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. For additional information and updates on this book, visit www.ams.org/bookpages/stml-85 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 Copying and reprinting. Individual readers of this publication, and nonprofit li- braries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for permission to reuse portions of AMS publication content are handled by the Copyright Clearance Center. For more information, please visit www.ams.org/ publications/pubpermissions. Send requests for translation rights and licensed reprints to reprint-permission @ams.org. c 2018 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 232221201918 For Marjorie 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.