237 Graduate Texts in Mathematics Editorial Board S.Axler K.A.Ribet Graduate Texts in Mathematics 1 TAKEUTI/ZARING.Introduction to 34 SPITZER.Principles ofRandom Walk. Axiomatic Set Theory.2nd ed. 2nd ed. 2 OXTOBY.Measure and Category.2nd ed. 35 ALEXANDER/WERMER.Several Complex 3 SCHAEFER.Topological Vector Spaces. Variables and Banach Algebras.3rd ed. 2nd ed. 36 KELLEY/NAMIOKAet al.Linear 4 HILTON/STAMMBACH.A Course in Topological Spaces. Homological Algebra.2nd ed. 37 MONK.Mathematical Logic. 5 MACLANE.Categories for the Working 38 GRAUERT/FRITZSCHE.Several Complex Mathematician.2nd ed. Variables. 6 HUGHES/PIPER.Projective Planes. 39 ARVESON.An Invitation to C*-Algebras. 7 J.-P.SERRE.A Course in Arithmetic. 40 KEMENY/SNELL/KNAPP.Denumerable 8 TAKEUTI/ZARING.Axiomatic Set Theory. Markov Chains.2nd ed. 9 HUMPHREYS.Introduction to Lie 41 APOSTOL.Modular Functions and Algebras and Representation Theory. Dirichlet Series in Number Theory. 10 COHEN.A Course in Simple Homotopy 2nd ed. Theory. 42 J.-P.SERRE.Linear Representations of 11 CONWAY.Functions ofOne Complex Finite Groups. Variable I.2nd ed. 43 GILLMAN/JERISON.Rings ofContinuous 12 BEALS.Advanced Mathematical Analysis. Functions. 13 ANDERSON/FULLER.Rings and 44 KENDIG.Elementary Algebraic Geometry. Categories ofModules.2nd ed. 45 LOÈVE.Probability Theory I.4th ed. 14 GOLUBITSKY/GUILLEMIN.Stable 46 LOÈVE.Probability Theory II.4th ed. Mappings and Their Singularities. 47 MOISE.Geometric Topology in 15 BERBERIAN.Lectures in Functional Dimensions 2 and 3. Analysis and Operator Theory. 48 SACHS/WU.General Relativity for 16 WINTER.The Structure ofFields. Mathematicians. 17 ROSENBLATT.Random Processes.2nd ed. 49 GRUENBERG/WEIR.Linear Geometry. 18 HALMOS.Measure Theory. 2nd ed. 19 HALMOS.A Hilbert Space Problem 50 EDWARDS.Fermat’s Last Theorem. Book.2nd ed. 51 KLINGENBERG.A Course in Differential 20 HUSEMOLLER.Fibre Bundles.3rd ed. Geometry. 21 HUMPHREYS.Linear Algebraic Groups. 52 HARTSHORNE.Algebraic Geometry. 22 BARNES/MACK.An Algebraic 53 MANIN.A Course in Mathematical Logic. Introduction to Mathematical Logic. 54 GRAVER/WATKINS.Combinatorics with 23 GREUB.Linear Algebra.4th ed. Emphasis on the Theory ofGraphs. 24 HOLMES.Geometric Functional Analysis 55 BROWN/PEARCY.Introduction to and Its Applications. Operator Theory I:Elements of 25 HEWITT/STROMBERG.Real and Abstract Functional Analysis. Analysis. 56 MASSEY.Algebraic Topology:An 26 MANES.Algebraic Theories. Introduction. 27 KELLEY.General Topology. 57 CROWELL/FOX.Introduction to Knot 28 ZARISKI/SAMUEL.Commutative Algebra. Theory. Vol.I. 58 KOBLITZ.p-adic Numbers,p-adic 29 ZARISKI/SAMUEL.Commutative Algebra. Analysis,and Zeta-Functions.2nd ed. Vol.II. 59 LANG.Cyclotomic Fields. 30 JACOBSON.Lectures in Abstract Algebra 60 ARNOLD.Mathematical Methods in I.Basic Concepts. Classical Mechanics.2nd ed. 31 JACOBSON.Lectures in Abstract Algebra 61 WHITEHEAD.Elements ofHomotopy II.Linear Algebra. Theory. 32 JACOBSON.Lectures in Abstract Algebra 62 KARGAPOLOV/MERIZJAKOV. III.Theory ofFields and Galois Theory. Fundamentals ofthe Theory ofGroups. 33 HIRSCH.Differential Topology. 63 BOLLOBAS.Graph Theory. (continued after index) To Rub´en, Amanda, Miguel, Lorena, Sof´ıa, Andrea, Jos´e Miguel Carol, Alan, Jeffrey, Michael, Daniel, Esther, Jeremy, Aaron, Margaret Rubén A. Martínez-Avendan˜o Peter Rosenthal An Introduction to Operators on the Hardy-Hilbert Space Rubén A.Martínez-Avendan˜o Peter Rosenthal Centro de Investigacio´n en Department of Mathematics Matemáticas University of Toronto, Universidad Auto´noma del Estado Toronto, ON M5S 2E4 de Hidalgo, Pachuca, Hidalgo 42184 Canada Mexico [email protected] [email protected] Editorial Board: S.Axler K.A.Ribet Department of Mathematics Department of Mathematics San Francisco State University University of California,Berkeley San Francisco,CA 94132 Berkeley,CA 94720-3840 USA USA [email protected] [email protected] Mathematics Subject Classification (2000):46-xx,47-xx,30-xx Library ofCongress Control Number:2006933291 ISBN-10:0-387-35418-2 e-ISBN-10:0-387-48578-3 ISBN-13:978-0387-35418-7 e-ISBN-13:978-0387-48578-2 Printed on acid-free paper. © 2007 Springer Science+Business Media,LLC All rights reserved.This work may not be translated or copied in whole or in part without the written permission ofthe publisher (Springer Science+Business Media,LLC,233 Spring Street, New York,NY 10013,USA),except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation,computer software,or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication oftrade names,trademarks,service marks,and similar terms,even if they are not identified as such,is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. 9 8 7 6 5 4 3 2 1 springer.com Preface The great mathematician G.H. Hardy told us that “Beauty is the first test: there is no permanent place in the world for ugly mathematics” (see [24, p. 85]). It is clear why Hardy loved complex analysis: it is a very beautiful partofclassicalmathematics.ThetheoryofHilbertspacesandofoperatorson themisalmostasclassicalandisperhapsasbeautifulascomplexanalysis.The studyoftheHardy–Hilbertspace(aHilbertspacewhoseelementsareanalytic functions), and of operators on that space, combines these two subjects. The interplay produces a number of extraordinarily elegant results. For example, very elementary concepts from Hilbert space provide simple proofs of the Poisson integral (Theorem 1.1.21 below) and Cauchy integral (Theorem 1.1.19) formulas. The fundamental theorem about zeros of func- tions in the Hardy–Hilbert space (Corollary 2.4.10) is the central ingredient of a beautiful proof that every continuous function on [0,1] can be uniformly approximated by polynomials with prime exponents (Corollary 2.5.3). The Hardy–Hilbert space context is necessary to understand the structure of the invariantsubspacesoftheunilateralshift(Theorem2.2.12).Conversely,prop- erties of the unilateral shift operator are useful in obtaining results on fac- torizations of analytic functions (e.g., Theorem 2.3.4)and on other aspects of analytic functions (e.g., Theorem 2.3.3). The study of Toeplitz operators on the Hardy–Hilbert space is the most natural way of deriving many of the properties of classical Toeplitz matri- ces (e.g., Theorem 3.3.18), and the study of Hankel operators is the best approach to many results about Hankel matrices (e.g., Theorem 4.3.1). Com- viii Preface positionoperatorsareaninterestingwayoflookingattheclassicalconceptof subordination of analytic functions (Corollary 5.1.8). And so on; you’ll have to read this entire book (and all the references, and all the references in the references!) to see all the examples that could be listed. Most of the material discussed in this text was developed by mathemati- cians whose prime interest was pursuing mathematical beauty. It has turned out, however, as is often the case with pure mathematics, that there are nu- merous applications of these results, particularly to problems in engineering. Although we do not treat such applications, references are included in the bibliography. The Hardy–Hilbert space is the set of all analytic functions whose power serieshavesquare-summablecoefficients(Definition1.1.1).ThisHilbertspace of functions analytic on the disk is customarily denoted by H2. There are Hp spaces (called Hardy spaces, in honor of G.H. Hardy) for each p ≥ 1 (and even for p ∈ (0,1)). The only Hp space that is a Hilbert space is H2, the most-studied of the Hardy spaces. We suggest that it should be called the Hardy–Hilbert space. There are also other spaces of analytic functions, including the Bergman and Dirichlet spaces. There has been much study of all of these spaces and of various operators on them. Ourgoalistoprovideanelementaryintroductionthatwillbereadableby everyone who has understood first courses in complex analysis and in func- tional analysis. We feel that the best way to do this is to restrict attention to H2 and the operators on it, since that is the easiest setting in which to in- troduce the essentials of the subject.We have tried to make the exposition as clear,asself-contained,andasinstructiveaspossible,andtomaketheproofs sufficiently beautiful that they will have a permanent place in mathematics. A reader who masters the material we present will have acquired a firm foun- dation for the study of all spaces of analytic functions and all operators on such spaces. Thisbookaroseoutoflecturenotesfromgraduatecoursesthatweregiven attheUniversityofToronto.Itshouldprovesuitableasatextbookforcourses offered to beginning graduate students, or even to well-prepared advanced undergraduates. We also hope that it will be useful for independent study by studentsandbymathematicianswhowishtolearnanewfield.Moreover,the exposition should be accessible to students and researchers in those aspects of engineering that rely on this material. Preface ix It is our view that a course based on this text would appropriately con- tribute to the general knowledge of any graduate student in mathematics, whatever field the student might ultimately pursue. In addition, a thorough understandingofthismaterialwillprovidethenecessarybackgroundforthose whowishtopursueresearchonrelatedtopics.Thereareanumberofexcellent books, listed in the references, containing much more extensive treatments of some of the topics covered. There are also references to selected papers of interest. A brief guide to further study is given in the last chapter of this book. Themathematicspresentedinthisbookhasitsoriginsincomplexanalysis, thefoundationsofwhichwerelaidbyCauchyalmost200yearsago.Thestudy of the Hardy–Hilbert space began in the early part of the twentieth century. Hankel operators were first studied toward the end of the nineteenth century, the study of Toeplitz operators was begun early in the twentieth century, and composition operators per se were first investigated in the middle of the twentieth century. There is much current research on properties of Toeplitz, Hankel, composition, and related operators. Thus the material contained in this book was developed by many mathematicians over many decades, and still continues to be the subject of research. Somereferencestothedevelopmentofthissubjectaregiveninthe“Notes andRemarks”sectionsattheendofeachchapter.Wearegreatlyindebtedto the mathematicians cited in these sections and in the references at the end of thebook.Moreover,itshouldberecognizedthatmanyothermathematicians havecontributed ideas that havebecomeso intrinsic to the subject that their history is difficult to trace. Our approach to this material has been strongly influenced by the books of Ronald Douglas [16], Peter Duren [17], Paul Halmos [27], and Kenneth Hoffman [32], and by Donald Sarason’s lecture notes [49]. The main reason that we have written this book is to provide a gentler introduction to this subject than appears to be available elsewhere. We are grateful to a number of colleagues for useful comments on prelim- inary drafts of this book; our special thanks to Sheldon Axler, Paul Bartha, JaimeCruz-Sampedro,AbieFeintuch,OliviaGutu´,FedericoMen´endez-Conde, Eric Nordgren, Steve Power, Heydar Radjavi, Don Sarason, and Nina Zor- boska. Moreover, we would like to express our appreciation to Eric Nordgren for pointing out several quite subtle errors in previous drafts. We also thank x Preface Sheldon Axler, Kenneth Ribet, and Mark Spencer for their friendly and en- couraging editorial support, and Joel Chan for his TEXnical assistance. It is very rare that a mathematics book is completely free of errors. We wouldbegratefulifreaderswhonoticemistakesorhaveconstructivecriticism would notify us by writing to one of the e-mail addresses given below. We anticipate posting a list of errata on the website http://www.math.toronto.edu/rosent Rub´en A. Mart´ınez-Avendan˜o Centro de Investigaci´on en Matem´aticas Universidad Auto´noma del Estado de Hidalgo Pachuca, Mexico [email protected] Peter Rosenthal Department of Mathematics University of Toronto Toronto, Canada [email protected] August 2006 Contents 1 Introduction............................................... 1 1.1 The Hardy–Hilbert Space................................. 1 1.2 Some Facts from Functional Analysis ...................... 21 1.3 Exercises ............................................... 31 1.4 Notes and Remarks...................................... 34 2 The Unilateral Shift and Factorization of Functions........ 37 2.1 The Shift Operators ..................................... 37 2.2 Invariant and Reducing Subspaces ......................... 43 2.3 Inner and Outer Functions................................ 49 2.4 Blaschke Products....................................... 51 2.5 The Mu¨ntz–Sza´sz Theorem ............................... 62 2.6 Singular Inner Functions ................................. 66 2.7 Outer Functions......................................... 80 2.8 Exercises ............................................... 91 2.9 Notes and Remarks...................................... 94 3 Toeplitz Operators ........................................ 95 3.1 Toeplitz Matrices........................................ 95 3.2 Basic Properties of Toeplitz Operators ..................... 98 3.3 Spectral Structure .......................................108 3.4 Exercises ...............................................119 3.5 Notes and Remarks......................................121 4 Hankel Operators..........................................123 4.1 Bounded Hankel Operators ...............................123