STUDENT MATHEMATICAL LIBRARY Volume 85

Volterra Adventures

Joel H. Shapiro 10.1090/stml/085

Volterra Adventures

STUDENT MATHEMATICAL LIBRARY Volume 85

Volterra Adventures

Joel H. Shapiro Editorial Board Satyan L. Devadoss John Stillwell (Chair) Erica Flapan Serge Tabachnikov

2010 Mathematics Subject Classification. Primary 46-01, 45-01.

Cover image: ESO/Igor Chekalin (http://www.fpsoftlab.com/gallery/index.htm) Licensed under Creative Commons Attribution 4.0 International (CC BY 4.0) https://creativecommons.org/licenses/by/4.0.

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Library of Congress Cataloging-in-Publication Data Names: Shapiro, Joel H., author. Title: Volterra adventures / Joel H. Shapiro. Description: Providence, Rhode Island : American Mathematical Society, [2018] | Series: Student mathematical library ; volume 85 | Includes bibliographical references and index. Identifiers: LCCN 2017052435 | ISBN 9781470441166 (alk. paper) Subjects: LCSH: Volterra equations. | Functional analysis. | Convolutions (Math- ematics) | AMS: Functional analysis – Instructional exposition (textbooks, tu- torial papers, etc.). msc | Integral equations – Instructional exposition (text- books, tutorial papers, etc.). msc Classification: LCC QA431 .S4755 2018 | DDC 515/.45–dc23 LC record available at https://lccn.loc.gov/2017052435

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Contents

Preface xi

List of Symbols xv

Part 1. From Volterra to Banach

Chapter 1. Starting Out 3 §1.1. A vector space 3 §1.2. A linear transformation 4 §1.3. Eigenvalues 6 §1.4. Spectrum 8 §1.5. Volterra spectrum 9 §1.6. Volterra powers 11 §1.7. Why justify our “formal calculation”? 13 §1.8. Uniform convergence 14 §1.9. Geometric series 16 Notes 19

Chapter 2. Springing Ahead 21 §2.1. An initial-value problem 21 §2.2. Thinking differently 24

vii viii Contents

§2.3. Thinking linearly 25 §2.4. Establishing norms 26 §2.5. Convergence 28 §2.6. Mass-spring revisited 32 §2.7. Volterra-type integral equations 35 Notes 35

Chapter 3. Springing Higher 37 §3.1. A general class of initial-value problems 37 §3.2. Solving integral equations of Volterra type 39 §3.3. Continuity in normed vector spaces 41 §3.4. What’s the resolvent kernel? 45 §3.5. Initial-value problems redux 49 Notes 51

Chapter 4. Operators as Points 53 §4.1. How “big” is a linear transformation? 54 §4.2. Bounded operators 56 §4.3. Integral equations done right 61 §4.4. Rendezvous with Riemann 63 §4.5. Which functions are Riemann integrable? 67 §4.6. Initial-value problemsalaRiemann ` 69 Notes 73

Part 2. Travels with Titchmarsh

Chapter 5. The Titchmarsh Convolution Theorem 81 §5.1. Convolution operators 81 §5.2. Null spaces 84 §5.3. Convolution as multiplication 86 §5.4. The One-Half Lemma 89 Notes 95 Contents ix

Chapter 6. Titchmarsh Finale 97 §6.1. The Finite Laplace Transform 97 §6.2. Stalking the One-Half Lemma 99 §6.3. The complex exponential 103 §6.4. Complex integrals 105 §6.5. The (complex) Finite Laplace Transform 107 §6.6. Entire functions 108 Notes 111

Part 3. Invariance Through Duality

Chapter 7. Invariant Subspaces 115 §7.1. Volterra-Invariant Subspaces 115 §7.2. Why study invariant subspaces? 117 §7.3. Consequences of the VIST 123 §7.4. Deconstructing the VIST 126 Notes 131

Chapter 8. Digging into Duality 133

§8.1. Strategy for proving Conjecture C0 133 §8.2. The “separable” Hahn-Banach Theorem 136 §8.3. The “nonseparable” Hahn-Banach Theorem 144 Notes 149

Chapter 9. Rendezvous with Riesz 155 §9.1. Beyond Riemann 155 §9.2. From Riemann & Stieltjes to Riesz 160 §9.3. Riesz with rigor 162 Notes 169

Chapter 10. V-Invariance: Finale 173 §10.1. Introduction 173 §10.2. One final reduction! 174 §10.3. Toward the Proof of Conjecture U 175 x Contents

§10.4. Proof of Conjecture U 178 Notes 180

Appendix A. Uniform Convergence 183

Appendix B. Complex Primer 185 §B.1. Complex numbers 185 §B.2. Some Complex Calculus 187 §B.3. Multiplication of complex series 188 §B.4. Complex power series 190

Appendix C. Uniform Approximation by Polynomials 195

Appendix D. Riemann-Stieltjes Primer 199 Notes 211

Bibliography 213

Index 217 Preface

This book guides mathematics students who have completed solid first courses in linear algebra and analysis on an expedition into the field of functional analysis. At the journey’s end they will have captured two famous theorems—often stated in graduate courses, but seldom proved, even there: (a) The Titchmarsh Convolution Theorem, which characterizes the null spaces of Volterra convolution operators, and which implies (in fact, is equivalent to): (b) The Volterra Invariant Subspace Theorem, which asserts that the only closed, invariant subspaces of the Volterra op- erator are the “obvious ones.” The pursuit of these theorems breaks into three parts. The first part (four chapters) introduces the Volterra operator, while gently induc- ing readers to reinterpret the classical notion of uniform convergence on the interval [a, b] as convergence in the max-norm, and to reimagine continuous functions on that interval as points in the Banach space C [a, b] . It exploits, at several levels, this “functions are points” par- adigm (often attributed to Volterra himself) in the process of solving integral equations that arise—via the Volterra operator—from the kinds of initial-value problems that students encounter in their be- ginning differential equations courses. At the conclusion of this part

xi xii Preface of the book, readers will be convinced (I hope) that even linear trans- formations can be thought of as “points in a space,” and that within this framework the proof that “Volterra-type” integral equations have unique solutions boils down to summation of a geometric series. In the process of tackling initial-value problems and integral equa- tions we naturally encounter Volterra convolution operators, which form the subject of the second part of the book (two chapters). It’s here that the problem of characterizing the null spaces of these oper- ators is introduced, and solved via the Titchmarsh Convolution The- orem. The final step in proving the Titchmarsh theorem involves Li- ouville’s theorem on bounded entire functions, for which just enough complex analysis (using only power series) is developed to give a quick proof. The final part of the book (four chapters) aims toward using Titchmarsh’s theorem to prove the Volterra Invariant Subspace The- orem. Here we encounter a pair of results that lie at the heart of func- tional analysis: the Hahn-Banach Theorem on separation by bounded linear functionals of closed subspaces from points not in them, and the Riesz representation of the bounded linear functionals on C [a, b] by means of Riemann-Stieltjes integrals. The Hahn-Banach theorem is derived from its extension form, which is proved in the usual way: extending by one dimension, then using some form of induction. This is done first for separable spaces, using ordinary mathematical induc- tion, and then in general by transfinite induction, which is carefully introduced. The Hahn-Banach extension theorem (nonseparable version!) then provides a quick proof of Riesz’s representation theorem. Here it’s hoped—but not assumed—that the reader has seen the Stieltjes extension of the Riemann integration theory. In any case, an ap- pendix covers much of the standard material on Riemann-Stieltjes integration, with proofs omitted where they merely copy those for the Riemann integral. The book’s final chapter completes the proof of the invariant subspace theorem for the Volterra operator. Preface xiii

Each chapter begins with an “Overview” and ends with a section of “Notes” in which the reader may find further results, historical ma- terial, and bibliographic references. Exercises are scattered through- out, most of them rather easy, some needed later on. Their purpose is twofold: first, to enhance the material at hand, and second (no less important) to emphasize the necessity of interacting actively with the mathematics being studied. I hope this book will expand its readers’ horizons, sharpen their technical skills, and for those who pursue functional analysis at the graduate level, enhance—rather than duplicate—that experience. In pursuit of this goal the book meanders through mathematics that is algebraic and analytic, abstract and concrete, real and complex, finite and transfinite. In this, it’s inspired by the words of the late Louis Auslander: “Mathematics is like a river. You just jump in someplace; the current will take you where you need to go.”1

Acknowledgments Much of the material presented here originated in lectures that I gave in beginning graduate courses at Michigan State University, and later in seminars at Portland State. Eriko Hiron- aka suggested that the notes from these lectures might serve as the basis for a book appropriate for advanced undergraduate students, and she provided much-needed encouragement throughout the result- ing project. Paul Bourdon and Jim Rulla read the manuscript, con- tributing vital corrections, improvements, and critical comments. The Fariborz Maseeh Department of Mathematics and Statistics at Port- land State University provided office space, library access, technical assistance, and a lively Analysis Seminar. Michigan State University provided electronic access to its library. To all of these people and institutions I am profoundly grateful. Above all, this project owes much to the understanding, patience, and encouragement of my wife, Jane; I couldn’t have done it without her.

Portland, Oregon September 2017

1Paraphrased by Anna Tsao in: Notices Amer. Math. Soc., Vol. 45, #3, 1998, page 393.

List of Symbols

N: The natural numbers; 1, 2, ... R:Therealnumbers C: The complex numbers “scalars”: R or C (usually your choice) ·: Symbol for a norm; usually the sup- or max-norm on a vector space of functions { n }∞ OrbT (v): T v 0 , the orbit of the vector v under the linear transformation T

MT (v): The closure of the linear span of OrbT (v); the smallest closed T -invariant subspace containing v C [a, b] : The space of scalar valued functions that are contin- uous on the compact interval [a, b] V : The Volterra operator on C [0,a]

Vκ: The “Volterra-type” operator with kernel κ 0: The constant functions taking value 0 1: The constant function taking value 1, or the least element of a well-ordered set (see page 146) VIST: The Volterra Invariant Subspace Theorem Cb: The subspace of C [0,a] consisting of functions that van- ish on the interval [0,b](0

xv xvi List of Symbols C0: The subspace of C [0,a] consisting of functions that van- ish at the origin. R [a, b] : The collection of scalar-valued functions on [a, b]that are Riemann integrable (f): The left-most support point of a function f ∈ C [0, ∞) C [0, ∞) : The space of continuous, scalar-valued functions on the half-line [0, ∞) spt f: The support of f ∈ C [0, ∞) ; closure of the set of points x for which |f(x)| =0

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Axiom of Choice, 147, 152 induced by norm, 28 dual space, 133 Banach -Tarski Paradox, 152, 153 function space, 30 entire, 108 Stefan, 35 piecewise continuous, 67

Cauchy-Riemann equations, 153 Helly closed, 30 Eduard, 150 compact, 16 Extension Lemma, 137 complete, 30 Intersection Theorem, 150 continuity Moment Problem, 170 absolute, 73 continuous Inequality at a point, 41 Cauchy-Schwarz, 28 piecewise, 67 Easy Titchmarsh, 84 uniformly, 42 Hard Titchmarsh, 84 convergence reverse triangle, 28 absolute, 31 triangle, 27 in normed vector space, 28 integral of series, 31 Lebesgue, 76 pointwise, 10 Riemann, 157 Riemann-Stieltjes, 158 deficiencies, 14  pointwise on C [0,a] ,33 intermediate points, 64 uniform, 15 invertible, 5 convolution boundedly, 62 as multiplication, 86 kernel operator, 81 resolvent, 14 distance Volterra, 39 from point to set, 134 217 218 Index

Laplace Transform, 97 relation, 145 Finite, 97 Riemann Lebesgue Bernhard, 74 and differentiation, 75 Hypothesis, 74 integral, 76 integrable, 64 measurable sets, 77 sum, 64 Riemann Integrability Theorem, Riesz 68 and Helly, 170 Leibniz rule, 23 Frigyes, 169 limit element, 146 Moment Theorem, 171 linear transformation, 5 Representation Theorem, 168 boundedly invertible, 62 Rota, Gian-Carlo, 111 finite dimensional, 5 invertible, 5 scalar field, 4 Liouville sequence Theorem, 97 Cauchy, 30 convergent, 28 matrix set Jordan block, 118 closed, 30 rotation, 7 compact, 16 measure zero, 68 dense, 118 metric, 28 space Banach, 30 N natural numbers: , 144 dual, 133 norm(s) normed vector, 27 1  -, 71 separable, 118 definition, 26 spectrum max-, 16 algebraic, 9 of a partition, 64 point, 9 Rn on ,27 topological, 73 supremum-, 60 successor, 146

One-Half Lemma, 89 Theorem implies Titchmarsh Theorem, 93 Hahn-Banach operator Extension, 135 bounded, 56 Separation, 134 multiplication, 9 Helly’s Intersection, 150 shift, 6 Liouville’s, 97, 109 similar, 8 meta, 124 Volterra, 4 Riesz Representation, 168 convolution, 81 Titchmarsh Convolution, 81, 82 powers, 11 Weierstrass Approximation, 93, ordering, 145 95 total, 145 Well-Ordering, 147, 151 well-, 145 theorem Titchmarsh Convolution, 84 partition (of interval), 64 Titchmarsh pointwise operations, 4 Convolution Theorem, 81, 82 Index 219

Edward C., 95

VIST, 116 Volterra -type integral equation, 14 and differentiation, 75 convolution, 81 examples, 82 convolution operator, 81 kernel, 39 life of, 181 operator, 4 Vito, 19

Weierstrass Approximation Theorem, 36, 93, 95, 195 M-test, 16 uniform convergence, 36 well-ordering, 144, 145 Selected Published Titles in This Series

85 Joel H. Shapiro, Volterra Adventures, 2018 84 Paul Pollack, A Conversational Introduction to Algebraic Number Theory, 2017 83 Thomas R. Shemanske, Modern Cryptography and Elliptic Curves, 2017 82 A. R. Wadsworth, Problems in Abstract Algebra, 2017 81 Vaughn Climenhaga and Anatole Katok, From Groups to Geometry and Back, 2017 80 Matt DeVos and Deborah A. Kent, Game Theory, 2016 79 Kristopher Tapp, Matrix Groups for Undergraduates, Second Edition, 2016 78 Gail S. Nelson, A User-Friendly Introduction to Lebesgue Measure and Integration, 2015 77 Wolfgang Kuhnel, ¨ Differential Geometry: Curves — Surfaces — Manifolds, Third Edition, 2015 76 John Roe, Winding Around, 2015 75 Ida Kantor, Jiˇr´ıMatouˇsek, and Robert S´ˇamal, Mathematics++, 2015 74 Mohamed Elhamdadi and Sam Nelson, Quandles, 2015 73 Bruce M. Landman and Aaron Robertson, Ramsey Theory on the Integers, Second Edition, 2014 72 Mark Kot, A First Course in the Calculus of Variations, 2014 71 Joel Spencer, Asymptopia, 2014 70 Lasse Rempe-Gillen and Rebecca Waldecker, Primality Testing for Beginners, 2014 69 Mark Levi, Classical Mechanics with Calculus of Variations and Optimal Control, 2014 68 Samuel S. Wagstaff, Jr., The Joy of Factoring, 2013 67 Emily H. Moore and Harriet S. Pollatsek, Difference Sets, 2013 66 Thomas Garrity, Richard Belshoff, Lynette Boos, Ryan Brown, Carl Lienert, David Murphy, Junalyn Navarra-Madsen, Pedro Poitevin, Shawn Robinson, Brian Snyder, and Caryn Werner, Algebraic Geometry, 2013 65 Victor H. Moll, Numbers and Functions, 2012 64 A. B. Sossinsky, Geometries, 2012 63 Mar´ıa Cristina Pereyra and Lesley A. Ward, Harmonic Analysis, 2012 62 Rebecca Weber, Computability Theory, 2012 61 Anthony Bonato and Richard J. Nowakowski, The Game of Cops and Robbers on Graphs, 2011 60 Richard Evan Schwartz, Mostly Surfaces, 2011 SELECTED PUBLISHED TITLES IN THIS SERIES

59 Pavel Etingof, Oleg Golberg, Sebastian Hensel, Tiankai Liu, Alex Schwendner, Dmitry Vaintrob, and Elena Yudovina, Introduction to Representation Theory, 2011 58 Alvaro´ Lozano-Robledo, Elliptic Curves, Modular Forms, and Their L-functions, 2011 57 Charles M. Grinstead, William P. Peterson, and J. Laurie Snell, Probability Tales, 2011 56 Julia Garibaldi, Alex Iosevich, and Steven Senger, The Erd˝os Distance Problem, 2011 55 Gregory F. Lawler, Random Walk and the Heat Equation, 2010 54 Alex Kasman, Glimpses of Soliton Theory, 2010 53 Jiˇr´ıMatouˇsek, Thirty-three Miniatures, 2010 52 Yakov Pesin and Vaughn Climenhaga, Lectures on Fractal Geometry and Dynamical Systems, 2009 51 Richard S. Palais and Robert A. Palais, Differential Equations, Mechanics, and Computation, 2009 50 Mike Mesterton-Gibbons, A Primer on the Calculus of Variations and Optimal Control Theory, 2009 49 Francis Bonahon, Low-Dimensional Geometry, 2009 48 John Franks, A (Terse) Introduction to Lebesgue Integration, 2009 47 L.D.FaddeevandO.A.Yakubovski˘i, Lectures on Quantum Mechanics for Mathematics Students, 2009 46 Anatole Katok and Vaughn Climenhaga, Lectures on Surfaces, 2008 45 Harold M. Edwards, Higher Arithmetic, 2008 44 Yitzhak Katznelson and Yonatan R. Katznelson, A(Terse) Introduction to Linear Algebra, 2008 43 Ilka Agricola and Thomas Friedrich, Elementary Geometry, 2008 42 C. E. Silva, Invitation to Ergodic Theory, 2008 41 Gary L. Mullen and Carl Mummert, Finite Fields and Applications, 2007 40 Deguang Han, Keri Kornelson, David Larson, and Eric Weber, Frames for Undergraduates, 2007 39 Alex Iosevich, A View from the Top, 2007 38 B. Fristedt, N. Jain, and N. Krylov, Filtering and Prediction: A Primer, 2007 37 Svetlana Katok, p-adic Analysis Compared with Real, 2007 36 Mara D. Neusel, Invariant Theory, 2007 35 J¨org Bewersdorff, Galois Theory for Beginners, 2006 34 Bruce C. Berndt, Number Theory in the Spirit of Ramanujan, 2006 33 Rekha R. Thomas, Lectures in Geometric Combinatorics, 2006 32 Sheldon Katz, Enumerative Geometry and String Theory, 2006 SELECTED PUBLISHED TITLES IN THIS SERIES

31 John McCleary, A First Course in Topology, 2006 30 Serge Tabachnikov, Geometry and Billiards, 2005 29 Kristopher Tapp, Matrix Groups for Undergraduates, 2005 28 Emmanuel Lesigne, Heads or Tails, 2005 27 Reinhard Illner, C. Sean Bohun, Samantha McCollum, and Thea van Roode, Mathematical Modelling, 2005 This book introduces functional analysis to undergraduate mathematics students who possess a basic background in analysis and linear algebra. By studying how the Volterra operator acts on vector spaces of continuous functions, its readers will sharpen their skills, reinterpret what they already know, and learn fundamental Banach-space techniques — all in the pursuit of two celebrated results: the Titchmarsh Convolution Theorem and the Volterra Invariant Subspace Theorem. Exercises throughout the text enhance the material and facilitate interactive study.

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