact Mz = S and we have already established that S is an isometric operator (Proposition 7.1.2). Moreover, since Mϕ S = SMϕ for all ϕ ∈ Mp , we see that Mp is a subset of the commutant of S, defined by {S} := {A ∈ B(pA ) : AS = SA}. In fact, we get more. Proposition 12.3.1 (Nikolskii). For p ∈ [1, ∞) we have {S} = Mp . Proof. We have already seen that Mp ⊆ {S} . Conversely, suppose that A ∈ {S} . Then for any analytic polynomial q we have (12.3.2)

A(q(S)1) = q(S)A(1),

equivalently, A(q) = qϕ, where ϕ = A(1). By the density of the polynomials in pA (Proposition 6.1.6) we can, for a given f ∈ pA , find a sequence of polynomials {qn }n1 such that qn → f in the norm of pA . Since point evaluations on D are continuous in the norm of pA (Proposition 6.4.5), we see that qn → f pointwise on D. Since Aqn → Af both in norm as well as pointwise on D, it follows from (12.3.2) that (Af )(z) = lim (Aqn )(z) = lim qn (z)ϕ(z) = f (z)ϕ(z), z ∈ D. n→∞

n→∞

In other words, Af = ϕf . Thus ϕ ∈ Mp and A = Mϕ .



To further explore the operator theory connections to multipliers, we now present an equivalent characterization of Mp . Given a sequence of complex numbers {an }n0 , define the infinite Toeplitz matrix A by ⎛ ⎞ a0 0 0 0 · · · ⎜a1 a0 0 0 · · ·⎟ ⎜ ⎟ ⎜ ⎟ (12.3.3) A := ⎜a2 a1 a0 0 · · ·⎟ . ⎜a3 a2 a1 a0 · · ·⎟ ⎝ ⎠ .. .. .. .. . . . . . . .  n ∈ Hol(D) and Proposition 12.3.4 (Nikolskii). Suppose ϕ(z) = n0 an z p ∈ (1, ∞). Then ϕ ∈ Mp if and only if the infinite matrix A from (12.3.3) defines a bounded operator on the sequence space p . In this case,

Proof. Let f (z) =



ϕMp = AB(p ) .

n0 bn z

(ϕf )(z) =

∈ pA and write b := {bn }n0 ∈ p . Then

n



n

cn z ,

n0

cn =

n 

ak bn−k .

k=0

Therefore, ϕ ∈ Mp if and only if  n0

for all b ∈  . p

|cn |p =

n

p 



ak bn−k < ∞

n0 k=0

12.3. Mp AS THE COMMUTANT

177

On the other hand, with c := {cn }n0 , the key point here is that b and c are related via the matrix identity (12.3.5)

Ab = c. pA

Thus ϕ is a multiplier for if and only if b ∈ p =⇒ Ab ∈ p . The latter is p equivalent to A ∈ B( ). For the equality of norms, note that by (12.3.5), ϕMp = sup{ϕf p : f ∈ pA , f p  1} = sup{Abp : bp  1} 

= AB(p ) . 

From (1.7.3) recall the bilinear pairing a, b between p and p . Proposition 12.3.4 implies the useful inequality 

| Ax, y|  ϕMp xp yp ,

(12.3.6)

x ∈  p , y ∈ p .

This next result relates the multipliers on the dual spaces, and is often an important reduction in our multiplier discussion. Proposition 12.3.7 (Nikolskii). For p ∈ (1, ∞) we have Mp = Mp with equal multiplier norms, that is, ϕMp = ϕMp for every multiplier ϕ. Proof. Define 

 (Z) := b = {bn }n∈Z : bp = p



|bn |

p

1

p

 1, where (12.6.1)

Γ(α, ξ) = {z ∈ D : |z − ξ|  α(1 − |z|)}

is a Stolz region anchored at eiθ with opening α (see Figure 1). Lebedev and

Figure 1. The Stolz domain Γ(2.5, 1) together with the unit circle. Olevski˘ı [108, 112, 113] observed that more can be said. Their work is based on the following technical details. Recall the definition of Ap (T) for p ∈ [1, 2] from (12.5.1) and its corresponding space of multipliers Mp (T). Theorem 12.6.2 (Lebedev and Olevski˘ı). If p ∈ [1, 2) and ψ ∈ Mp (T), there is function Ψ on T that is continuous almost everywhere such that ψ = Ψ almost everywhere on T. The proof of this theorem is lengthy and so we refer the reader to the original paper for the details. However, we will prove the following interesting consequence. Corollary 12.6.3. For p ∈ [1, 2) and ϕ ∈ Mp , the unrestricted limit lim ϕ(z)

z→ζ

exists for almost every ζ ∈ T. Before getting to the proof, we need some standard details from harmonic analysis. Recall that Pz (ξ) =

1 − |z|2 , |ξ − z|2

z ∈ D,

ξ ∈ T,

12.6. BOUNDARY PROPERTIES OF MULTIPLIERS

is the Poisson kernel. It is well known that (12.6.4) Pz (ξ) dm(ξ) = 1, T

and that for any h ∈ L∞

183

z ∈ D,

z →

T

Pz (ξ)h(ξ) dm(ξ)

is a harmonic function on D with lim Prζ (ξ)h(ξ) dm(ξ) = h(ζ) r→1

T

for almost every ζ ∈ T. Furthermore, if h ∈ H ∞ we have the formula Pz (ξ)h(ξ) dm(ξ), z ∈ D. (12.6.5) h(z) = T

Lemma 12.6.6. Let h ∈ L∞ be continuous at ξ0 ∈ T. Then the unrestricted limit lim Pz (ξ)h(ξ)dm(ξ) z→ξ0

T

exists and is equal to h(ξ0 ). Proof. Let > 0 be given and let I0 be an open subarc of T containing ξ0 such that |h(ξ) − h(ξ0 )| < , ξ ∈ I0 . By (12.6.4) we have









P (ξ)h(ξ) dm(ξ) − h(ξ ) Pz (ξ)(h(ξ) − h(ξ0 )) dm(ξ)

z 0 =

T T  Pz (ξ)|h(ξ) − h(ξ0 )| dm(ξ) T Pz (ξ)|h(ξ) − h(ξ0 )| dm(ξ)  I0 + Pz (ξ)|h(ξ) − h(ξ0 )| dm(ξ) T\I 0 Pz (ξ)|h(ξ) − h(ξ0 )| dm(ξ).  + T\I0

By the Dominated Convergence Theorem the above integral goes to zero as z → ξ0 . The result now follows.  Proof of Corollary 12.6.3. From Proposition 12.5.3 we know that the almost everywhere defined radial boundary function ϕ∗ belongs to Mp (T) and, by Theorem 12.6.2, there is a function Φ on T that is continuous almost everywhere and equal to ϕ∗ almost everywhere. Since ϕ∗ is the radial boundary function for ϕ and ϕ ∈ H ∞ , we can use (12.6.5) to get ϕ(z) = Pz (ξ)ϕ∗ (ξ) dm(ξ) = Pz (ξ)Φ(ξ) dm(ξ), z ∈ D. T

T

Lemma 12.6.6 says that the unrestricted limit Pz (ξ)Φ(ξ) dm(ξ) = Φ(ζ) lim z→ζ

T

12. MULTIPLIERS OF p A

184

at every point ζ of continuity for Φ. However, Φ = ϕ∗ almost everywhere and this verifies the claim.  The essential feature of Corollary 12.6.3 is that z ∈ D tends freely to ξ ∈ T and it is not obliged to stay in a fixed Stolz domain (recall (12.6.1)). It is often said that the multipliers for a Banach space of analytic functions are better behaved near the boundary than generic functions in the space, and this is an instance of this phenomenon. 12.7. Isometric Multipliers We now explore the multipliers ϕ which satisfy ϕf p = f p ,

(12.7.1)

f ∈ pA .

These are known as the isometric multipliers. Once again the case p = 2 is exceptional. A quick computation shows that ϕf 2 = f 2 for all f ∈ H 2 = 2A whenever ϕ is an inner function. This follows from the simple observation |ϕf |2 dm = |ϕ|2 |f |2 dm = |f |2 dm. T

T

T

Interestingly enough, the converse is also true. Proposition 12.7.2. If ϕ is an isometric multiplier of 2A , then ϕ is an inner function. Proof. Proposition 12.2.4 and (12.7.1) show that |ϕ(z)|  1 for every z ∈ D. Now use Parseval’s identity to rewrite (12.7.1) in integral form as |f |2 (1 − |ϕ|2 ) dm = 0, f ∈ H 2 , T

which holds if and only if |ϕ| = 1 almost everywhere. In other words, ϕ is inner.



When p = 2 the story is different. Observe that (12.7.3)

z n f (z)p = f p ,

f ∈ pA ,

n  0.

In other words, the monomials z are isometric multipliers for pA . n

Theorem 12.7.4 (Nikolskii). If p ∈ (1, ∞)\{2} and ϕ is an isometric multiplier for pA , then ϕ(z) = γz n for some n  0 and unimodular constant γ. The proof is based on two sets of elementary inequalities due to Bernoulli [157, p. 31]. In the following x  0, y  0 and t  −1. (i) For α ∈ (0, 1) (a) we have (x + y)α  xα + y α and equality holds if and only if x = 0 or y = 0; (b) we have (1 + t)α  1 + αt and equality holds if and only if t = 0. (ii) For α ∈ (1, ∞) (a) we have (x + y)α  xα + y α and equality holds if and only if x = 0 or y = 0; (b) we have (1 + t)α  1 + αt and equality holds if and only if t = 0. Parts (a) above also follow from Proposition 1.5.2.

12.7. ISOMETRIC MULTIPLIERS

185

Proof of Theorem 12.7.4. We treat the case p ∈ (1, 2) for which the first set of inequalities above is used. The other case is similar. We may write ϕ(z) = γz n ϕ1 (z), where γ is unimodular, n is the order of the zero of ϕ at the origin, and ϕ1 (0) > 0. By (12.7.3), we have ϕ1 f p = f p ,

f ∈ pA .

We will now show that ϕ1 ≡ 1. Hence, considering the above reduction, assume that ϕ(0) > 0 and ϕf p = f p for all f ∈ pA . Take f (z) = 1 + eiθ z,  where we treat θ as a free parameter. If ϕ(z) = n0 an z n , the isometric identity can be rewritten as  |an+1 + an eiθ |p = 2. (12.7.5) |a0 |p + n0

We integrate both sides with respect to θ. First, due to periodicity, we have 2π 2π



|a| + |b|eiθ p dθ. |a + beiθ |p dθ = 0

0

Second, an elementary calculation reveals that



|a| + |b|eiθ 2 = |a|2 + |b|2 + 2|ab| cos θ = (|a|2 + |b|2 ) (1 + s cos θ), where 2|ab| . + |b|2 Note that s ∈ [0, 1] and thus t := s cos θ  −1. Third, by the above Bernoulli inequalities we have 2π 2π  p 2 2 p



2 1 2

|a| + |b|eiθ p dθ = (|a| + |b| ) 1 + s cos θ dθ 2π 0 2π 0 2π   ps |a|p + |b|p cos θ dθ 1+  2π 2 0 = |a|p + |b|p , s=

|a|2

and the equality holds if and only if a = 0 or b = 0. Returning to (12.7.5), we get   2  |a0 |p + (|an+1 |p + |an |p ) = 2 |an |p . n0

n0

If we plug f ≡ 1 into the isometric identity ϕf p = f p we see that ϕp =



|an |p

1

p

= 1.

n0

Hence, in the above relation, equality holds, which in turn implies that equality holds in all preceding identities. In particular, since a0 = 0, the case n = 0 implies a1 = 0. We now repeat the above procedure with the function f (z) = 1 + eiθ z 2 and deduce that a2 = 0. Continuing in this fashion, we have an = 0 for all n  1. Since a0 > 0 and n0 |an |p = 1, we conclude that ϕ ≡ a0 = 1. 

12. MULTIPLIERS OF p A

186

12.8. Smooth Multipliers The family of functions which are analytic in a disk larger than the open unit disk is denoted by Hol(D). From Young’s Inequality (Theorem 12.1.2) we see that 1A ⊆ Mp ,

p ∈ [1, ∞)

with equality when p = 1. Thus certainly we have Hol(D) ⊆ 1A ⊆ Mp ,

p ∈ [1, ∞).

We present below an alternative proof of this fact from which we can obtain further information. Our main tool is Schur’s Test [76, p. 24]: Let A = [akj ]k,j0 be an infinite matrix, and let p ∈ (1, ∞). Assume that there are positive constants α and β, and positive sequences {pk }k1 and {qj }j1 , such that 

|akj | ppk  α qjp ,

j  1 and

k0







|akj | qjp  β ppk ,

k  1.

j0

Then A is a bounded operator on p and moreover, 1

1

AB(p )  α p β p .

(12.8.1)

For each f ∈ Hol(D) with Taylor series expansion f (z) = exist an R > 1 and a c > 0 such that |an | 

(12.8.2)

c , Rn

 n0

an z n , there

n  0.

The constants R and c depend on f , but work uniformly with respect to n. This exponential decay plays a major role in establishing the following result. Theorem 12.8.3. If p ∈ (1, ∞), then Hol(D) ⊆ Mp . Proof. By Proposition 12.3.4, it is enough to show that the matrix A formed with the coefficients of ϕ, according to recipe (12.3.3), is a bounded operator on p . We apply Schur’s test with pk = qk = tk , where t is a positive parameter to be determined momentarily. Fixing j, by (12.8.2), we have  k0

|akj | ppk =



|ak−j | tkp

kj

 tkp Rk−j kj   tp k = ctjp R c

k0

c p = p q , 1 − tR j

j  0.

12.8. SMOOTH MULTIPLIERS

187

Similarly, fixing k, by (12.8.2), we have 



|akj | qjp =

k 



|ak−j | tjp

j=0

j0

c

 k  tjp Rk−j j=0

k   1 j tp R j=0

= ctkq =

c 1−



1 tp R

ppk .

Schur’s test ensures that AB(p ) 



  c c p  p . tp 1 1− R 1 − tp R 1

1



The above geometric series are convergent provided that tp < R and tp > 1/R. Hence, the acceptable range of t is 1 < t < R1/p . 1/p R  Therefore we can say that  1  c  1    c p p (12.8.4) AB(p )  inf : t ∈ (R−1/p , R1/p ) . tp 1 1− R 1 − tp R In particular, with t = 1 we get AB(p ) 

(12.8.5)

c 1−

1 R

, 

which is enough for our applications.

In the above proof, we took t = 1 in (12.8.4). Is it possible to get a better bound by choosing another value of t? In other words, what is the optimal value of t? The proof of Theorem 12.8.3 contains more information than presented in the theorem. By a closer look, we obtain the following convergence result. Corollary 12.8.6. Let p ∈ (1, ∞) and ϕ ∈ Hol(D) and denote its Taylor polynomial of degree n by ϕn . Then lim ϕn − ϕMp = 0.

n→∞

Moreover, the rate of decay is exponential. Proof. Since ϕ(z) − ϕn (z) =



ak z k = z n+1

kn+1

by (12.7.3), we have



ak+n+1 z k ,

k0

    ϕn − ϕMp =  ak+n+1 z k  k0

Mp

.

12. MULTIPLIERS OF p A

188

Thus, according to Proposition 12.3.4, ϕn − ϕMp = An B(p ) , where the matrix An is given by ⎛ an+1 ⎜an+2 ⎜ ⎜ An := ⎜an+3 ⎜an+4 ⎝ .. .

0

0 0

an+1 an+2 an+3 .. .

an+1 an+2 .. .

0 0 0 an+1 .. .

⎞ ··· · · ·⎟ ⎟ · · ·⎟ ⎟. · · ·⎟ ⎠ .. .

By (12.8.2) we have the estimate |ak+n+1 | 

c Rk+n+1

=

c/Rn+1 , Rk

k  0.

Therefore by (12.8.5), c/Rn+1 , 1 − R1 exponentially decreases to zero.

An B(p )  which reveals that ϕn − ϕMp



The conclusion of Corollary 12.8.6 does not hold for an arbitrary multiplier. For example, we saw that M2 = H ∞ . However if ϕ ∈ M2 satisfies ϕn − ϕM2 = ϕn − ϕ∞ → 0, then ϕ is continuous on D. But, for example, if ϕ is a singular inner function, this cannot hold. 12.9. 1A Embeds Contractively in Mp In this section, we provide a result which contains Theorem 12.8.3 as a special case. However, we stated that theorem separately since the estimate in (12.8.4) should provide a better bound for the norm of the narrower class of multipliers Hol(D). Theorem 12.9.1. For p ∈ (1, ∞) we have 1A ⊆ Mp and hMp  h1 ,

h ∈ 1A .

Proof. Of course the result follows immediately from Theorem 12.1.2 but we include a proof using the tools developed above. We again appeal to Proposition 12.3.4. Hence, it is enough to show that the matrix A formed with the coefficients of h according to the recipe in (12.3.3) is a contraction on p . We apply the simplest version of Schur’s test, i.e., with pk = qk = 1. Fixing j, we have    |akj | ppk = |ak−j | = |al | = h1 . k0

kj

l0

Similarly, fixing k, we obtain  j0

|akj | qjq =

k  j=0

|ak−j | =

k 

|al |  h1 .

l=0

Thus we can let α = β = h1 . Schur’s test (12.8.1) says that AB(p )  h1 .



12.10. QUOTIENTS OF MULTIPLIERS

189

Corollary 12.9.2. Let p ∈ (1, ∞) and for h ∈ 1A denote its Taylor polynomial of degree n by hn . Then lim hn − hMp = 0. n→∞

If the coefficients of h are all nonnegative, then Theorem 12.9.1 is reversible.  Theorem 12.9.3. Let p ∈ (1, ∞). If h(z) = n0 an z n ∈ Mp and an  0 for all n  0, then h ∈ 1A . Proof. In the inequality (12.3.6), take x = y = (1, 1, . . . , 1, 0, 0, . . . ) 1 23 4 n+1

to get k n  

aj  (n + 1)hMp .

k=0 j=0

After rearranging the sums we obtain n   k  ak  hMp . 1− n+1 k=0

Let n → ∞ and use the Monotone Convergence Theorem to deduce  ak  hMp .



k0

12.10. Quotients of Multipliers From Proposition 12.3.7, every f ∈ H 2 can be written as a/b where a, b ∈ H ∞ , the multipliers of H 2 . In general, for a Banach space X of analytic functions on D, one can explore whether any function from X is the quotient of two multipliers of X . In some cases the answer is known. Besides the Hardy space H 2 , the answer is affirmative for the classical Dirichlet space, as well as reproducing kernel Hilbert spaces satisfying the complete Pick property. We now explore whether every f ∈ pA can be written as the quotient of two multipliers. When p = 2, we are in the classical setting of 2A = H 2 and thus, as seen above, the answer is affirmative. The case p = 1 is also trivial since 1A is itself an algebra, and thus it coincides with its multiplier algebra. For any other p ∈ (1, ∞) \ {2}, the answer is negative. Corollary 12.10.1. For each p ∈ (1, ∞) \ {2} there are functions in pA which are not the quotient of two multipliers of pA . Proof. Fix p ∈ (1, ∞). Let B be a Blaschke product with exponential zeros (recall Definition 6.2.10) that accumulate on all of T. Such a Blaschke product will belong to pA for all p ∈ (1, ∞) (Theorem 6.2.8). However, B will not be the quotient of two multipliers. Indeed, if B = f /g where f and g are multipliers of pA , then by Corollary 12.6.3, f and g will have unrestricted limits almost everywhere on T. Furthermore, the unrestricted limits of f and g will be nonzero almost everywhere (Theorem 5.1.7). Thus B = f /g will have an unrestricted (and nonzero) limit almost everywhere on T. However, since the zeros of B accumulate on all of T, the unrestricted limit of B will be zero everywhere. This contradicts the property that |B| = 1 almost everywhere. 

12. MULTIPLIERS OF p A

190

Using the fact that the Bergman spaces Ap are not contained in the Nevanlinna class N and that the multipliers of Ap are precisely H ∞ , we also see that not every Ap function is a quotient of two multipliers for Ap . 12.11. Inner Multipliers A worthy set of functions to test as possible multipliers for pA are the inner functions (by which we mean “inner” in the traditional sense, and not p-inner). We know that the monomials ϕn (z) = z n are multipliers, in fact isometric multipliers. Are there any other inner multipliers? Certainly any finite Blaschke product is an inner multiplier since they are analytic in an open neighborhood of D (Theorem 12.8.3). There are some infinite Blaschke products which are multipliers for all of the pA classes. To present this result, we need the domains Ω(r, α) := {z : |z| < r} \ {z : | arg(z − 1)|  α}, where r ∈ (1, ∞) and 0  α
0,

θ ∈ (−π, π),

12.12. NOTES

191

we have

|ψα (z)| = |eiα log(1−z) | = |eiα(log r+iθ) | = e−αθ  eπα , and therefore ψα is bounded on C \ [1, ∞). Thus by Theorem 12.11.1, & Mp . ψα ∈ p∈(1,2)

We can use Corollary 12.6.3 to eliminate certain classes of inner functions as multipliers. If h is an inner function, then its boundary spectrum is   (12.11.2) Σb (h) := ζ ∈ T : lim |h(z)| = 0 , z→ζ

where the above lim is unrestricted in D. Equivalently [66, p. 154], if h = BSν is the decomposition of h as the product of a Blaschke factor B with zeros {zn }n1 and a singular inner function Sν with the singular measure ν, then Σb (h) is precisely the union of the support of ν and the accumulation points of {zn }n1 on T. Theorem 12.11.3. Let h be an inner function such that m(Σb (h)) > 0. Then h ∈ Mp for any p ∈ (1, ∞) \ {2}. Proof. Since Mp = Mp (Proposition 12.3.7) we can assume p ∈ (1, 2). If h ∈ Mp , then, by Corollary 12.6.3, h(ζ) = 0 for almost all ζ ∈ Σb (h). Since at the same time |h| = 1 almost everywhere on T, we obtain a contradiction.  Surprisingly, the criterion for a singular inner function to belong to Mp is still an open question. According to Theorem 12.11.3, the condition that m(Σb (h)) = 0 is necessary for being a multiplier. However, this condition is not sufficient. For example, Verbitski˘ı [163] showed that the simplest singular inner functions  1 + z sa (z) = exp − a , a > 0, 1−z which have Σ(sa ) = {1}, do not belong to any Mp , for p ∈ (1, ∞) \ {2}. 12.12. Notes Multipliers of function spaces have been studied quite extensively [122]. Much of this chapter is based on a survey paper of Cheng, Ross, and Mashreghi [34]. There is a sizable literature concerning multipliers from the convolution perspective. The papers of Hirschman [85, 86] and Hormander [88] are some examples of this. Propositions 12.3.1, 12.3.4, and 12.3.7, along with Theorem 12.7.4, come from Nikolskii [133]. Theorem 12.6.2 is from Lebedev and Olevski˘ı [108, 112, 113]. The literature on inner multipliers of pA is quite large with many wonderful examples and we were quite selective in what we presented. Theorem 12.11.1 appears in Vinogradov [165]. We refer the reader to the papers of Lebedev [109, 110], Verbitski˘ı [163], and Vinogradov [165, 166] for more on the inner multipliers of pA . See Lebedev’s paper [111] for yet another class of multipliers.

10.1090/ulect/075/13

CHAPTER 13

The Wiener Algebra The Wiener algebra 1A has appeared briefly at various places in this book. In this final chapter, we will cover a selection of interesting and sometimes unexpected properties of this Banach algebra of functions. As some of these results are quite advanced, we will not always give detailed proofs. The end notes will direct the reader to the original sources. 13.1. Some Inhabitants of the Wiener Algebra A space related to the Wiener algebra 1A is the disk algebra A of continuous functions on D that are analytic on D. One can endow A with the supremum norm f ∞ = max{|f (z)| : z ∈ D},

f ∈A,

and the classical Maximum Modulus Principle says that f ∞ = max{|f (ξ)| : ξ ∈ T},

f ∈A.

For each integer n  0 one also has the algebra of functions A n := {f ∈ A : f (j) ∈ A , 0  j  n},

(13.1.1) with norm

f A n :=

n 

f (j) ∞ ,

f ∈ A n.

j=0

Obviously A ⊆ A when n  m  0. As an indication of the types of functions that belong to 1A , we have the following. n

m

Proposition 13.1.2. A 1  1A  A . 1 1 Proof. To prove the inclusion A   A n, we will need Hardy’s Inequality [54, p. 48] which says that if g(z) = n0 gn z belongs to the Hardy space H 1 , then  |gn |  πgH 1 . n+1 n0  Thus if f (z) = n0 an z n ∈ A 1 , i.e., f  ∈ A , then certainly f  ∈ H 1 . Moreover, after re-indexing,  (n + 1)an+1 z n . f  (z) = n0

Apply Hardy’s Inequality to the function g = f  to see that  |an+1 |  πf  H 1 , n0 193

194

13. THE WIENER ALGEBRA

which shows that f ∈ 1A . The containment A 1 ⊆ 1A is strict since  (−1)n zn f (z) = (z + 1) log(z + 1) − z = n(n − 1) n2

belongs to 1A (its Taylor series coefficients are summable), but f  (z) = log(1 + z) does not belong to A since it is unbounded on D. The containment 1A ⊆ A follows from the absolute and uniform convergence of the Taylor series on D of elements of 1A (see (6.6.1)). However, the containment is strict with specific examples from [67, p. 226] or [107, p. 68].  Looking carefully at the proof of Proposition 13.1.2, one will notice that A 1 can be replaced by the slightly bigger class {f ∈ Hol(D) : f  ∈ H 1 } (see [54, p. 78]). The following well-known result of Bernstein is a specific case of Proposition 6.2.3 and provides further examples of functions in 1A . Proposition 13.1.3 (Bernstein). Let b ∈ ( 12 , 1]. Suppose that f ∈ H 2 satisfies 2π

i[θ+t]

2

f (e ) − f (eiθ ) dθ  C|t|2b , t ∈ R, (13.1.4) 0

for some C > 0. Then f ∈ 1A . We point out that although the functions in 1A have continuous boundary regularity, they can still be quite wild as shown by the following Peano curve result (see Theorem 13.1.5 below). In fact, many examples of analytic functions with erratic boundary behavior are functions with lacunary Taylor expansions. These are the analytic functions f on D with Taylor series of the form  nk+1 f (z) = ak z nk ,  λ, k  0, nk k0

for some constant λ > 1. In fact, the standard examples of analytic functions on D which fail to have an analytic continuation across any point of T come from the well-known gap theorems of Hadamard and Fabry. Any f ∈ 1A is continuous on T and so f (T) is a compact subset of C. This following Peano curve theorem uses lacunary series to show that f (T) can be quite large. Theorem 13.1.5 (Salem and Zygmund). There exists an f ∈ 1A with lacunary Taylor expansion such that f (T) is a square [a, b] × [c, d]. As it turns out, there is a whole class of these 1A lacunary  series f for which f (T) is a square. The only requirement is that the series k0 |ak | converges slowly enough (see the end notes for the references). These types of examples can be extended to create 1A functions with other wild properties. One can also factor 1A functions via convolutions. For f, g ∈ L2 define the convolution (f ∗ g)(ξ) = f (ζ)g(ξζ) dm(ζ), ξ ∈ T. T

Fubini’s Theorem reveals the following identity on the Fourier coefficients: (f ∗ g)n = fn gn , 1

n ∈ Z,

where for an L function h we use hn to denote the nth Fourier coefficient.

13.2. WIENER’S 1/f THEOREM

195

Proposition 13.1.6. If f, g ∈ H 2 , then f ∗ g ∈ 1A and f ∗ g1  f 2 g2 . Conversely, every h ∈ 1A can be written as h = f ∗ g for some f, g ∈ H 2 . Proof. If f, g ∈ H 2 , then (f ∗ g)n = fn gn for all n ∈ Z. In particular, this means that (f ∗ g)n = 0 for all n < 0. Moreover, by the Cauchy-Schwarz Inequality, 1   1    2 2 |(f ∗ g)n | = |fn ||gn |  |fn |2 |gn |2 . n0

n0

n0

n0

This shows that f ∗ g ∈ and f ∗ g1  f 2 g2 . For the converse, suppose that  h(z) = hn z n ∈ 1A . 1A

n0

Define

⎧ ⎨ !hn if hn = 0 |hn | an = ⎩ 0 if hn = 0, ! bn = |hn | and note that {an }n0 and {bn }n0 ∈ 2 . Thus the functions   f (z) = an z n and g(z) = bn z n n0

n0

2

belong to H . Finally, by our choice of an and bn above, we have hn = fn gn for all n  0, and so h = f ∗ g.  13.2. Wiener’s 1/f Theorem For the disk algebra A , it is easy to determine whether f ∈ A is invertible in A , that is to say, 1/f ∈ A . Just check that f has no zeros in D. It seems more challenging to characterize the invertible elements of 1A . If f ∈ 1A has no zeros in D, one can always guarantee that 1/f is analytic on D and continuous on D. The following theorem of Wiener makes it easy to say whether 1/f has absolutely summable Taylor coefficients. Theorem 13.2.1 (Wiener). Suppose that f ∈ 1A and f (z) = 0 for any z ∈ D. Then 1/f ∈ 1A . We will prove the following stronger result. Theorem 13.2.2 (L´evy). If f ∈ 1A and F is an analytic function on a domain containing f (D), then F ◦ f ∈ 1A . Observe that Wiener’s theorem is L´evy’s theorem with F (z) = 1/z. Since it is of no extra effort, we will prove L´evy’s theorem for the set W of Fourier series   an ξ n , f W := |an | < ∞. f (ξ) = n∈Z

n∈Z

One can show that W is a Banach algebra of continuous functions on T that contains 1A as a subspace. The version of L´evy’s theorem in this context is the following. Theorem 13.2.3. If f ∈ W and F is an analytic function on a domain containing f (T), then F ◦ f ∈ W .

196

13. THE WIENER ALGEBRA

We follow an elegant proof of Newman [128] and begin with a Fourier series result. Lemma 13.2.4. If f ∈ C ∞ (T) and  an einθ f (θ) =

θ ∈ [−π, π],

n∈Z

then



|an |  max |f (θ)| + 2 max |f  (θ)|. θ

n∈Z

Proof. Note that



|a0 | =

θ

π

f (θ)

−π

dθ

 max |f (θ)| θ 2π

and, by the Cauchy-Schwarz Inequality and Parseval’s Theorem,  2   1 2 n|an | |an | = n n =0

n =0

 1   · n2 |an |2 n2 n =0 n =0 dθ π2 π  = |f (θ)|2 3 −π 2π π2 max |f  (θ)|2 3 θ  4 max |f  (θ)|2 . 

θ

Thus



|an | = |a0 | +

n∈Z



|an |  max |f (θ)| + 2 max |f  (θ)|.

n =0

θ

θ



which is the desired inequality.

Proof of Theorem 13.2.3. Let F ∈ Hol(Ω) and let K := f (T). The hypotheses of the theorem say that K is a compact subset of Ω. Define δ := dist(z, ∂Ω) and, for t ∈ (0, δ), define Kt := {z : dist(z, K)  t}. If C(z, r) = {z : |z − a| = r} the Cauchy Integral formula says that ' F (ζ) n! F (n) (a) = dζ, n  0, a ∈ Kδ/5 . 2πi C(a,2δ/5) (ζ − a)n+1 If M = max{|F (z)| : z ∈ K3δ/5 } then (13.2.5)

|F (n) (a)| 

n!M , (2δ/5)n

n  0,

a ∈ Kδ/5 .

The trigonometric polynomials are dense in W and so there is a trigonometric polynomial p with (13.2.6)

f − pW  δ/5.

13.2. WIENER’S 1/f THEOREM

197

This means that |f (ξ) − p(ξ)|  f − pW < δ/5,

ξ ∈ T,

and so p(ξ) ∈ Kδ/5 ,

(13.2.7)

ξ ∈ T.

The next step is to show that the series  F (n) ◦ p (f − p)n n!

(13.2.8)

n0

converges in the norm of W . Let

d

M  = max p(eiθ )

θ dθ

(13.2.9)

By the Banach algebra property of W and (13.2.6), (13.2.10)

(f − p)n W  f − pnW  (δ/5)n .

Using (13.2.5) and (13.2.7) we have |(F (n) ◦ p)(ξ)| = |F (n) (p(ξ))| 

n!M , (2δ/5)n

ξ ∈ T.

Moreover, from (13.2.9),

d







(F (n) ◦ p)(eiθ ) = d p(eiθ )

(F (n+1) ◦ p)(eiθ )

dθ dθ (n + 1)!M  M · , θ ∈ [−π, π]. (2δ/5)n+1 These previous two estimates, along with Lemma 13.2.4, yield n!M (n + 1)!M M  +2 n (2δ/5) (2δ/5)n+1  n! 2M M  (n + 1)  · = M+ . (2δ/5) (2δ/5)n

F (n) ◦ pW  (13.2.11)

The inequalities from (13.2.11) and (13.2.10) say that   F (n) ◦ p 1   · (f − p)n   F (n) ◦ pW f − pnW  n! n! W 2M M  (n + 1)  n! 1 M+ · = · (δ/5)n n! (2δ/5) (2δ/5)n n  C n. 2 This proves that the series in (13.2.8) converges and so h :=

 F (n) ◦ p (f − p)n ∈ W . n!

n0

198

13. THE WIENER ALGEBRA

For any ξ ∈ T we have z := f (ξ) ∈ K, a := p(ξ) ∈ Kδ/5 , and h(ξ) =

 F (n) (p(ξ)) (f (ξ) − p(ξ))n n!

n0

=

 F (n) (a) (z − a)n n!

n0

= F (z) = F (f (ξ)). Thus F ◦ f ∈ W .

 13.3. Composition

We had a preliminary investigation of composition operators on 1A in Section 7.5 which we now unpack further. In Section 7.5 we saw that if an analytic selfmap ϕ of D induces a composition operator Cϕ f = f ◦ ϕ on pA , p ∈ [1, ∞), then necessarily sup{ϕn p : n  0} < ∞. Moreover, when p = 1, this condition is sufficient. Let us extend this. Suppose that ϕ ∈ Hol(D). For the moment, we are not assuming any other conditions on ϕ. If f ◦ ϕ ∈ 1A for all f ∈ 1A we can see, applying this condition to f (z) = z n , n  0, that ϕn ∈ 1A . Since 1A is a Banach algebra, we can compute the spectrum of ϕ as σ(ϕ) := {λ ∈ C : ϕ − λ is not invertible in 1A }. Wiener’s Theorem (Theorem 13.2.1) now says that σ(ϕ) = ϕ(D). Standard theory of Banach algebras, similar to the results for operators on Banach spaces, yields the following version of the Spectral Radius Formula: (13.3.1)

1/n

lim ϕn 1

n→∞

= max{|λ| : λ ∈ σ(ϕ)}.

Thus if ϕn 1  M for all n  0, we have 1/n

lim ϕn 1

n→∞

 lim M 1/n = 1. n→∞

Hence the condition that ϕn 1 is uniformly bounded in n implies that ϕ is a self-map of the disk. We summarize this discussion with the following result. Proposition 13.3.2. Suppose that ϕ ∈ Hol(D). (i) If f ◦ ϕ ∈ 1A for every f ∈ 1A , then ϕ ∈ 1A and ϕ(D) ⊆ D. (ii) The condition that ϕ ∈ 1A and sup{ϕn 1 : n  0} < ∞ is equivalent to the condition that f ◦ ϕ ∈ 1A for every f ∈ 1A . We have already seen from Theorem 7.5.5 that one needs to read this proposition carefully since there are analytic self-maps ϕ, even very simple ones, which do not induce composition operators on 1A . In fact, one sees from the proof of Theorem 7.5.5 that any non-rotational disk automorphism fails to induce a composition operator on 1A . So what types of self-maps ϕ yield composition operators on 1A ? Using (13.3.1) we see that if max{|ϕ(z)| : z ∈ D} < 1 then ϕ induces a composition operator on 1A . The real interest then becomes when max{|ϕ(z)| : z ∈ D} = 1, in other words, the closure of ϕ(D) touches the unit circle. If we are in the extreme case when |ϕ(ξ)| = 1 for every ξ ∈ T, then a classical theorem of Fatou [67, p. 49] says that

13.4. ZERO SETS

199

ϕ must be a finite Blaschke product. However, not every finite Blaschke product induces a composition operator on 1A . Theorem 13.3.3 (Hedstrom-Newman). A finite Blaschke product B induces a composition operator on 1A if and only if B(z) = ζz n for some ζ ∈ T and integer n  0. When max{|ϕ(z)| : z ∈ D} = 1 but ϕ is not a finite Blaschke product, then there is an equivalent criterion for the condition that sup{ϕn 1 : n  0} < ∞ that involves the notion of ordinary maximum points. The interested reader can consult the references in the end notes for the details, including the following two examples. The polynomials 1 + z − z2 12 + 16z − 3z 2 √ and ϕ2 (z) = 25 5 define analytic self-maps of D for which ϕ1 (z) =

max{|ϕj (z)| : z ∈ D} = 1,

j = 1, 2.

However, sup{ϕn1 1 : n  0} < ∞ while

sup{ϕn2 1 : n  0} = ∞.

Thus ϕ1 induces a composition operator on 1A whereas ϕ2 does not. We end this section with a classification of the invertible composition operators on 1A . Corollary 13.3.4. The only invertible composition operators on 1A are those induced by the rotations f (z) →  f (ξz), where ξ ∈ T. The proof of this result, from [127], relies on the fact that if a composition operator Cϕ on 1A is onto, then the symbol ϕ must be an analytic bijection of D onto D. An application of the Schwarz Lemma [67, p. 3] shows that z−a ϕ(z) = ξ 1 − az for some ξ ∈ T and a ∈ D. Now apply Theorem 13.3.3 to see that ϕ(z) = ξz. 13.4. Zero Sets Certainly any set Z ⊆ D for which Z = f −1 ({0}) for some f ∈ A \ {0} has the properties that Z ∩ D is a Blaschke sequence and Z ∩ T has Lebesgue measure zero. However, the boundary zero sets for 1A can be quite mysterious. Theorem 13.4.1 (Kaufman). There exists a closed set E ⊆ T for which E = f −1 ({0}) for some f ∈ A \ {0}, but for which ϕ(E) is not an 1A zero set for any non-rotational disk automorphism ϕ. For such E and f we see that f ◦ ϕ ∈ 1A for any non-rotational disk automorphism ϕ. A more advanced result from [90] says that such f are actually dense in 1A . We have already seen in Theorem 5.1.7 that if f ∈ 1A \ {0} and f |K ≡ 0 for some K ⊆ T, then m(K) = 0. As we will see in Theorem 13.5.3 below, any closed subset of T with measure zero is a zero set for some function in A \ {0}. This next result of Carleson shows there is a difference between the boundary zero sets for A and those for 1A .

200

13. THE WIENER ALGEBRA

Proposition 13.4.2 (Carleson). There exists a closed set K ⊆ T with m(K) = 0 for which K = f −1 ({0}) for any f ∈ 1A . From our discussion surrounding Theorem 9.7.2, a closed set E ⊆ T for which log dist(ξ, E) dm(ξ) > −∞ T

is a boundary zero set for

1A

since it is the boundary zero set for a function in A 1 . 13.5. Ideals

Recall from Theorem 6.6.13 that 1A is a Banach algebra; that is, 1A is a Banach space with the additional property that f g ∈ 1A for all f, g ∈ 1A and f g1  f 1 g1 . Definition 13.5.1. An ideal of 1A is a subspace I of 1A for which f g ∈ I whenever f ∈ I and g ∈ 1A . Certainly any ideal I of 1A is an S-invariant subspace, that is to say, SI ⊆ I , where (Sf )(z) = zf (z) is the shift operator on 1A . It turns out that the converse is also true. Proposition 13.5.2. If I is an S-invariant subspace of 1A , then I is an ideal

in

1A .

Proof. Let f ∈ I and g ∈ 1A . By Proposition 6.1.6 there are polynomials gn with gn −g1 → 0. Since I is S-invariant, we have gn f ∈ I , n  1. Furthermore, f g ∈ 1A and f g − f gn 1  f 1 gn − g1 → 0, n → ∞. Now use the fact that I is closed to see that f g ∈ 1A and so I is an ideal.



A complete description of the ideals of 1A is unknown. To give these complexities some context, let us review the ideals of the disk algebra A for which a complete description was given by Rudin. As with 1A , the disk algebra A is a commutative Banach algebra, which is an important feature of Rudin’s analysis. If K is a closed subset of T, consider the set AK := {f ∈ A : f |K ≡ 0} and observe that AK is a subspace of A , and in fact an ideal in A . Of course we need to settle the issue of whether AK = {0}. Certainly if m(K) > 0 then AK = {0}. Theorem 13.5.3 (Rudin). If K ⊆ T is closed with m(K) = 0, then there is an f ∈ A for which {ξ ∈ T : f (ξ) = 0} = K. Thus AK = {0} if and only if m(K) = 0. There are other types of ideals in A . Recall the factorization of an inner function into its Blaschke factor and its singular inner factor (Theorem 5.1.9). Also recall the definition of the boundary spectrum Σb (u) of an inner function u from (12.11.2). Proposition 13.5.4. Let K be a closed subset of T with m(K) = 0 and let u be an inner function such that Σb (u) ⊆ K. Then uAK is a nonzero ideal of A .

13.5. IDEALS

201

As we shall see below, such uAK are all of the closed ideals of A . Thus given an ideal I of A , one must produce the closed set K and the inner function u. To produce the K, define for any f ∈ A \ {0}, ZT (f ) = {ξ ∈ T : f (ξ) = 0} and note that m(ZT (f )) = 0 (Theorem 5.1.7). Thus for our given nonzero ideal I of A , the set & K := ZT (f ), f ∈I \{0}

the boundary zero set for the ideal I , is a closed subset of T with m(K) = 0. To obtain the associated inner function u, consider the family of inner functions F(I ) consisting of all of the inner factors of functions from the ideal I . One can define u to be the greatest common inner divisor of the family F(I ). This is the unique (up to a multiplicative unimodular constant) inner function f for which f /u ∈ H ∞ for every f ∈ F(I ) and that if f /u1 ∈ H ∞ for every f ∈ F(I ), then u/u1 ∈ H ∞ . With this boundary zero set K and the inner function u produced above, we can now proceed to describe the ideal I . Define f  (13.5.5) J := :f ∈I u and observe that f /u always belongs to A (stemming from the fact that dividing a disk algebra function by its inner factor keeps the quotient in the disk algebra [87, Ch. 6]) and so J ⊆ A . One can also see that in fact J is an ideal of A whose greatest common inner factor is a constant function and whose boundary zero set, as defined above, is equal to K (the boundary zero set for I ). The main result here is that J = AK and so I = uAK . This is summarized in the following theorem. Theorem 13.5.6. Every nonzero ideal of A takes the form uAK for some closed set K ⊆ T with m(K) = 0 and some inner function u with Σb (u) ⊆ K. Using this theorem, along with some general properties of ideals in a general commutative Banach space, one can discuss the maximal ideals of A . An ideal I of A is maximal if there are no proper ideals containing I . A consequence of the description of the ideals of A (Theorem 13.5.6) is the following. Corollary 13.5.7. Every maximal ideal of A takes the form {f ∈ A : f (λ) = 0} for some λ ∈ D. An ideal I of A is principal if there is an f ∈ A such that I = {f g : g ∈ A }. Corollary 13.5.8. Any ideal I of A is principal. Proof. For a nonzero ideal I Theorem 13.5.6 says that I = uAK . Now use Theorem 13.5.3 to produce a g ∈ A whose zeros are precisely K. If G is the outer factor of g (which belongs to A and has the same zero set as g), then one can argue that I = {uGh : h ∈ A } = uGA .  From our description of the maximal ideals, one can also phrase a corona condition for A .

202

13. THE WIENER ALGEBRA

Corollary 13.5.9. If f1 , . . . , fn belong to A and n & {z ∈ D : fj (z) = 0} = ∅, j=1

then there are g1 , g2 , . . . , gn in A such that f1 g1 + f2 g2 + · · · + fn gn ≡ 1. Proof. Argue that the set of functions n  fj gj , gj ∈ A , j=1

forms an ideal in A . This ideal must contain the constant function f ≡ 1, since if not, Zorn’s Lemma would imply this ideal would be contained in a maximal ideal. Corollary 13.5.7, along with the assumption that the functions f1 , . . . , fn have no common zeros, will provide a contradiction.  Let us now discuss the ideals of 1A . Despite having a few of the characteristics of the ideals of the disk algebra, in general, they turn out to be complicated and their description seems impossible. Before getting to this, let us explore the complications that arise when applying Rudin’s analysis of the ideals of A to those of 1A . The first problem is that we are lacking a version of Theorem 13.5.3 for 1A and thus do not have a clear understanding of the boundary zero sets. Indeed, we saw some of the stark differences between the boundary sets for A (closed sets of measure zero) and those for 1A (Theorem 13.4.1 and Proposition 13.4.2). In addition, important to Rudin’s analysis (see (13.5.5)) was the fact that given an f ∈ A , we can divide by the inner factor of f and the quotient belongs to A . As it turns out, this property no longer holds for 1A . Theorem 13.5.10 (Gurari˘ı). There exists an f ∈ 1A and an inner function u such that f /u ∈ H ∞ but f /u ∈ 1A . Despite these differences, the ideals of A and 1A both share the following index one property from Definition 10.3.6. Theorem 13.5.11 (Richter). If I is a nonzero ideal of 1A or A , then dim(I /SI ) = 1. Proof. We follow a proof from [12]. Suppose there is a ϕ ∈ I for which ϕ(0) = 1. We will show in a moment that (13.5.12)

I = Cϕ + SI .

Assuming this, one can see that the quotient space I /SI = (Cϕ + SI )/SI is isomorphic to the one dimensional vector space C. To prove (13.5.12) we note that the ⊇ inclusion is clear. For the reverse inclusion, let f ∈ I and find functions v and g ∈ 1A for which f − f (0)ϕ = Sg,

1 − ϕ = Sv.

Since I contains the functions ϕ and Sg = f − f (0)ϕ and I is an ideal, then g = ϕg + vSg ∈ I which implies that f ∈ I . This proves (13.5.12). In the above argument we assumed there was a ϕ ∈ I with ϕ(0) = 1. If this were not the case then an argument will produce an ideal J of 1A with the

13.6. NOTES

203

property that there is a ϕ ∈ J for which ϕ(0) = 1 and such that I = S n J for some positive integer n. We have already shown that the quotient space J /SJ is one dimensional. Now use the fact that S is an isometry to see that I /SI and J /SJ have the same dimension.  Even though they have index one, the ideals of 1A are still very complicated in the following sense. For an ideal I we say that a set X ⊆ I generates I if I = {f g : f ∈ X, g ∈ 1A }. In other words, X generates I when the smallest ideal of 1A containing X is equal to I . We define the multiplicity of I , denoted by mult(I ), to be the cardinality of the smallest set that generates I . We see from Corollary 13.5.8 that mult(I ) = 1 for every nonzero ideal I of the disk algebra A . This next result says that even though dim(I /SI ) = 1 for every nonzero ideal I of 1A , the multiplicity can be anything. Theorem 13.5.13 (Atzmon). Given any n ∈ N ∪ {∞}, there is an ideal I of 1A with mult(I ) = n. Despite the complexity of the ideals of 1A , which are markedly different from those of A , there are some similarities. The next assertion is the analog of Corollary 13.5.7. Theorem 13.5.14. The maximal ideals of 1A are {f ∈ 1A : f (λ) = 0}, where λ ∈ D. The proof of this is more complicated than in the disk algebra case since we do not have a complete description of the ideals of 1A . However, there is the general Gelfand theory which characterizes the maximal ideals in a variety of settings, including this one. This next result is the analog of Corollary 13.5.9, with the same proof. Corollary 13.5.15. If f1 , . . . , fn belong to 1A and have no common zeros in D, then there are g1 , g2 , . . . , gn in 1A such that f1 g1 + f2 g2 + · · · + fn gn ≡ 1. 13.6. Notes Proposition 13.1.3 is due to Bernstein (see [171, Vol. I, p. 240]) and is sharp in the sense that it no longer holds when a = 12 . With some additional hypotheses, however, it can be extended to all a ∈ (0, 1) [100, p. 32]. Peano curve type results for analytic functions have a long history and appear in papers of Belov [18], Kahane and Weiss [98], Bagemihl and Piranian [15], Schaeffer, Piranian, Titus, and Young [135], and Schaeffer [146]. Theorem 13.1.5 is from a paper of Salem and Zygmund [145]. Some of our treatment of composition operators follows the work of Newman [127]. Theorem 13.3.3 was proved independently by Newman [127] and Hedstrom [84]. Theorem 13.4.1 comes from Kaufman [101] and its extension is from Indlekofer [90]. Theorem 13.4.2 and other related boundary zero set results appear in Carleson’s paper [23]. See Kahane and Katznelson [97] for other boundary zero set results. Our discussion of the ideals of the disk algebra closely follows [87]. Just before Corollary 13.5.9, we made a passing comment about the Corona Theorem. This celebrated result of Carleson is covered in the text [68]. The ability to divide out the inner factor and remain in the space is often called “the F -property” and holds in many spaces of analytic functions [156] but not all of

204

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them. The failure of the F -property for 1A (Theorem 13.5.10) is a result of Gurari˘ı [72]. The index one property for both A and 1A (Theorem 13.5.11) is a consequence of a general result of Richter concerning ideals of certain Banach algebras [136]. The fact that the ideals of 1A can have arbitrary multiplicity (Theorem 13.5.13) is from Atzmon [12]. Some of this chapter follows a related treatment of these topics from two papers of Mortini [125, 126]. The ideals of algebras of analytic functions is a well studied topic. For many algebras, including the class A n from (13.1.1), their ideals can be completely described. See [156] for a thorough survey on this.

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Author Index Duffin, R., 28 Duren, P., ix, 72, 87, 99, 122, 124, 148, 194 Dyakonov, K., 86, 88

Abakumov, E., 157 Abakumov, E., ix, xiii, 156, 169, 171 Abramovich, Y., 156 Albiac, F., ix Aleksandrov, A., 72 Aleman, A., xii, 72, 156, 157 Aliprantis, C., 156 Alonzo, J., 40 Alp´ar, L., ix, 96, 100 Arsenovi´c, M., ix, 99 Atkinson, F., 151 Atzmon, A., 202, 204 Axler, S., ix, xiii, 72, 156

Egervary, E., 122 El-Fallah, O., ix Elaydi, S., 112, 114 Fabry, E., xiii, 171, 194 Fatou, P., 198 Fej´er, L., 122 Forelli, F., 100 Fricain, E., 156 Friedrichs, K., 40

Bagemihl, F., 203 Beals, R., 15 Beauzamy, B., 36, 40 Belov, A., 203 Bernstein, S., 75, 203 Beurling, A., xii, xiii, 70, 72, 154, 157 Birkhoff, G., 40 Borichev, A., ix, xiii, 156, 157 Bourdon, P., xiii, 156 Bynum, W., 61

Gaier, D., 194 Gamelin, T., 72 Gao, J., 62 Garcia, S., 157, 162, 171, 191, 194, 199 Garnett, J., ix, 72, 122, 203 Girela, D., ix, 148 Gleason, J., 157 Grafakos, L., 81 Gurari˘ı, V., 87, 202, 204

Carleson, L., 199, 203 Carothers, N., ix, 15, 62 Cheng, R., ix, 40, 62, 122, 148, 191 Christensen, O., 28 Cima, J., 72, 171 Clarkson, J., 40, 62 Conway, J., 156, 165 Cowen, C., 100

H¨ormander, L., 191 Hadamard, J., xiii, 168, 171, 194 Hal´asz, G., ix, 100 Halmos, P., 186 Han, D., 28 Hanche-Olsen, H., 40 Hanner, O., ix, 40 Hardy, G., xii Harris, C., 62 Havin, V., 88 Hedenmalm, H., 157

Day, M., ix, 15, 30, 43, 72 de Leeuw, K., 100 Douglas, R., 70, 72, 160, 171 Dragas, J., ix, 122, 148 Drew, J., 61 213

214

AUTHOR INDEX

Hedenmalm, H., ix, xii, 72, 115, 122, 145, 148 Hedstrom, G., 199, 203 Hirschman, I., 191 Hoffman, K., 72, 201, 203 Horowitz, C., 145, 148 Indlekofer, K., 199, 203 James, R., 40 Jevti´c, M., ix, 99 Jiao, H., 62 Jordan, P., 41, 61 K¨othe, G., ix Kaewkhaom A., 62 Kahane, J., ix, 87, 100, 203 Kalton, N., ix Katznelson, Y., 203 Kaufman, R., 199, 203 Kellay, K., ix Khavinson, S., 122 Klenke, A., 37 Koosis, P., ix, 72 Korenblum, B., ix, xii, 72 Kornelson, K., 28 L´evy, P., 195 Lamperti, J., 93, 99 Landau, E., 194 Larson, D., 28 Lau, K., 62 Lebedev, V., ix, 182, 191 Li, C., ix, 100 Lindenstrauss, J., 47 Littlewood, J., xii, 83, 87, 95 Liu, B., 28 Liu, R., 28 MacCluer, B., 100 Macintyre, A. J., 122 Marcus, A., 28 Marden, M., 122 Martini, H., 40 Mashreghi, J., ix, 62, 122, 148, 156, 157, 162, 171, 191, 194, 199 Mazya, V., 191 Meir, A., 40 Miamee, A., 40, 62 Milman, D., 40

Mortini, R., 204 Nevanlinna, R., xii Newman, D., ix, 76, 77, 80, 87, 100, 196, 199, 203 Nikolskii, N., ix, 156, 171, 176, 177, 184, 191 Olevski˘ı, A., ix, 182, 191 Pang, B., 62 Pel´ aez, J., ix, 148 Pettis, B., 40 Piranian, G., 203 Pourahmadi, M., 40, 62 Ransford, T., ix Richter, S., 157 Richter, S., xii, 72, 156, 157, 202 Riesz, M., xii Roberts, D., 40 Rogosinski, W., 122 Romberg, B., 72 Ross, W., 157 Ross, W., ix, 62, 72, 122, 148, 162, 171, 191, 194, 199 Royden, H., 40 Rudin, W., 75, 100, 169, 200 Salem, R., 194, 203 Sch¨affer, J., 62 Schaeffer, A., 28, 203 Schuster, A., ix, 72, 122, 148 Seip, K., xii, 148 Shamoyan, F., 88 Shapiro, H., ix, xii, xiii, 70, 72, 76, 77, 80, 87, 122, 148, 160, 171 Shapiro, J., 16, 100 Shaposhnikova, T., 191 Shields, A., xii, 70, 72, 148, 160, 171 Shioji, N., 40 Shirokov, N., ix, 88, 148, 204 So, W., ix, 100 Spielman, D., 28 Srivastava, N., 28 Steele, J., 184 Stein, E., 100 Stiles, W., 16 Stoeva, D., 28 Sundberg, C., 157

INDEX

Sundberg, C., xii, 171 Taylor, B., 148 Terpigoreva, V., 122 Titus, C., 203 Verbitski˘ı, I., ix, 191 Vinogradov, S., ix, 123, 148, 190, 191 von Neumann, J., 41, 61 von Renteln, M., 204 Vukoti´c, D., ix, 99 Wang, F., 62 Weber, E., 28

215

Weiss, G., 203 Weiss, M., 203 Wermer, J., 100 Wiener, N., 195 Williams, D., 148 Wu, S., 15, 40 Yang, C., 62 Young, G., 203 Zheng, B., 28 Zhu, K., ix, 72, 115, 122, 148 Zygmund, A., 194, 203

Subject Index duality of weak parallelogram laws, 48

adjoint, 70, 89, 159, 178 analysis operator, 17, 20 annihilator, 70, 159 Atkinson’s Theorem, 151

embedding, 9, 64, 74, 188 essential spectrum, 152 essential supremum, 2 evaluation functional, 82, 86, 87, 107, 139, 174 exponential sequence, 78 extended exterior disk, 71, 160 extra zeros, 115 extremal problem, 102, 109, 127

backward shift, 70, 159, 162, 177 Banach space, ix, xi, 1, 5 Bergman space, 68 Beurling’s Theorem, 157 Beurling’s Theorem, 70, 154 biorthogonal sequence, 26 Birkhoff-James orthogonality, 34, 38–40, 105 Blaschke condition, 65, 123, 129 Blaschke product, 65, 77 boundary function, 64, 84 boundary spectrum, 191 bounded mean oscillation, 72 bounded type, 66, 71, 161

F-property, 204 factorization, 65, 69, 122 Feichtinger conjecture, 27 finite co-dimensional, 149 finite point extremal functions, 111, 113 Fourier coefficient, 65 Fourier multiplier, 173, 180 frame, 17, 19, 146 Fredholm operator, 151

Calkin algebra, 151 canonical pairing, 10 Clarkson’s Inequalities, 31, 32, 40, 44, 61 closed unit ball, 29 commutant, 176 compact operator, 93 completeness, 5 composition operator, 95, 198 convex, 29, 40 convolution, 173, 194 Corona Theorem, 203 cyclic vector, 70, 71, 159, 161

Gˆateaux derivative, 36 generator of an ideal, 203 Hadamard gap series, 168 Hadamard multiplier, 92 Hanner’s Inequality, 40, 52, 61 Hardy space, 63, 81, 160 Hardy’s Inequality, 193 Hausdorff-Young Inequalities, 81 ideal, 200 index of an invariant subspace, 154 inner, 66, 69, 105, 106, 122, 127, 154, 190 invariant subspaces for S, 149

difference quotient operator, 91 disk algebra, 193 division property, 153 dual basis, 26 dual space, 10, 67, 69, 82 217

218

SUBJECT INDEX

invariant subspaces for S ∗ , 160 isometric multipliers, 184 isometry, 93

principal ideal, 201 pseudocontinuation, 71, 161 Pythagorean Inequalities, 55, 122

James constant, 62 Jensen’s Formula, 125 Jordan-von Neumann Theorem, 41

quasinorm, 7

kernel of an operator, 151 Kronecker delta, 10 L´evy’s Theorem, 195 lacunary series, 194 linear functional, 10 Littlewood Subordination Theorem, 95 maximal ideal, 201, 203 McShane’s Lemma, 56 metric projection, 34, 35, 60, 106 Milman-Pettis Theorem, 40 Minkowski’s Inequality, 3 modulus of convexity, 47 modulus of smoothness, 47 multiplicity of an ideal, 203 multiplier, 67, 69, 173 multiplier norm, 174 Nevanlinna class, 66 Newman condition, 80, 123 non-tangential limit, 182 normal family, 164 norming functional, 14, 36–38, 56, 108 nullity-rank theorem, 151 one point extremal function, 111 optimal frame bounds, 20 optimal Riesz basis constants, 25 optimal weak parallelogram constants, 44, 50 order of a zero, 101 outer, 66, 69, 122 p-inner, 105, 106, 127 Parallelogram Law, 41 Parseval frame, 17, 19, 21 Peano curve, 194, 203 pointwise multiplier, 173 pre-zero set, 102, 126

radial limit, 64, 83 range of an operator, 151 reflexive, 12, 40 repeated according to multiplicity, 101 Reverse H¨older’s Inequality, 8 Reverse Triangle Inequality, 7 Riesz basic sequence, 27 Riesz basis, 18, 24 Riesz Representation Theorem, 10, 11 Riesz-Fischer Theorem, 15 sampling sequence, 145 Schauder basis, 10 Schur’s Test, 186 semi-Fredholm operator, 151 sesquilinear pairing, 67 shift operator, 69, 70, 89 signed power, 15 singular inner function, 66 Smirnov class, 66 Smirnov’s Theorem, 66 smooth, 36, 40 Stolz region, 182 strictly convex, 29, 36 subspace, 102 synthesis operator, 18, 20 tight frame, 17, 19, 23 Toeplitz matrix, 176 uniform convexity, 29, 31, 33–35, 47 uniformly separated, 77 uniformly smooth, 47 unit ball, 29 Van Der Corput Lemma, 100 von Neumann-Jordan constant, 62 weak parallelogram law, 43, 48, 50 Wiener algebra, xi, xii, 87, 193 Wiener’s Theorem, 194 Young’s Convolution Inequality, 173

INDEX

Young’s Inequality, 2

219

zero set, 144

Selected Published Titles in This Series 75 Raymond Cheng, Javad Mashreghi, and William T. Ross, Function Theory and p Spaces, 2020 74 Leonid Polterovich, Daniel Rosen, Karina Samvelyan, and Jun Zhang, Topological Persistence in Geometry and Analysis, 2020 73 Armand Borel, Introduction to Arithmetic Groups, 2019 72 Pavel Mnev, Quantum Field Theory: Batalin–Vilkovisky Formalism and Its Applications, 2019 71 Alexander Grigor’yan, Introduction to Analysis on Graphs, 2018 70 Ian F. Putnam, Cantor Minimal Systems, 2018 69 Corrado De Concini and Claudio Procesi, The Invariant Theory of Matrices, 2017 68 Antonio Auffinger, Michael Damron, and Jack Hanson, 50 Years of First-Passage Percolation, 2017 67 Sylvie Ruette, Chaos on the Interval, 2017 66 Robert Steinberg, Lectures on Chevalley Groups, 2016 65 Alexander M. Olevskii and Alexander Ulanovskii, Functions with Disconnected Spectrum, 2016 64 63 62 61

Larry Guth, Polynomial Methods in Combinatorics, 2016 Gon¸ calo Tabuada, Noncommutative Motives, 2015 H. Iwaniec, Lectures on the Riemann Zeta Function, 2014 Jacob P. Murre, Jan Nagel, and Chris A. M. Peters, Lectures on the Theory of Pure Motives, 2013

60 William H. Meeks III and Joaqu´ın P´ erez, A Survey on Classical Minimal Surface Theory, 2012 59 Sylvie Paycha, Regularised Integrals, Sums and Traces, 2012 58 Peter D. Lax and Lawrence Zalcman, Complex Proofs of Real Theorems, 2012 57 Frank Sottile, Real Solutions to Equations from Geometry, 2011 56 A. Ya. Helemskii, Quantum Functional Analysis, 2010 55 Oded Goldreich, A Primer on Pseudorandom Generators, 2010 54 John M. Mackay and Jeremy T. Tyson, Conformal Dimension, 2010 53 John W. Morgan and Frederick Tsz-Ho Fong, Ricci Flow and Geometrization of 3-Manifolds, 2010 52 Marian Aprodu and Jan Nagel, Koszul Cohomology and Algebraic Geometry, 2010 51 J. Ben Hough, Manjunath Krishnapur, Yuval Peres, and B´ alint Vir´ ag, Zeros of Gaussian Analytic Functions and Determinantal Point Processes, 2009 50 John T. Baldwin, Categoricity, 2009 49 J´ ozsef Beck, Inevitable Randomness in Discrete Mathematics, 2009 48 Achill Sch¨ urmann, Computational Geometry of Positive Definite Quadratic Forms, 2008 47 Ernst Kunz, David A. Cox, and Alicia Dickenstein, Residues and Duality for Projective Algebraic Varieties, 2008 46 Lorenzo Sadun, Topology of Tiling Spaces, 2008 45 Matthew Baker, Brian Conrad, Samit Dasgupta, Kiran S. Kedlaya, and Jeremy Teitelbaum, p-adic Geometry, 2008 44 Vladimir Kanovei, Borel Equivalence Relations, 2008 43 Giuseppe Zampieri, Complex Analysis and CR Geometry, 2008 42 Holger Brenner, J¨ urgen Herzog, and Orlando Villamayor, Three Lectures on Commutative Algebra, 2008

For a complete list of titles in this series, visit the AMS Bookstore at www.ams.org/bookstore/ulectseries/.

The classical p sequence spaces have been a mainstay in Banach spaces. This book reviews some of the foundational results in this area (the basic inequalities, duality, convexity, geometry) as well as connects them to the function theory (boundary growth conditions, zero sets, extremal functions, multipliers, operator theory) of the associated spaces Ap of analytic functions whose Taylor coefficients belong to p . Relations between the Banach space p and its associated function space are uncovered using tools from Banach space geometry, including Birkhoff-James orthogonality and the resulting Pythagorean inequalities. The authors survey the literature on all of this material, including a discussion of the multipliers of Ap and a discussion of the Wiener algebra A1 . Except for some basic measure theory, functional analysis, and complex analysis, which the reader is expected to know, the material in this book is self-contained and detailed proofs of nearly all the results are given. Each chapter concludes with some end notes that give proper references, historical background, and avenues for further exploration.

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