Table Of Content

A Short Course on Spectral Theory William Arveson Springer 209 Graduate Texts in Mathematics Editorial Board S. Axler F.W. Gehring K.A. Ribet This page intentionally left blank William Arveson A Short Course on Spectral Theory WilliamArveson DepartmentofMathematics UniversityofCalifornia,Berkeley Berkeley,CA94720-0001 USA [email protected] EditorialBoard S.Axler F.W.Gehring K.A.Ribet MathematicsDepartment MathematicsDepartment MathematicsDepartment SanFranciscoState EastHall UniversityofCalifornia, University UniversityofMichigan Berkeley SanFrancisco,CA94132 AnnArbor,MI48109 Berkeley,CA94720-3840 USA USA USA MathematicsSubjectClassification(2000):46-01,46Hxx,46Lxx,47Axx,58C40 LibraryofCongressCataloging-in-PublicationData Arveson,William. Ashortcourseonspectraltheory/WilliamArveson. p.cm.—(Graduatetextsinmathematics;209) Includesbibliographicalreferencesandindex. ISBN0-387-95300-0(alk.paper) 1.Spectraltheory(Mathematics) I.Title. II.Series. QA320.A832001 515′.7222–dc21 2001032836 Printedonacid-freepaper. 2002Springer-VerlagNewYork,Inc. All rights reserved. This work may not be translated or copied in whole or in part without the writtenpermissionofthepublisher(Springer-VerlagNewYork,Inc.,175FifthAvenue,NewYork, NY10010,USA),exceptforbriefexcerptsinconnectionwithreviewsorscholarlyanalysis.Use inconnection withany formof informationstorageand retrieval,electronic adaptation,computer software,orbysimilarordissimilarmethodologynowknownorhereafterdevelopedisforbidden. Theuseofgeneraldescriptivenames,tradenames,trademarks,etc.,inthispublication,evenifthe formerarenotespeciallyidentified,isnottobetakenasasignthatsuchnames,asunderstoodby theTradeMarksandMerchandiseMarksAct,mayaccordinglybeusedfreelybyanyone. ProductionmanagedbyFrancineMcNeill;manufacturingsupervisedbyJacquiAshri. Photocomposedcopypreparedfromtheauthor’sAMSLaTeXfiles. PrintedandboundbyMaple-VailBookManufacturingGroup,York,PA. PrintedintheUnitedStatesofAmerica. 9 8 7 6 5 4 3 2 1 ISBN0-387-95300-0 SPIN10838691 Springer-Verlag NewYork Berlin Heidelberg AmemberofBertelsmannSpringerScience+BusinessMediaGmbH To Lee This page intentionally left blank Preface This book presents the basic tools of modern analysis within the context of what might be called the fundamental problem of operator theory: to cal- culatespectraofspecificoperatorsoninfinite-dimensionalspaces,especially operators on Hilbert spaces. The tools are diverse, and they provide the basisformorerefinedmethodsthatallowonetoapproachproblemsthatgo well beyond the computation of spectra; the mathematical foundations of quantum physics, noncommutative K-theory, and the classification of sim- ple C∗-algebras being three areas of current research activity that require mastery of the material presented here. The notion of spectrum of an operator is based on the more abstract notion of the spectrum of an element of a complex Banach algebra. Af- ter working out these fundamentals we turn to more concrete problems of computing spectra of operators of various types. For normal operators, this amounts to a treatment of the spectral theorem. Integral operators require the development of the Riesz theory of compact operators and the ideal L2 of Hilbert–Schmidt operators. Toeplitz operators require several important tools;inordertocalculatethespectraofToeplitzoperatorswithcontinuous symbol one needs to know the theory of Fredholm operators and index, the structureoftheToeplitzC∗-algebraanditsconnectionwiththetopologyof curves, and the index theorem for continuous symbols. I have given these lectures several times in a fifteen-week course at Berkeley (Mathematics 206), which is normally taken by first- or second- yeargraduatestudentswithafoundationinmeasuretheoryandelementary functional analysis. It is a pleasure to teach that course because many deep and important ideas emerge in natural ways. My lectures have evolved sig- nificantlyovertheyears,buthavealwaysfocusedonthenotionofspectrum and the role of Banach algebras as the appropriate modern foundation for such considerations. For a serious student of modern analysis, this material is the essential beginning. Berkeley, California William Arveson July 2001 vii This page intentionally left blank Contents Preface vii Chapter 1. Spectral Theory and Banach Algebras 1 1.1. Origins of Spectral Theory 1 1.2. The Spectrum of an Operator 5 1.3. Banach Algebras: Examples 7 1.4. The Regular Representation 11 1.5. The General Linear Group of A 14 1.6. Spectrum of an Element of a Banach Algebra 16 1.7. Spectral Radius 18 1.8. Ideals and Quotients 21 1.9. Commutative Banach Algebras 25 1.10. Examples: C(X) and the Wiener Algebra 27 1.11. Spectral Permanence Theorem 31 1.12. Brief on the Analytic Functional Calculus 33 Chapter 2. Operators on Hilbert Space 39 2.1. Operators and Their C∗-Algebras 39 2.2. Commutative C∗-Algebras 46 2.3. Continuous Functions of Normal Operators 50 2.4. The Spectral Theorem and Diagonalization 52 2.5. Representations of Banach ∗-Algebras 57 2.6. Borel Functions of Normal Operators 59 2.7. Spectral Measures 64 2.8. Compact Operators 68 2.9. Adjoining a Unit to a C∗-Algebra 75 2.10. Quotients of C∗-Algebras 78 Chapter 3. Asymptotics: Compact Perturbations and Fredholm Theory 83 3.1. The Calkin Algebra 83 3.2. Riesz Theory of Compact Operators 86 3.3. Fredholm Operators 92 3.4. The Fredholm Index 95 Chapter 4. Methods and Applications 101 4.1. Maximal Abelian von Neumann Algebras 102 ix