C* - Algebras and Numerical Analysis

C*-ALGEBRAS AND NUMERICALANALYSIS

PURE

AND APPLIED

MATHEMATICS

A Program of Monographs, Textbooks, and Lecture Notes

EXECUTIVE EDITORS Zuhair Nashed University of Delaware Newark, Delaware

EarlJ. Taft Rutgers University New Brunswick, New Jersey

EDITORIAL M. S. Baouendi University of California, San Diego Jane Cronin Rutgers University

Georgia Institute

Jack K. Hale of Technology

BOARD Anil Nerode Cornell University Donald Passman University of Wisconsin, Madison Fred S. Roberts Rutgers University

S. Kobayashi University of California, Berkeley

David L. Russell Virginia Polytechnic Institute and State University

Marvin Marcus University of California, Santa Barbara

Walter Schempp Universitgit Siegen

W. S. Masse)) Yale University

Mark Teply University of Wisconsin, Milwaukee

MONOGRAPHS AND TEXTBOOKS IN PURE AND APPLIED MATHEMATICS 1. K. Yano,Integral Formulas in Riemannian Geometry (1970) 2. S. Kobayashi,HyperbolicManifoldsandHolomorphic Mappings (1970) of Mathematical Physics(A. Jeffrey, ed.; A. Littlewood, 3. V. S. Vladimirov,Equations trans.) (1970) 4. B. N. Pshem’chnyi, Necessary Conditionsfor an Extremum (L. Neustadt,translation ed.; K. Makowski, trans.) (1971) 5. LoNarici et al., Functional AnalysisandValuation Theory(1971) Infinite GroupRings(1971) 6. S.S. Passman, 7. L. Domhoff,GroupRepresentation Theory.Part A: OrdinaryRepresentation Theory. Part B: ModularRepresentation Theory(1971,1972) W. Boothby and G. L. Weiss, eds., Symme~c Spaces (1972) 8. 9. Y. Matsushima, DifferentiableManifolds (E. T. Kobayashi, trans.) (1972) 10. L. E. Ward,Jr., Topology (1972) 11. A. Babakhanian, Cohomological Methods in GroupTheory(1972) t2. R. Gilmer,Multiplicative Ideal Theory(1972) 13. J. Yeh,StochasticProcesses andthe WienerIntegral (1973) 14. J. Barros-Neto, Introductionto the Theoryof Distributions(1973) 15. R. Larsen,FunctionalAnalysis(1973) 16. K. YanoandS. Ishihara, TangentandCotangent Bundles(1973) 17. C. Procesi,Ringswith Polynomial Identities (1973) 18. R. HetTnann,Geometry, Physics, andSystems (1973) 19. N.R. Wallach, HarmonicAnalysis on Homogeneous Spaces(1973) 20. J. Dieudonnd, Introductionto the Theoryof FormalGroups (1973) 21. I. Vaisman, Cohomology andDifferential Forms(1973) 22. B.-Y. Chert, Geometry of Submanifolds (1973) 23. M.Marcus,Finite Dimensional MultilinearAlgebra(in twoparts) (1973,1975) 24. R. Larsen,Banach Algebras(1973) 25. R. O. KujalaandA. L. Vitter, eds., ValueDistributionTheory:Part A; Part B: Deficit andBezoutEstimatesby WilhelmStoll (1973) 26. K.B. Stolarsky, AlgebraicNumbers andDiophantineApproximation (1974) Rings(1974) 27. A.R. Magid,TheSeparableGalois Theoryof Commutative Finite Ringswith Identity (1974) 28. B.R. McDonald, 29. J. Satake,LinearAlgebra(S. Kohet al., trans.) (1975) Rings(1975) 30. J.S. Go/an,Localization of Noncommutative 31. G. Klambauer, Mathematical Analysis(1975) 32. M. K, Agoston,AlgebraicTopology(1976) 33. K.R. Goodear/,Ring Theory(1976) 34. L.E. Mansfield,LinearAlgebrawith Geometric Applications(1976) 35. N.J. Pullman,MatrixTheoryandIts Applications(1976) 36. B.R. McDonald, GeometricAlgebraOverLocal Rings(1976) 37. C. W.Groetsch,Generalized Inversesof LinearOperators(1977) andJ, L. Gersting,AbstractAlgebra(1977) 38. J. E. Kuczkowski 39. C. O. Chdstenson andW.L. Voxman, Aspectsof Topology(1977) 40. M. Nagata,Field Theory(1977) 41. R.L. Long,Algebraic Number Theory(1977) 42. W.F, Pfeffer, Integrals andMeasures (1977) 43. R.L. Wheeden andA. Zygmund, MeasureandIntegral (1977) of a Complex Variable(1978) 44. J.H. Curtiss, Introductionto Functions 45. K. Hrbacek andT. Jech,Introductionto Set Theory(1978) 46. W.S. Massey,Homology andCohomology Theory(1978) 47. M. Marcus,Introduction to Modem Algebra(1978) 48. E.C. Young,VectorandTensorAnalysis(1978) 49. S.B.Nadler,Jr., Hyperspaces of Sets(1978) 50. S.K. Segal,Topicsin GroupKings(1978) 51. A. C. M. van Rooij, Non-Archimedean FunctionalAnalysis(1978) 52. L. CorwinandR. Szczarba,Calculusin VectorSpaces(1979) 53. C. Sadosky, Interpolationof Operators andSingularIntegrals(1979) 54. J. Cronin,DifferentialEquations (1980) 55. C. W.Groetsch,Elements of ApplicableFunctionalAnalysis(1980)

56. 57. 58. 59. 60. 61. 62.

L Vaisman,Foundations of Three-Dimensional EuclideanGeometry (1980) H, I. Freedan,DeterministicMathematical Modelsin PopulationEcology(1980) S.B. Chae,Lebesgue Integration (1980) C.S.Reeset al., TheoryandApplicationsof Fouder Analysis(1981) L. Nachbin, Introductionto FunctionalAnalysis(R. M.Aron,trans.) (1981) G. OrzechandM. Ot-zech,PlaneAlgebraicCurves(1981) R. Johnsonbaugh and W.E. Pfaffenberger, Foundationsof MathematicalAnalysis (1981) 63. W.L. Voxman andR.H. Goetschel,AdvancedCalculus (1981) 64. L. J. CorwinandR. H. Szczarba,MultivariableCalculus(1982) 65. V.I. Istr~tescu,Introductionto LinearOperatorTheory(1981) 66. R.D.J~rvinen,Finite andInfinite Dimensional LinearSpaces (1981) 67. J. K. Beem andP. E. Ehrlich, GlobalLorenlzianGeomet~ (1981) 68. D.L. Armacost,TheStructure of Locally Compact AbelianGroups(1981) 69. J. W.BrewerandM. K. Smith, eds,, Emmy Noether:A Tribute (1981) 70. K.H. K/m,BooleanMatrix TheoryandApplications(1982) 71. T. W. Wieting, TheMathematicalTheoryof ChromaticPlaneOmaments (1982) 72. D. B.Gau/d,Differential Topology (1982) 73. R. L. Faber,Foundations of EuclideanandNon-Euclidean Geometry (1983) 74. M. Carmeli,Statistical TheoryandRandom Matrices(1983) 75. J.H. Canutheta/., TheTheoryof TopologicalSemigroups (1983) 76. R.L. Faber,Differential Geometq/and Relativity Theory(1983) 77. S. Bamett,PolynomialsandLinear ControlSystems (1983) 78. G. Karpilovsky, Commutative GroupAlgebras(1983) 79. F. VanOystaeyen andA.Verschoren,Relative Invadantsof Rings(1983) 80. /. Vaisman, A First Course in Differential Geometry (1964) 81. G. W.Swan,Applicationsof OptimalControlTheoryin Biomedicine (1964) 82. T. PetdeandJ.D. Randa/I,Transformation Groupson Manifolds(1984) andNonexpansive 83. K. GoebelandS. Reich, UniformConvexity,HyperbolicGeomet~, Mappings(1964) RelativeFinitenessin ModuleTheory(1984) 84. T. AlbuandC. N~st~sescu, 85. K. Hrbacek andT. Jech,Introductionto Set Theory:Second Edition (1964) andA.Verschoren, Relative Invariants of Rings(1964) 86. F. VanOystaeyen 87. B.R. McDonald,LinearAIgebraOverCommutative Rings(1964) Geometry of Projective AlgebraicCurves(1984) 88. M. Namba, 89. G.F. Webb,Theoryof NonlinearAge-Dependent PopulationDynamics (1985) et al., Tablesof Dominant WeightMultiplicities for Representations of 90. M. R. Bremner SimpleLie Algebras(1985) 91. A. E. Fekete,RealLinearAlgebra(1985) andCalculus in Normed Spaces(1985) 92. S.B. Chae,Holomorphy 93. A.J. Je~,Introductionto Integral Equations with Applications(1985) 94. G. Karpi/ovsky,ProjectiveRepresentations of Finite Groups (1985) 95. L. Nar~ciandE. Beckenstein, TopologicalVectorSpaces(1985) 96. J. Weeks,The Shapeof Space(1985) of OperationsResearch (1985) 97. P.R. GdbikandK. O. Kortanek,ExtremalMethods 98. J.-A. Chaoand W.A. Woyczynski,eds., Probability TheoryandHarmonicAnalysis (1986) 99. G.D.Crowneta/., AbstractAlgebra(1986) 100. J.H. Carruthet al., TheTheoryof TopologicalSemigroups, Volume 2 (1986) 101. R. S. DoranandV. A. Belfi, Characterizations of C*-Algebras (1986) 102. M. W. Jeter, Mathematical Programming (1986) 103. M. Airman,A Unified Theoryof Nonlinear Operatorand Evolution Equationswith Applications(1986) 104. A. Verschoren, Relative Invadantsof Sheaves (1987) 105. R.A. Usmani,AppliedLinear Algebra(1987) andDifferential Equations in Characteristicp > 106. P. B/assandJ. Lang,Zariski Surfaces 0 (1987) 107. J.A. Reneke et al., StructuredHereditarySystems (1987) and B. B. Phadke,Spaceswith DistinguishedGeodesics (1987) 108. H. Busemann LinearOperators (1988) 109. R. Harte,Invertibility andSingularityfor Bounded 110. G. S. Laddeet al., Oscillation Theoryof Differential Equationswith DeviatingArguments(1987) 111. L. Dudkinet al., Iterative Aggregation Theory (1987) (1987) 112. T. Okubo,Differential Geometry

113.D. L. StandandM. L. Stancl, RealAnalysiswith Point-SetTopology (1987) 114.T. C. Gard,Introductionto Stochastic Differential Equations (1988) 115. S. S, Abhyankar,Enumerative Combinatodcs of YoungTableaux(1988) 116. H. StradeandR. Famsteiner,ModularUeAlgebrasandTheir Representations (1988) 117. J.A. Huckaba,Commutative Ringswith Zero Divisors (1988) 118. W.D.Wallis, CombinatorialDesigns(1988) 119. W.Wi~slaw,TopologicalFields (1988) 120. G. Karpilovsky,Field Theory(1988) 121. S. Caenepeel andF. VanOystaeyen,BrauerGroupsand the Cohomology of Graded Rings(1989) 122. W.Kozlowski,ModularFunctionSpaces(1988) 123. E. Lowen-Colebunders, FunctionClassesof Cauchy Continu6usMaps(1989) of PattemRecognition(1989) 124. M. Pavel,Fundamentals 125. V. Lakshmikantham et al., Stability Analysisof NonlinearSystems (1989) 126. R. Sivaramakdshnan, TheClassicalTheoryof Arithmetic Functions(1989) 127.N. A, Watson, ParabolicEquations onan Infinite Strip (1989) 128. K.J. Hastings,Introductionto the Mathematics of Operations Research (1989) 129. B. Fine, AlgebraicTheoryof the BianchiGroups (1989) 130. D. N. Dikranjanet aL, TopologicalGroups (1989) 131. J. C. Morgan II, Point Set Theory(1990) 132. P. BilerandA.Witkowski,Problems in Mathematical Analysis(1990) 133. H.J. Sussmann, NonlinearControllability andOptimalControl(1990) 134.J.-P. Florenset al., Elements of Bayesian Statistics (1990) 135. N. Shell, TopologicalFieldsandNearValuations(1990) 136. B. F. Doolin andC. F. Martin, Introduction to Differential Geometry for Engineers (1990) 137. S.S. Holland,Jr., AppliedAnalysisby the Hilbert Space Method (1990) 138. J. Okninski,Semigroup Algebras(1990) 139. K. Zhu,OperatorTheoryin FunctionSpaces(1990) 140. G.B.Pdce,AnIntroduction to Multicomplex SpacesandFunctions(1991) 141. R.B. Darst, Introductionto LinearProgramming (1991) 142.P.L. Sachdev, NonlinearOrdinaryDifferential Equations andTheir Applications(1991) 143. T. Husain,OrthogonalSchauder Bases(1991) 144. J. Foran,Fundamentals of RealAnalysis(1991) 145. W.C.Brown,Matdcesand Vector Spaces(1991) 146. M. M. RaoandZ. D. Ren,Theoryof OdiczSpaces(1991) 147. J.S. GolanandT, Head,Modules andthe Structuresof Rings(1991) 148.C. Small,Arithmeticof Finite Fields(1991) 149. K. Yang,Complex Algebraic Geometry (1991) 150. D. G. Hoffman et al., CodingTheory(1991) 151. M. O. Gonz~lez, Classical Complex Analysis (1992) 152. M. O. Gonzdlez,Complex Analysis (1992) 153. L. W.Baggett,FunctionalAnalysis(1992) 154. M. Sniedovich, DynamicProgramming (1992) 155. R. P. Agarwal,DifferenceEquations andInequalities (1992) 156.C. Brezinski,Biorthogonality andIts Applicationsto Numerical Analysis(1992) 157. C. Swartz,AnIntroductionto FunctionalAnalysis(1992) 158. S.B. Nadler,Jr., Continuum Theory(1992) 159. M.A.AI-Gwaiz,Theoryof Distributions (1992) 160. E. Perry, Geometry: Podomatic Developments with Problem Solving(1992 161. E. Castillo andM. R. Ruiz-Cobo, FunctionalEquationsandModellingin Scienceand Engineering(1992) 162. A. J. Jerd, Integral andDiscrete Transforms with ApplicationsandError Analysis (1992) 163.A. CharlieretaL, Tensors andthe Clifford Algebra(1992) 164.P. BilerandT. Nadzieja,Problems andExamples in Differential Equations(1992) 165.E. Hansen, GlobalOptimizationUsingInterval Analysis(1992) 166. S. Guerre-Delabfi~re,Classical Sequences in Banach Spaces(1992) 167. Y.C. Wong,Introductory Theoryof TopologicalVectorSpaces (1992) 168. S.H. KulkamiandB. V. Limaye,Real Function Algebras(1992) 169. W.C. Brown,MatdcesOverCommutative Rings(1993) 170. J. LoustauandM. Dillon, Linear Geometry with Computer Graphics(1993) 171. W.V. Petryshyn,Approximation-Solvability of NonlinearFunctionalandDifferential Equations(1993)

172. E. C. Young,VectorandTensorAnalysis:Second Edition (1993) 173. T.A. Bick, ElementaryBoundary ValueProblems(1993) 174. M. PaveI, Fundamentals of PattemRecognition:Second Edition (1993) 175. S. A. Albevedo et aL, Noncommutative Distributions (1993) 176. W.Fulks, Complex Variables(1993) 177. M.M.Rao,ConditionalMeasures andApplications (1993) 178. A. Janicki andA. Weron,SimulationandChaotic Behaviorof s-Stable Stochastic Processes(1994) 179. P. Neittaanm~ki andD. ~ba,OptimalControlof NonlinearParabolicSystems (1994) Edition 180. J. Cronin,Differential Equations:IntroductionandQualitativeTheory,Second (1994) 181. S. Heikkil~ andV. Lakshmikantham, Monotone Iterative Techniques for Discontinuous Nonlinear Differential Equations (1994) 182. X. Mao,Exponential Stability of Stochastic Differential Equations (1994) 183. B.S. Thomson, Symmetric Propertiesof RealFunctions(1994) 184. J.E. Rubio,OptimizationandNonstandard Analysis(1994) 185. J.L. Bueso et al., Compatibility,Stability, andSheaves (1995) 186. A. N. MichelandK. Wang,Qualitative Theoryof Dynamical Systems (1995) (1995) 187. M.R.Dame/,Theoryof I.attica-Ordered Groups 188. Z. NaniewiczandP. D. Panagiotopoulos,MathematicalTheoryof Hemivadational InequalitiesandApplications(1995) 189. L.J. CorwinandR. H. Szczarba,Calculusin VectorSpaces:Second Edition (1995) for Functional Differential Equations (1995) 190. L.H.Erbeet al., OscillationTheory 191. S. Agaianet al., BinaryPolynomial Transforms andNonlinearDigital Filters (1995) 192. M.I. Gil’, NormEstimationsfor Operation-Valued FunctionsandApplications(1995) 193. P.A. Gdllet, Semigroups: AnIntroductionto the StructureTheory(1995) 194. S. Kichenassamy, NonlinearWaveEquations(1996) 195. V.F. Krotov, GlobalMethods in OptimalControlTheory(1996) Identities (1996) 196. K.I. Beidaret al., RingswithGeneralized 197. V. I. Amautov et al., Introduction to the Theoryof TopologicalRingsandModules (1996) 198. G. Sierksma,Linear andInteger Programming (1996) 199. R. Lasser,Introductionto FourierSedes (1996) 200. V. Sima,Algorithms for Linear-Quadratic Optimization(1996) 201. D. Redmond, NumberTheory(1996) 202. J.K. Beem et al., GlobalLorentzianGeometry: Second Edition (1996) 203. M. Fontanaet al., Pr0fer Domains (1997) 204. H. Tanabe, FunctionalAnalyticMethods for Partial Differential Equations (1997) 205. C. Q. Zhang,Integer FlowsandCycleCoversof Graphs (1997) 206. E. SpiegelandC. J. O’Donnell,Inddence Algebras(1997) 207. B. JakubczykandW. Respondek, Geometry of Feedback and OptimalControl (1998) et al., Fundamentals of Domination in Graphs (1998) 208. T. W.Haynes eta/., Domination in Graphs:Advanced Topics(1998) 209. T. W.Haynes 210. L. A. D’Alotto et al., A Unified SignalAlgebraApproach to Two-Dimensional Parallel Digital SignalProcessing (1998) 211. F. Halter-Koch,Ideal Systems (1998) 212. N.K. Govilet al., Approximation Theory(1998) 213. R. Cross,MultivaluedLinearOperators(1998) 214. A. A. Martynyuk,Stability by Liapunov’sMatrix FunctionMethodwith Applications (1998) 215. A. Favini andA. Yagi, Degenerate Differential Equationsin Banach Spaces (1999) 216. A. I/lanes andS. Nadler, Jr., Hyperspaces:Fundamentals and RecentAdvances (1999) 217. G. KatoandD.Struppa,Fundamentals of AlgebraicMicrolocalAnalysis(1999) 218. G.X.-Z.Yuan,KKM TheoryandApplicationsin NonlinearAnalysis(1999) 219. D. MotreanuandN. H. Pave/, Tangency,FlowInvadancefor Differential Equations, andOptimizationProblems(1999) 220. K. Hrbacek andT. Jech,Introductionto Set Theory,Third Edition (1999) 221. G.E. Kolosov,OptimalDesignof Control Systems(1999) 222. N. L. Johnson,SubplaneCoveredNets(2000) 223. B. Fine andG. Rosenberger, AlgebraicGeneralizations of DiscreteGroups (1999) 224. M.V~th,Volterra andIntegral Equations of VectorFunctions(2000) 225. S. S. Miller andP. T. Mocanu, Differential Subordinations (2000)

226. R. Li et aL, Generalized DifferenceMethods for Differential Equations:Numerical Analysisof Finite Volume Methods (2000) 227. H. Li andF. VanOystaeyen, A Pdmer of AlgebraicGeometry (2000) 228. R. P. Agarwal,DifferenceEquationsandInequalities: Theory,Methods,andApplications, Second Edition (2000) 229. A.B.KharaTJshvi/i, StrangeFunctionsin RealAnalysis(2000) 230. J.M.Appellet al., Partial Integral.Operators andIntegro-DifferentialEquations (2000) 231. A. I. Pdlepkoe! al., Methods for SolvingInverse Problems in Mathematical Physics (2O00) AlgebraicGeometry for AssociativeAlgebras(2000) 232. F. VanOystaeyen, 233. D. L. Jagennan, DifferenceEquations with Applicationsto Queues (2000) 234. D. R, Hankerson, D. G. Hoffman,D. A. Leonard,C.C. Lindner, K.T. Phelps,C. A. Rodger, J. R. Wall Coding Theoryand Cryptography: The Essentials, Second Edition, RevisedandExpanded (2000) 235. S. D~sc~lescu et al. HopfAlgebras:AnIntroduction(2001) 236. R. Hagen et al. C*-Algebras andNumericalAnalysis(2001) 237. Y. Talpaert, Differential Geometry:With Applications to Mechanics and Physics (2001") Additional Volumes in Preparation

C*-ALGEBRAS AND NUMERICALANALYSIS Roland Hagen Freies Gymnasium Penig Penig, Germany

Steffen Roch TechnicalUniversity of Darmstadt Darmstadt, Germany

BerndSilbermann TechnicalUniversity of Chemnitz Chemnitz, Germany

MARCEL DEKKER, INC.

NEW YORK- BASEL

To the memoryof Siegfried PrSBdorf (1939 - 1998)

ISBN: 0-8247-0460-6 This bookis printed on acid-free paper. Headquarters Marcel Dekker,Inc. 270 Madison Avenue, NewYork, NY10016 tel: 212-696-9000;fax: 212-685-4540 Eastern HemisphereDistribution Marcel Dekker AG Hutgasse 4, Postfach 812, CH-4001Basel, Switzerland tel: 41-61-261-8482;fax: 41-61-261-8896 World Wide Web http://www.dekker.com Thepublisher offers discounts on this bookwhenordered in bulk quantities. For moreinformation, write to Special Sales/Professional Marketingat the headquartersaddress above. Copyright© 2001 by Marcel Dekker, Inc. All Rights Reserved. Neither this book nor any part maybe reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying,microfilming, and recording, or by any information storage and retrieval system, without permission in writing from the publisher. Currentprinting (last digit): 109 8 765 4 3 2 l PRINTED IN THE UNITED STATES OF AMERICA

Preface This book is devoted to C*-algebras as a tool in numerical analysis. Some readers might consider the use of C*-algebras to study properties of approximation methods as unusual and exotic. Wewould like to encourage them to read and see for themselves the powerof such techniques both for the investigation of very concrete discretization proceduresand for establishing the theoretical foundation of numerical analysis. For a general overview of the fruitful interplay between C*-techniques, concrete operator theory, and numericalanalysis and, thus, of the contents of this book, we refer the reader to the Introduction. The book is adressed to a wide audience. Wehope that it proves to be of use both for the student whowants to see applications of functional analysis and to learn numerical analysis, and for the mathematician and the engineer interested in theoretical aspects of numericalanalysis. Wewish to express our sincere appreciation to our friends and colleagues, Albrecht Btittcher, Torsten Ehrhardt, Peter Junghanns, and MarkoLindner, who read the bulk of the manuscript very carefully and not only made many corrections but also offered constructive criticism to improve the book substantially. Our students, Michael Ehrenberger and Florian Meyer, did an excellent job in drawingthe figures and performingthe test calculations. Oneof the authors (S. R.) was supported by a DFGHeisenberg grant while working the manuscript. He is grateful to the GermanResearch Foundation for this support, as well as to Bernd Kirstein and WolfgangWendlandand their staffs for their hospitality during that time. Finally, we are pleased to express our gratitude to the publisher, Marcel Dekker, Inc., and to the mathematicsseries editor, Prof. Zuhair Nashed,for inviting us to write this monograph and for their careful work on the book. Roland Hagen Steffen Roch Bernd Silbermann

Contents Preface

3

0

Introduction analysis ....................... 0.1 Numerical 0.2 Operator chemistry ....................... 0.3 The algebraic language of numerical analysis ........ 0.4 Microscoping .......................... 0.5 A few remarks on economy .................. of the contents ................ 0.6 Brief description

11 11 14 15 18 21 22

1

The algebraic language of numerical analysis 1.1 Approximation methods .................... 1.1.1 Basic definitions .................... 1.1.2 Projection methods .................. 1.1.3 Finite section method ................. 1.2 Banach algebras and stability ................. 1.2.1 Algebras, ideals and homomorphisms ......... 1.2.2 Algebraization of stability ............... Small perturbations .................. 1.2.3 1.2.4 Compact perturbations ................ 1.3 Finite sections of Toeplitz operators with continuous generating function .......................... 1.3.1 Laurent, Toeplitz and Hankel operators ....... 1.3.2 Invertibility and Fredholmnessof Toeplitz operators 1.3.3 The finite section method ............... C*-algebras of approximation sequences ........... 1.4 1.4.1 C*-algebras, their ideals and homomorphisms.... 1.4.2 The Toeplitz C*-algebra and the C*-algebra of the finite section method for Toeplitz operators .....

25 25 26 28 31 34 35 36 39 39 45 45 48 49 52 53 56

CONTENTS Stability of sequences in the C*-algebra of the finite section method for Toeplitz operators ........ 1.4.4 Symbolof the finite section method for Toeplitz operators ........................ 1.5 Asymptotic behaviour of condition numbers ......... of an operator ............. 1.5.1 The condition of norms ................. 1.5.2 Convergence 1.5.3 Condition numbersof finite sections of Toeplitz operators ........................ 1.6 Fractality of approximation methods ............. 1.6.1 Fractal homomorphisms,fractal, algebras, fractal sequences ........................ 1.6.2 Fractal algebras, and convergence of norms ..... Notes and references ...................... 1.4.3

60 61 62 63 64 65 66 67 71 73

Regularization of approximation methods 75 76 2.1 Stably regularizable sequences ................. 2.1.1 Moore-Penrose inverses and regularizations of matrices ......................... 76 2.1.2 Moore-Penrose inverses and regularization of operators ........................ 80 85 2.1.3 Stably regularizable approximation sequences .... 2.2 Algebraic characterization of stably regularizable sequences 89 89 2.2.1 Moore-Penrose invertibility in C*-algebras ...... 2.2.2 Stable regularizability, and Moore-Penroseinvertibility in ~/G ........................ 92 2.2.3 Finite sections of Toeplitz operators and their stable regularizability ..................... 97 100 2.2.4 Convergence of generalized condition numbers .... with Moore-Penrose stability ....... 103 2.2.5 Difficulties 104 Notes and references ...................... Approximation of spectra 105 105 3.1 Set sequences .......................... 3.1.1 Limiting sets of set functions ............. 106 3.1.2 Coincidence of the partial and uniform limiting set . 108 3.2 Spectra and their limiting sets ................ 110 3.2.1 Limiting sets of spectra of norm convergent sequences ........................ 112 3.2.2 Limiting sets of spectra: the general case ...... 114 sequences ............. 3.2.3 The case of fractal 117 3.2.4 Limiting sets of singular values ............ 119

CONTENTS

4

7

3.3 Pseudospectra and their limiting sets ............. ...................... 3.3.1 e-invertibility 3.3.2 Limiting sets of pseudospectra ............ sequences ............. 3.3.3 The case of fractal 3.3.4 Pseudospectra of operator polynomials ........ 3.4 Numerical ranges and their limiting sets ........... 3.4.1 Spatial and algebraic numerical ranges ........ 3.4.2 Limiting sets of numerical ranges ........... 3.4.3 The case offractal sequences ............. Notes and references ......................

119 119 125 127 128 134 134 136 140 143

Stability 4.1 Local 4.1.1 4.1.2 4.1.3

145 146 146 149

4.2

4.3

4.4

4.5

4.6

analysis for concrete approximation methods principles ......................... Commutative C*-algebras ............... The local principle by Allan and Douglas ...... Fredholmness of Toeplitz operators with piecewise continuous generating function ............ Finite sections of Toeplitz operators generated by a piecewise continuous function ....................... 4.2.1 The lifting theorem ................... 4.2.2 Application of the local principle ........... 4.2.3 Galerkin methods with spline ansatz for singular integral equations ..................... Finite sections of Toeplitz operators generated by a quasicontinuous function ....................... 4.3.1 Quasicontinuous functions ............... 4.3.2 Stability of the finite section method ......... 4.3.3 Someother classes of oscillating functions ...... Polynomial collocation methods for singular integral operators with piecewise continuous coefficients .......... 4.4.1 Singular integral operators .............. 4.4.2 Stability of the polynomial collocation method . . . 4.4.3 Collocation versus Galerkin methods ......... Paired circulants and spline approximation methods .... 4.5.1 Circulants and paired circulants ........... theorem ................. 4.5.2 The stability Finite sections of band-dominated operators ......... 4.6.1 Multidimensional band dominated operators .... 4.6.2 Fredholmness of band dominated operators ..... 4.6.3 Finite sections of band dominated operators ..... Notes and references ......................

151 158 158 163 167 169 169 173 175 177 178 183 187 188 190 191 197 197 198 200 204

CONTENTS Representation theory 207 5.1 Representations ......................... 208 5.1.1 The spectrum of a C*-algebra .............. 208 ideals ..................... 5.1.2 Primitive 210 5.1.3 The spectrum of an ideal and of a quotient ..... 212 5.1.4 Representations of some concrete algebras ...... 213 5.2 Postliminal algebras ...................... 222 algebras ........... 5.2.1 Liminal and postliminal 223 5.2.2 Dual algebras ...................... 226 5.2.3 Finite sections of Wiener-Hopfoperators with almost periodic generating function .............. 230 theory ......... 5.3 Lifting theorems and representation 238 5.3.1 Lifting one ideal .................... 238 5.3.2 The lifting theorem ................... 239 5.3.3 Sufficient families of homomorphisms ......... 243 Structure of fractal lifting homomorphisms ..... 249 5.3.4 Notes and references ....................... 254 6

Fredholm sequences 255 6.1 Fredholm sequences in standard algebras ........... 256 6.1.1 The standard model .................. 256 sequences ................. 6.1.2 Fredholm . 258 6.1.3 Fredholm sequences and stable regularizability . . . 259 6.1.4 Fredholm sequences and Moore-Penrose stability . . 260 6.2 Fredholm sequences and the asymptotic behavior of singular values .............................. 264 265 6.2.1 The main result .................... 6.2.2 A distinguished element and its range dimension . . 266 of dimImII,~ ............. 269 6.2.3 Upper estimate 270 6.2.4 Lower estimate of dimImII~ ............. 6.2.5 Some examples ..................... 276 Fredholm theory .................. 6.3 A general 282 6.3.1 Centrally compact and Fredholm sequences ..... 282 6.3.2 Fredholmness modulo compact elements ....... 288 6.3.3 Fredholm sequences in standard algebras ...... 297 6.4 Weakly Fredholm sequences .................. 305 6.4.1 Sequences with finite splitting property . ...... 305 6.4.2 Properties of weakly Fredholm sequences ...... 305 6.4.3 Strong limits of weakly Fredholm sequences ..... 307 6.4.4 Weakly Fredholm sequences of matrices ....... 313 ....................... 6.5 Some applications 314 6.5.1 Numerical determination of the kernel dimension... 314

CONTENTS 6.5.2 Around the finite section method for Toeplitz operators ........................ 6.5.3 Discretization of shift operators ............ Notes and references ......................

9

315 317 322

Self-adjoint approximation sequences 323 The spectrum of a self-adjoint approximation sequence . . . 323 7.1 7.1.1 Essential and transient points ............. 323 Fractality of self-adjoint sequences .......... 327 7.1.2 7.1.3 Arveson dichotomy: band operators ......... 333 7.1.4 Arveson dichotomy: standard algebras ........ 338 7.2 SzegS-type theorems ...................... 339 7.2.1 F¢lner and Szeg5 algebras ............... 340 7.2.2 SzegS’s theorem revisited ............... 346 7.2.3 A further generalization of SzegS’s theorem ..... 348 7.2.4 Algebras with unique tracial state .......... 352 Notes and references ...................... 354 Bibliography

357

Index

373

Chapter 0

Introduction 0.1

Numerical

analysis

The goal of functional analysis is to solve equations with infinitely many variables, and that of linear algebra to solve equations in finitely many variables. Numerical analy§is builds a bridge between these fields. The subject of numerical analysis - as we understand it - is the investigation and theoretical foundation of approximation methods for operator equations. An operator is a mapping A which associates with every element of a set X one element of a set Y, A:X~x

~Ax

E Y.

Wewill exclusively consider linear operators, that is we suppose X and Y to be linear spaces over the complexfield ~2 and require that A(~lxl ÷c~2x2) (~lAxl ÷ ~2Ax2 for all elements xl,x2 of X and complex numbers ~,~. Manyof the concrete applications of mathematics in science and technology lead finally to one of the following three basic problems for linear operator equations. Problem A. To solve equations Ax = y which are uniquely solvable. Thus, we are given an operator A : X -~ Y as well as an element y E Y, and we have to find the element x ~ X which satisfies the equality Ax = y.

(1)

If the solution of (1) exists and is unique for every right hand side y ~ Y, then A is invertible, and the solution of (1) is given by x = A-~y. 11

12

CHAPTER O.

INTRODUCTION

Problem B. To solve equations Ax =-- y which are not. uniquely solvable. Thus, either there is no x E X solving Ax = y, or there are too many solutions. In both cases, one can look for elements x E X which are generalized or weak solutions of (1) in a certain sense which has to specified. For example, if X = Y are Hilbert spaces, then a distinguished generalized solution of Ax = y, the least square solution, can be obtained as follows: amongall x ~ X which minimize IIAx-yll choose that one with minimal Ilxll. Again, the least square solution does not exist in general, but the conditions for its existence are evidently much weaker than that one for the unique solvability of (1) for all y. Indeed, if A is bounded, then the equation Ax = y has a unique least square solution for every, right hand side y if and only if the range Im A = AX of A is closed. In this case, A possesses a generalized inverse, its Moore-Penroseoinverse A+, such that the least square solution of Ax = y is given by x = A+y. Problem C. To compute eigenvalues and eigenvectors o] A. Here X = Y, and one looks for all £ ~ (2 - the eigenvalues of A - for which there exist non-zero solutions - the associated eigenvectors - of the equation Ax = Ax. A related problem is to computethe spectrum of A, i.e., the set of all A ~ C for which A - AI (with I referring to the identity operator on X) is not invertible. Clearly, every eigenvalue belongs to the spectrum. Other related problems are the computation of pseudospectra, singular values, numerical ranges, or operator determinants. A direct solution of Problems A - C (which would require knowledge of A-1 or A+, for instance) is impossible for most of the problems of practical relevance, but it is often possible (at least this will be our point of view here) for operators acting betweenlinear spaces of finite dimension, i.e. for matrices. Consequently, we will try to solve Problems A - C approximately by replacing the operator A by matrices An in a suitable manner. In order to be able to speak about approximate solutions or approximation of operators, we have to be able to measure distances in X and Y. Hence, we assume X and Y to be normed spaces, and we also suppose that the operator A maps bounded subsets of X onto bounded subsets of Y, i.e. that A is bounded. The boundedness (or continuity) of A does not automatically imply that of A-1, but if the normed spaces X and Y are complete, i.e. Banach spaces, then it does, as a theorem by Banach asserts. Observe that the boundedness of A and A-1 (and, thus, the choice of appropriate norms in X and Y) is of fundamental importance also for numerical purposes: it guarantees that small errors (caused, e.g., by the finite accuracy of the computer) remain small also after application of -1. or A

0.1.

NUMERICAL ANALYSIS

13

For the approximate solution of Ax = y, one chooses a sequence (Yn) Y of vectors which approximate the right hand side y, and a sequence (An) of operators which approximate the operator A, and one replaces (1) the approximation equations Anxn = Yn, n = 1, 2, ...,

(2)

the solutions Xn of which are sought in X (or in certain subspaces Xn of X) again. Approximation of y by Yn usually means that [[y - Yn[[v -~ 0 as n --~ oo. It is tempting to suppose that the operators An also approximate A in the norm: ][A - An][L(X,y) --~ 0 as n --~ c~, but this assumption does not work in practice. The point is that usually A acts between infinitedimensional spaces, whereas one will, of course, try to choose the An as acting between spaces of finite dimension, i.e. as finite matrices. But the only operators which can be approximated in the operator norm by finite rank operators, are the compactones, and so the restriction to norm convergence would exclude many important operator equations (and, in fact, almost all equations we will discuss in this book). The kind of approximation which fits much better to the purposes of numerical analysis is that of pointwise or strong convergence. The sequence (An) converges strongly to the operator A if I[Ax - Anx[[y -~ 0 for every x E X. This notion of convergence is on the one hand weak enough to include most of the approximations of practical interest, and it is on the other hand strong enough to make the connections between A and An not too loose. In place of the Problems A - C for the operator A, we now obtain analogues of these problems for the approximation operators An: Problem A’. Are the equations (2) uniquely solvable, and do their solutions xn converge to a solution x of equation (1)? Observe that the invertibility of (2) for small n will be without any importance since the approximation of A by An will be quite coarse for these n. So, a more precise formulation of Problem A’ would be: Does there exist an no such that the equations (2) are uniquely solvable for all n _> no and all right hand sides y, and do their solutions converge to a solution of (1)? Related questions are: Do the condition numbers condAn := IIAnll IIA~lll remain uniformly bounded? Do they even converge? Problem B~ Do the equations (2) possess unique least square solutions for all sufficiently large n and for all right hand sides y, and do these solutions converge to the least square solution of (1)? A related question is: What can be said about the convergence of the generalized condition numbers condAn:= IlAnll IIA+~II?

14

CHAPTER O.

INTRODUCTION

Problem C’ Do the eigenvalues of An (or the singular values, the points in the pseudospectrum, the Rayleigh quotients ...) approximate the eigenvalues (singular values, points in the pseudospectrum, Rayleigh quotients ...) of A? Or: What is the asymptotic behavior of the determinants of the matrices A,~? To avoid any misunderstandings, our objective here is not to ask whether, say, A127is invertible, and also not to compute the inverse of A127 on a computer, but we will ask whether Anis invertible for all sufficiently large n, and whether A~1 converges to A-1 strongly. Our objective will not be the determination of the eigenvalues of A127, but we will ask what happens if we plot all eigenvalues of At, A~,A3,... onto a commonsheet of paper. Aroundwhich sets these eigenvalues will cluster? Is this cluster set related with the spectrum of A? In which way? In other words: we are not interested in the properties of a single approximation operator An, we are only interested in the properties of the sequence (An) of approximation operators as a whole. Thus, we define: Definition 0.1 An approximation method for a linear bounded operator A : X ~ Y is a sequence (An) of operators A,~ : X -~ Y which converges to A strongly. Definition 0.2 Let (An) be an approximation method for A. This method is applicable i] there is an no such that the equations Anxn = y,~ possess unique solutions ]or all n >_ no and all convergent sequences (Yn) with limit y, and i] their solutions x,~ converge to a solution x o] Ax = y.

0.2

Operator

chemistry

In chemistry, one splits molecules into their elementary parts - the atoms - in order to create newmolecules. Similarly, it is often useful to think of a ’complicated’ operator (an operator molecule) as being composed by more elementary operators (the operator atoms). Here ’being composed’ means that the complicated operator arises by addition, multiplication, inversion, taking limits, or by other operations from the elementary operators. Consider, for example, the singular integral operator (Au)(t)

= a(t)u(t)

1 /_~ b(s)u(s) + ~r-’~ oo -~---~ ds, t e

(3)

(which is called singular due to the singularity of the kernel for s = t). It is convenient to think of A as the composed operator A = aI + SbI, where aI and bI are the operators of multiplication by the functions a and

0.3.

ALGEBRAIC LANGUAGE OF NUMERICAL ANALYSIS

15

b, and where S is the operator of singular integration along the real line: oo ~8-t ds, t E (-c~, o~). The latter operator can be further (Su)(t) = ~ 1f~-~o decomposedinto S -- FsF-1 , where F is the operator of Fourier transform on L2(IR) and s refers to the sign function. This decomposition provides a lot of information almost at once: So it is immediate for instance that $2=I. ¯ A system of operators which is closed with respect to addition and multiplication of operators and to multiplication with complex numbers is called an algebra. Thus, the singular integral operator (3) is an element of the algebra whose atoms (or, as we will say now, whose generating elements) are the multiplication operators and the operator S. But the very same operator can also be viewed of as an element of the algebra which is generated by the multiplication operators and the operators of direct and inverse Fourier transform or (if a and b are bounded functions) as element of the algebra L(L2(IR)) of all linear boundedoperators on L2(IR). The appropriate choice of the algebra is decisive. Clearly, the algebra should not be too large, but on the other hand the application of certain techniques (see Section 0.4 below) often requires algebras which are not too small (it is seldom wise to consider the operator A as an element of the algebra whoseonly generator is the operator A itself).

0.3

The algebraic language of numerical analysis

Weconsider again the singular integral operator, but nowwith integrating against the unit circle ~’. If a and b are boundedon ~’, the operator aI÷SbI is boundedon L2(~’). For the approximate solution of the integral equation

(4)

Au := (aI + Sbl)u =

we choose an orthogonal basis of L2(~?), the functions {zn}nez say, and seek approximate solutions u,~ of (4) of the form un = ~-~]k[o)m>obe a sequence of elements of G which converge to the sequence (G~)n>0 E ~ in the norm of ~ as n -~ c~. Thus, given ~ > 0 there is an m0such that supn IIG~L,~ - G(~m°)Lnll < ~/2. Choose N such that IIG(~m°)L,~II < ~/2 for all n _> g. Then, for all n _> N, tlG~Lnll < IIG(n’n°)Lnll + ]IG~Ln- G(n’~°)Lnll _< ~, from which it follows that (G~) belongs to 6. Consider the quotient algebra 9~/G, the elements of which are the cosets (An) ÷ 6 of sequences (An) E $’. This algebra is a Banach algebra due Propositions 1.13 and 1.14. The following theorem reveals that the algebra ~’/6 indeed provides a perfect frame to study stability problems in an algebraic way. Theorem1.15 (Kozak) A sequence (An) ~ J: is stable i] and only i] its coset (An) + 6 is invertible in the quotient algebra Proof. If (A,~)n>o is a stable sequence, then the sequence (A~l)n>_~o is bounded for some sufficiently large no by definition. Wemake to a sequence (Bo, B~,..., Bno- 1, A~o~, A~o~+l,...) in ~" by freely choosing

38

CHAPTER 1.

THE LANGUAGE OF NUMERICAL ANALYSIS

operators Bi E Im Li. It is evident that this sequence is an inverse of (An) modulo G. Let, conversely, (An) + ~ be an invertible coset in 9~/~. Then there are sequences (Bn) E ~" as well as (Gn) and (Hn) in 6 such AnBn = In + Gn and B,~An = In + Hn with In = IlImL". If n is large enough, then IIGnll < 1/2 and IIHnll < 1/2, and a Neumann series argument yields the invertibility of the operators In + Gn and In + Hn as well as the uniform boundedness of the norms of their inverses by 2. Hence, AnBn(In + Gn) -1 = In, (In + Hn)-lBnAn = In, and the norms of Bn(In + Gn)-1 and of (In + Hn)-lBn are uniformly hounded. Thus, the operators An are invertible for all sufficiently large n, and their inverses are uniformly bounded. ¯ Weclose our first acquaintance with the algebra ~/~ by a nice expression for the norm of a coset (An) + in~’/ ~. Recall tha t, by definition,

It(An)+GII=(a~)~g inf II(An)+ (Gn)ll~ = (a.)~a inf suPllA,~Ln Proposition

÷GnL,dlL(X)

1.16 For all (An)

II(A~)+ ~ll~/g= limsup IIAnLnll. Proof. Let (An) ~ Y. Then, for every sequence (Gn) ~

limsupIIAnLnll - lira supIIAnLn + GnL,~II _< supIIAnLn + whencethe estimate lim sup IIAnLnll~ II(An)+ ~ll follows. For the reverse inequality, let ¢ > 0, and choose no such that IIAnL~I I _< lira sup for all n > no. If we set Gn:=

-An 0

if if

n<no n>_no,

then the sequence (Gn) belongs to G, and

II(A~)÷ GII~ m2. n

Now fix m0 >_ max{m~,m2} and choose N such that IIW(A(~’~°))xA(,m°)Lnxll < ¢ for all n _> g. Then, for all n >_ N, 2¢11xll + ¢, which proves our claim. The strong convergence of the adjoint sequence can be checked analogously. Hence, the limit sequence (A~) belongs to ~c. ¯ Clearly, the ideal G of 9v is contained in the algebra 9rc, and it closed ideal of ~-c. But this algebra possesses a muchlarger ideal roughly speaking, consists of all possible compactperturbations of imation sequences. Theorem 1.19 The set GC o/all sequences (LnKLn+Gn)n>owith

forms a which, approx(Gn)

G and K compact on X is a closed ideal o/ Proof. It is easy to see that every sequence (LnKLn + Gn) belongs to ~-c and that W(LnKLn+ Gn) = K. It is further clear that Gc is a linear space. For a proof of the ideal property, observe that for every sequence An(nngnn + Gn) = (Annn - LnW(An))KLn + LnW(An)KLn Evidently, the sequence (AnGn) lies in G, and since W(An)Kis again compact for compact K, one has (LnW(An)KLn) e Gc. Finally, AnLn LnW(An) -4 st rongly, fr om which, vi a Le mma 1.5 (b ), fo llows th [I(AnLn - LnW(An))KII -~ and, he nce, (( dnLn - nnW(An))Knn) Thus, (An)(LnKLn + Gn) C,and simi larly one checks that (LnKLn +

an)(An)c.

For a proof of the closedness of G°, let ((LnK(m)L,~ + G(~m))n>_o)m>_o be a sequence of elements in Gc which converges in the norm of ~’. Then K(’~) = W(LnK(m)Ln+G(~’~)), and from (1.11) we infer that the sequence (K(ra)) converges in the operator norm to an operator K°%Being the norm limit of compact operators, the operator K¢~ is compact again, hence, the sequence (LaK~°Lu) belongs to Gc. With this sequence, we have [I(LnK(~)Ln) - (LnK°°in)[]z = sup I]Ln(g (’0 - KC~)Lnll n

where C := sup IIL,~II. Consequently, the sequence ((LnK(m)Ln)n>o)m>o converges in the norm of ~" to (LnK°°Ln) as m --~ ~o, which implies

42

CHAPTER 1.

THE LANGUAGE, OF NUMERICAL ANALYSIS

that the sequence ((G(nm))n_>o)m_>0is convergent, too. Its limit belongs to G by Proposition 1.14. It is now evident that the sequence ((LnK(m)Ln G(nm))n_>0)~nk0 co nverges to (LnK°°L~ + G~)which belongs to the ideal Gc as we have seen. . That it is possible to include all compactperturbations into one ideal (although in the smaller algebra 9re rather than in ,T) shows that the compact perturbations form indeed a quite natural class of perturbations. The following theorem is a key result. It does not only describe the influence of compact perturbations, but is moreover a first step on a way which will finally lead us to an important tool of analyzing algebras of concrete approximation methods: the so-called lifting theorems (see Chapter 4 and Section 5.3). Theorem 1.20 Let the approximation method (An) belong to the algebra 3:c. The sequence (An) is stable if and only i] its strong limit W(An) invertible and i] its coset (An) + ~c is invertible in the quotient ~zc /6c. Weprepare the proof of Theorem1.20 by a lemma. Recall that an operator A E L(X) is normally solvable if its range is closed, and that A is bounded below or an operator of regular type if there is a C > 0 such that Ilxll _~ C IIAxll for all x E X. Lemma 1.21 Let X be a Banach space. The operator A E L(X) boundedbelow if and only i/it is normally solvable and i/its kernel consists o/the zero element only. The assertion of Lemma1.21 becomes obvious if X has finite dimension, and we already used it in this special form in the proof of Theorem1.10. Proof. If Im A is closed and Ker A = {0}, then A is invertible as operator from X onto Im A. By Banach’s theorem, there is a bounded linear operator B : Im A -~ X such that BA -- I on X. Thus, for every x E X,

Let, conversely, A be bounded below, and let (Yn) C_ Im A be a convergent sequence. Choose elements Xn ~- X with Axn -~ Yn. The inequalities

imply that (Xn) is a Cauchy sequence in X, hence convergent. Set x lira xn. Since A is continuous, one has y,~ = Axn ~. Ax, i.e. the limit of (Yn) belongs to Im A. Thus, A is normally solvable, and the injectivity of A is obvious. ¯

1.2. BANACH ALGEBRAS AND STABILITY

43

Proof of Theorem 1.20. Let (An) E re be a st able se quence an d se t C := supn>,~o IIA~ILnlI. Then, for every x E X and all sufficiently large n, IILnxll = IVA’(~IAnLnxll _O. Clearly, (PnV~P~ V~_, P,) = (PnV’P,)(P, P1Pn)(P~V~_,P~), and the sequences (P~V~P~) and (PnV~_IP~) belong to 8(C) since V~ = T(a~) and V_Jl = T(aJ__l) where a(t) = and a-l(t) = -1 . Further, P~V = P~VP~ and V-~Pn = PnV-1Pn, which implies that (PnPIPn) = (Pn(I - VV-~)Pn) = (Pn) -- (PnVPn)(PnV-1Pn) is in 8(C), too.

1.4. C*-ALGI~BRAS

59

OF APPROXIMATION SEQUENCES

Similarly, to prove that (R,~LR,~) is in S(C) for every compact L, we have to check whether RnV~P1V~_IR,~)e ,9(C) for every i,j >_ O. In this case one has (R,~V~P1VJ_IR,~)= (R,~V~P,~P1P,~V~_IR,~) = (RnV~Rn)(RnP1Rn)(RnVJ_IRn), and the sequences (R,~V~Rn) and (RnVJ_IRn) are in $(C) since RnViRn = RnT(ai)Rn and R,~VJ_IRr~ = RnT(aJ__I)R,~ (also recall (1.22)). Furthermore, (R,~P~R,~) = (P,~) - (R,~VV_IR,~) = (P,~) - (R,~VP,~V_IR,~) = (Pn) - (RnVRn)(R,~V-~Rn) ¯ Finally, a little thought showsthat, to verify the inclusion G _C S(C), it sufficient to showthat, for every fixed no, every se.quence of the form where Cn = 0 for n ¢ no and C,~o = P,~oVIP1V~_IP,~owith 0 _< i,j < no belongs to $(C). But this is a consequence of the identity

and of what has already been shown. It remains to prove the closedness of the set $1- First observe that, given a sequence (An) ¯ ,~, the strong limits W(A~) := s-lim AnPn and lYd(A~) := s-lim RnAR,~exist and that W(PnT(a)Pn + P,~KP,~ + R,~LR~ + G,~) = T(a)

(1.23)

17V(P,~T(a)P,~ + P,~KP,~ + RnLRn+ Gn) = T(a)

(1.24)

Indeed, for (1.23) one has to show that R,~LR,~ ~ 0 strongly if L is compact

(1.25)

which can be most easily seen by approximating L by a linear combination of operators of the form ViPIV~_~; then (1.24) is a consequence of (1.22) and (1.25). Thus, if ((PnT(a(m))P,~ + P,~K(m)Pn+ R,~L(m)R,~ + G(nrn))n>_o)ra>_ois a sequence of elements of S~ which converges in ~’, then the sequences (T(a(m)) + K(rn))m>_Oand (T(5(m)) + L(m))m>_oare convergent, too. From Theorem1.51 we knowthat there is a continuous function a as well as compact operators K, L such that limr,-~oo T(a(’~))) + K(’~) = T(a) + and limm-~ooT(?z(m)) + L(m) = T(~t) + L. Thus, the sequence ( (PnT(a(’~))Pn

60

CHAPTER 1.

THE LANGUAGE OF NUMERICAL ANALYSIS

PnK(m) pn + RnL(m) R,~)n>_o)m>_otends to (PnT(a)Pn + PnKPn+ RnLRn), which shows that the sequence ((G(nm))n>_o),~_>0 is also convergent. limit (Gn) of this sequence belongs to ~ because ~ is closed in $" (Proposition 1.14). Nowit is clear that the limit of the sequence under consideration is just (PnT(a)P,~ + PnKPn+ R~LRn + Gn) which belongs to 81. Hence, S1 is closed, and S1 = S(C).

1.4.3

Stability of sequences in the C*-algebra of the finite section method for Toeplitz operators

Nowwe are going to examine the stability of an arbitrary sequence in the algebra 8(C). At the first glance, it might seem to be quite strange to consider something like (PnT(a)Pn ÷ RnLRn) with compact L as an approximation sequence for the Toeplitz operator T(a). But observe that many difference methods lead to Toeplitz matrices which are perturbed in the left upper and right lower corner. The perturbations of the right lower corner correspond exactly to the sequences (RnLTln) with compact L, whereas the sequences (P,~I(Pn) with compact K describe perturbations in the left upper corner. The importance of the following result for the purpose of describing the algebra $(C) will becomeclear in the next section. Theorem1.54 Let (An) e ~(C). The sequence (An) is stable if and i] both operators W(An) := s-lim AnPn and I~V(An) := s-lim RnAnRnare invertible. Proof. The necessity of the invertibility of W(An) and l/~d(An) can shown as in the proof of Theorem 1.20. Here is yet another proof, which takes advantage of the C*-property of $(C). If the sequence (An) S(C) is stable the n the coset (An) +~ is i nvertible in 9~/~ (Kozak’s theorem). Since S(C) is a C*-algebra which contains the closed ideal 6 (Theorem 1.53) we find that 8(C)/6 is a C*-subalgebra of ~/~. From the inverse closedness (Theorem 1.49) we further infer that (An) + 6 is even invertible in 3(C)/6. In other words: there are sequences (Bn) E $(C) (Gn), (Ha) such th at (AnBn) = (IlIrnP~)

+ (Gn), (Bndn) = (IlImP~) + (Hn).

Applying both homomorphismsWand l~d to these equalities, we obtain the invertibility of W(An)and I?d(An). (Recall that the existence of strong limits W(Bn) and I~(Bn) for arbitrary (Bn) S(C) was ve rified in the proof of Theorem1.53.) Now let (An) (PnT(a)Pn + PnKPn + R, ~LRn + Gn) ~ ,S and su ppose that the operators W(An) = T(a) + andI~V(An) = T(a)+ L are in -

1.4. C*-ALGEBRAS

OF APPROXIMATION

61

SEQUENCES

vertible. The invertibility of T(a) ÷ implies vi a Corollary 1. 36 the st ability of (PnT(a)Pn + PnKPn). Then, clearly, the sequence (Rn(PnT(a)Pn PnKPn)Rn) = (PnT(5)Pn + RnKRn) is stable, too. For the sequence (PnT(5)Pn + RnKRn+ PnLPn), which is nothing but a compact perturbation of (PnT(h)Pn+RnKRn), we have W(PnT(a)P~+RnKRn+PnLPn) T(5) +L. The invertibility of this operator together with the (perturbation) Theorem 1.20 yields the stability of the sequence (PnT(5)Pn + RnKRn PnLPn), which is evidently equivalent to the stability of (An). 1.4.4

Symbol of the operators

finite

section

method

for

Toeplitz

Let us start with reformulating the stability result of the preceding section. Write L(/2) × L(/2) for the product of the C*-algebra L(/2) with itself, for the set of all ordered pairs (B1, B2) of operators B1,B2 E L(12). Provided with elementwise operations, elementwise involution, and the maximumnorm II(B1,B2)II := max{llBlll , IIB~II}, this product becomes a C*algebra. Next take the *-homomorphismsW, l~d : S(C) -~ L(/2) and glue them together to obtain one homomorphism,say smb°, acting via smb°: S(C) --~ L(l 2) × L(12), (An) ~+ (W(A,~), IYV(An)). It is furthermore clear that the ideal G of 8(C) lies in the kernel of the homomorphism smb°. Thus, the quotient homomorphism smb : 8(C)/6 -~ L(l ~) × L(l~),

(An) + ~ ~ smb°(A~)

(1.26)

is correctly defined, and we can restate Theorem1.54 as follows: Theorem 1.55 Let (An) ~ S(C). The coset (An) + ~ is invertible S(C)/G i] and only i] smb((An)+ is invertible in L(/2) x L(12). Thus, smb is a symbol mapping for S(C)/~ in the following sense. Definition 1.56 Let A, B be unital Banach algebras and S : A -~ B a unital homomorphism.S is a symbol mappingfor A if, for arbitrary a ~ A, the invertibility o] S(a) in B implies the invertibility o] a in ,4. I] S is symbol mapping, then S(a) is called the symbol o] a. Since the invertibility of a ~ A implies the invertibility of S(a) in B for every unital homomorphismS : ,4 -~ B, one can also characterize symbol mappings as homomorphismswhich preserve spectra. In the general (Banach algebra) case, this is almost all what can be said about symbol

62

CHAPTER 1.

THE LANGUAGE OF NUMERICAL ANALYSIS

mappings, and the construction of a symbol mapping is often the ultimate goal of any analysis of Banach algebras of approximation sequences. But in the C*-case, and when S is a *-symbol mapping (i.e. a symmetric homomorphism),then the symbol mapping is not only responsible for invertibility, but moreoverreflects the algebraic and metric properties of the algebra A exactly. Theorem 1.57 Let A and be a *-symbol mapping. Then S is a *-isomorphism (hence, an isometry) from A onto the C*-algebra Im Proof. In view of Theorems 1.44(d) and 1.45, it remains to verify that every *-symbol mapping is one-to-one. Let a E ,4 and Sa = O. Then, by the C*-axiom, 0 --lISa[[ 2 = I[(Sa)*(Sa)[] = I[S(a*a)[[~

(1.27)

For selfadjoint elements b of a C*-algebra/3, the norm []b][ and the spectral radius p(b) coincide. Hence, by (1.27), p(S(a*a)) = 0: But S preserves spectra and, in particular, spectral radii, which implies that p(a*a) ~- 0, too. Nowwe get as in (1.27) that 0 = p(a*a) = I[a*a[[ = [[a[[ 2, i.e. a = 0. ¯ Combining Theorems 1.55 and 1.57 we obtain: Corollary 1.58 The mapping smb in (1.26) is a *-isomorphism (and, hence, an isometry) from $(C)/6 onto the C*-subalgebra L(/ 2) x L (/ which consists of all pairs (W(An),IV(An)) with (An) running

s(c). Thus, one need not distinguish between the algebra ~q(C)/g and its image in L(/2) x L(/2) and, consequently, instead of working with the elements S(C)/6 (i.e. with cosets of infinite sequences of approximation operators), we will simply deal with their symbols (i.e. with pairs of operators), which is much more pleasant. For applications of the description of 8(C)/~ via its symbol mapping we refer to Chapters 2 and 3 and also to the forthcoming section, whereas the whole Chapter 4 is devoted to the construction of symbol mappings for more involved subalgebras of

1.5

Asymptotic bers

behaviour

of condition

num-

As a first application of the complete description of the algebra $(C)/~ obtained in the previous section via constructing a symbol mapping, we are

1.5.

ASYMPTOTIC BEHAVIOUR OF CONDITION NUMBERS

63

nowgoing to prove that the sequence of the condition numbersof the finite sections of an invertible Toeplitz operator in S(C) is convergent. (That this sequence remains bounded is evident from the stability of the finite section method.) The proof of this fact motivates the introduction of the notion of a ]ractal approximation sequence in the forthcoming section. 1.5.1

The condition

of an operator

The condition number condA := IIAII IIA-Xll of an invertible operator A is a measure for the sensitivity of the dependenceof the solution u of the equation Au = f from the right hand side f. Namely, if u +/~u is the solution of the equation A(u +/ku) = f +/kf with perturbated right hand side, then the relative errors of u resp. f satisfy the estimate [[/ku[[

< condA

Clearly, cond A is never less than 1, and for selfadjoint and positive definite operators A one has condA = sup{A:A E a(A)} inf {A: A E a(A)} Let now (An) be an approximation method for the operator A. For computational purposes, the asymptotic behaviour of the condition numbers cond An as n tends to infinity is of great interest. If (An) is a stable sequence, then the sequence (cond An) is bounded: sup condAn= sup [[An[[ [[A~X[] _< sup [IA,[[ sup [[A~I[] < cx~. As we shall see later (Corollary 1.72), there are additional but quite natural conditions for the sequence (An) which guarantee that the sequence (cond A,~) is even convergent. For the moment,we restrict our attention a first example, namely, to the behavior of the condition numbers cond An for sequences (An) in the algebra $(C) of the finite section method for Toeplitz operators. The sequence of the norms[IPnT (a) Pn [[ is monotonically increasing and bounded from above by []T(a)[[. Hence, lim [[P,~T(a)Pn[[ exists. Further, the Banach-Steinhaus theorem gives the estimate [[T(a)[[ = [[s-lim PnT(a)P,~[[ Then the Toeplitz operator T(a) is invertible, and IIT(a)-~ll ~_ (1 + V/1 - d2/llall2~)/d < 2/d.

(1.33)

Proof. There is a ~/¯ ~" such that ~a(~l’) is contained in the set {z¯C: Rez>_d, Iz] no. Consider the sequence r/(n) := n + no. Because Wis fractal,

=

=

=

and, consequently,

(recall that assertion.

IlWoll_0,

and set ~ := diag (71, ..., a-n). Then the matrix B := V~U*satisfies the axioms (2.3). For a proof of uniqueness suppose that both B~ and B2 satisfy (2.3) place of B. Then B1 = Sl AB1 = B1 (ABI)* = B~ B~ A* = BIB~ (A* B~ = BI(B~A )(B2A ) B~(AB~)*(AB2)* = BI (ABIA)B2 =

BL

CHAPTER 2.

78

REGULARIZATION

and, similarly, B2 = BLAB2, whence B1 = B2. It remains to check whether B = V~U* coincides with the MoorePenrose inverse of A = UEV*,i.e. whether Bf is the least square solution of Au = f. By the unitarity of U and V, one has [IABf - f[I = [IUEV* Y~U*f - fl[

= [[E~V*f - U’f[ I.

(2.4)

Clearly, E~ is an orthogonal projection matrix, and for every orthogonal projection P E C~×’~ and arbitrary x,y E C~ the estimate IIPx - xl] _< IIPy - xll holds. With P := E~, x := U*f and y := EV*u this gives

IIABf - fll - 0. Set Cn := IIAn,~ - Anll and 1/d := supn IIAn+,~ll. Then d > 0 and ¢n -} 0 as n -~ cx~, and it remains to showthat a2(An) C_ [0,~n] (A [d, o~). From(2.10) we conclude that ~n = [[An,~ - An[[ = [[Rn,~ - Rn[[ = [[E(¢)Rn[[ = sup{ A: A e a(E(~)Rn)} which in combination with Theorem 2.8 yields a(Rn) [0,¢] = a~(An) N [0,¢] C_[0, ¢n].

(2.14)

Further we infer from Theorem 2.5 that

IIA+~,AI: sup(l/a: a e a2(A,~,e) \ {0} d, and since a:(An,~) -- ae(Rn,~) = a((I-E(e))R,~)

C_ a(Rn)V~[¢,~) :- a2(An)N[~,cx~)

by Theorem2.8 again, we finally arive at ae(An)N [~, oo) C_[d, oo).

(2.15)

From (2.14) and (2.15) we obtain the desired inclusion a2(An)C_[0, en] f~ [d, oo).

(2.16)

If, conversely, (2.16) is satisfied with certain numbers d > 0, and en _> tending to zero then choose ~ E [0, d) arbitrarily, and let A,~,~ denote the e-regularization of An. By repeating the above arguments it is not hard to check that [[A,~,, - An[[ _< en and [[A~+,~[I _< l/d, which implies the stable regularizability of (An). ¯ Let us once more emphasize that the stable regularizability of a sequence (An) is equivalent to the splitting of the set a2(An) into two parts: tending to zero with n going to infinity, and one remaining bounded away from zero by a positive constant for all n. For a deeper investigation of this splitting property (which will concern the numberof the singular values in [0, cn]) we refer to Chapter 6.

2.2. ALGEBRAIC

2.2

CHARACTERIZATION

89

Algebraic characterization ularizable sequences

of stably reg-

The main result of this section relates the stable regularizability of a sequence (A,~) E ~ with a property of thecoset (An) +~ E 9v/G, viz. with its Moore-Penroseinvertibility. Westart with recalling some facts concerning Moore-Penroseinvertibility in C*-algebras. 2.2.1

Moore-Penrose

invertibility

in

C*-algebras

Let /~ be a C*-algebra. An element a ~ B is said to be Moore-Penrose invertible if there is a b ~ B such that aba = a, bab-~ b, (ab)* = ab, and (ba)* = ba. (2.17) The proof of Theorem2.1 shows that the element b is unique (if it exists) and we call b the Moore-Penroseinverse of a and denote it by a+. One easily checks that a and a* are Moore-Penroseinvertible only simultaneously and that (a*) + = (a+) * and (a’a) + = a+(a*) +, (2.18) + + whereas the identity (ab) = b+a is wrong in general. The following theorem summarizes some equivalent conditions for the Moore-Penrose invertibility in unital C*-algebras. Recall that an element p of a C*-algebra is a projection if it is idempotent(i.e. p2 = p) and self-adjoint. Theorem2.15 Let I~ be a C*-algebra with identity e. The ]ollowing conditions are equivalent for every element a o] B: (a) The element a is Moore-Penroseinvertible. (b) The element a*a is invertible or 0 is an isolated point o] a(a*a). (c) There is a projection p in alg (e, a’a) (--- the smallest closed subalgebra o] B which contains e and a’a) such that a*ap = 0 and a*a+pis invertible. (d) There is a projection q in l~ such that aq = 0 and a*a ÷ q is invertible. I] one o] these conditions is satisfied, then q is uniquely determined, and

a+ = + as well as II +ll = sup{1/ : e \ {0} }.

Proof. (a) ~ (b): Let a be Moore-Penrose invertible and set b +. If a = 0, then 0 is an isolated point ofa(a*a). Ifa ~ 0, then b ~ 0, and e-)~bb* is invertible in/~ for every complex A with 0 < ]~1 < Ilbb*l1-1 (Neumann series). A straightforward calculation shows that (e-)~bb*)-lbb * - 1/A(e ba) is the inverse of a’a- Ae, i.e. either 0 ~ a(a*a) or 0 is an isolated point of that spectrum. (b) ~ (c): a*ais i nvertible, choo se p = 0. Solet 0 b e an is olated

9O

CHAPTER 2.

REGULARIZATION

point of a(a*a). The C*-algebra alg (e, a’a) is commutative; thus, by the Gelfand-Naimark theorem, this algebra is *-isomorphic to the C*-algebra C(X) where X is a certain compact, and every element c E alg (e, a’a) corresponds to a certain continuous function ~ on X. Thereby, at3(c) aalg(e,a*~)(c) = ac(x)(~) = ~(Z) for every c e alg(e,a*a). Set X0 := {x e X : (a*a)(x) = 0} and X1 := X \ X0. Assumption (b) guarantees that sets X0 and X1 are both open and closed subsets of X. Hence, :

X--+C

x~

0 [

if

x~X1 1 if x~Xo

defines a continuous function on X. Let p denote the (uniquely determined) element of aig (e, a’a) which corresponds to this function. One easily checks (by considering functions instead of elements again) that p is subject to the condition (c). (c) ~ (d): a*ap = 0, then 0 = ]]pa*ap]l = ]] (ap)*(ap)ll = Ha 2, hence one can choose q := p. (d) =~ (a): It is straightforward to Check that (a*a + q)-la* is the MoorePenrose inverse of a. Let us show the uniqueness of q. From aq = 0 we obtain that (a* a + q)q = and, he nce, (a * a + q)-i q =

(2.19)

whereas the identity a + = (a*a + q)-la* involves (a*a + q)-la*a -= a+a.

(2.20)

Addition of (2.19) and (2.20) yields e = q + a+a or q = e - a+a, which shows the uniqueness. Finally, the norm identity follows from

2 = Ila÷(a÷)*ll Ila÷ll2 = II(a÷)*ll = Ila÷(a*)÷ll _-II(a*a)÷ll by employing the *-isomorphy (which is actually aig (e, a’a) and C(X) again.

an isometry) between ¯

Wedenote the (uniquely determined) projection q in (d) n, and call a rI the Moore-Penroseprojection associated with a. Example 2.16 Operators on Hilbert space. Let A be a bounded linear operator on a Hilbert space H, and suppose that A is Moore-Penrose invertible (i.e. Moore-Penroseinvertible as element of the C*-algebra L(H)). Let B ~ L(H) be the Moore-Penrose inverse of A, and denote the orthogonal projections AB and BA by P and I - Q, respectively. From Im A = Im ABA C_ Im AB C_ Im A,

2.2. ALGEBRAIC

CHARACTERIZATION

91

Ker A _C Ker BA C_ Ker ABA = Ker A we conclude that ImA=ImP

and

KerA=Ker(I-Q)

which in particular showsthat the range of A is closed and that the MoorePenrose projection of A is actually the orthogonal projection from H onto the kernel of A. If, conversely, A E L(H) is an operator with closed range and with orthogonal projection Q from H onto the kernel of A, then AQ = 0, and the operator A*A+ Q is invertible which can be seen as follows: Let x belong to the kernel of A*A ÷ Q. Then, since AQ = 0 and QA* = O, one has (I - Q)A*A(I - Q)x + Qx whic h immediately give s Qx =0 and (I - Q)A*A(I - Q)x = Thelatt er equa lity impl ies ((I Q)A*A(I Q)x, x) = (A(I - Q)x, A(I - = 0, i. e. A(I - Q)x = 0. S in ce Q is the orthogonal projection onto the kernel of A, one has (I - Q)x -- 0 and, consequently, x -- 0. Further, A*A÷ Q is self-adjoint and, hence, Ker (A*A + Q) Im(A*A + Q) = H So it remains to show that the range of A*A+Qis closed. From A*A+Q= (I - Q)A*A(I - Q) and the fact that proje ctions are n ormally solva ble we conclude that A*A ÷ Q is normally solvable if and only if A*A is so. Thus, let z = lim A*Ax,~. Since A* is normally solvable whenever A is so, the vector z must belong to the range of A*, that is, z = A*y with some y E H. If further P denotes the orthogonal projection onto the (closed) range of A then PA = A and A*P = A*. Consequently, there is an x ~ H such that y = Ax and z = A*Ax. The closedness of Im A*A implies that of Im (A*A ÷ Q), which, via the preceding theorem, yields the Moore-Penrose invertibility of A. . Example 2.17 Self-adjoint elements. A self-adjoint element a of a unital C*-algebra B is Moore-Penroseinvertible if and only if it is invertible or if 0 is an isolated point of the spectrum of a. This follows from the equivalence of (a) and (b) in the preceding theorem and from the identity a(a*a) = a(a2) = a(a) 2. Further, repeating the arguments of the implication (b) =~ (c) with the Moore-Penroseinvertible element a in place a’a, one obtains the existence of a projection p in alg (e, a) such that ap = 0 and a ÷ p is invertible. Conversely, if p is a projection possessing these properties, then it is straightforward to check that (a + p)-i (e - p) is Moore-Penrose inverse of a. Combiningthese observations with those from Example 2.16 above, one easily gets that the Moore-Penrose inverse of a self-adjoint and normally solvable operator A ~ L(H) is A+ (A+P~/er -1

92

CHAPTER 2.

RE, GULARIZATION

As we have already observed, one peculiarity of C*-algebras is their inverse closedness with respect to usual invertibility, which simplifies the study of invertibility problems in C*-algebras essentially. It is an immediate consequence of Theorem2.15 that C*-algebras are also inverse closed with respect to Moore-Penroseinvertibility. Corollary 2.18 Let B be a C*-algebra with identity and C be a C*- subalgebra o] 13 which contains the identity. If c E C is Moore-Penroseinvertible in 13, then c + ~ C. Indeed, the inverse closedness of C*-algebras with respect to the usual invertibility gives a~(c*c) = ac(c*c), and the equivalence of (a) and (b) Theorem 2.15 yields the assertion. ¯ As a first application verified analogously.

we will prove Theorem 2.12. Theorem 2.13 can be

Proof of Theorem 2.12 . The implication (a) :=~ (b) is an immediate consequence of the Banach-Steinhaus theorem. For the reverse implication, consider the C*-algebra ~ of all bounded sequences (A,~) of operators A,~ L(Hn), and let "C denote it s su bset co nsisting of all sequences (An)for which the strong limits s-lim AnPnand solim A~Pnexist. The set 5re is a C*-subalgebra of ~- which contains the identity of ~’. Without loss of generality we now suppose that (An) is a sequence such that all An are Moore-Penrose invertible and sup IIAn+ll < ~. Then (An) belongs to ~-c, and the sequence (An+) belongs to 9t- and is just the Moore-Penrose inverse of (An) in ~’. From Corollary 2.18 we infer that (An+) also belongs to ~-c, i.e. the strong limits B s-lim A+~P, and B* := s-lim (An+)*Pnexist. Letting in AnAn+A~= A,~, An+A,~A+,~= A+~, (A,~A+~)* = A,~A+~, (A+~An)* = A+~A,~ n go to infinity we obtain ABA = A, BAB = B, (AB)* +. i.e. A is Moore-Penroseinvertible, 2.2.2

Stable regularizability, vertibility in ~’/G

---

AB and (BA)*

= BA, ¯

and B --- A and

Moore-Penrose

in-

Let H be a Hilbert space, (Hn) be a sequence of subspaces of H, v t he C*-algebra of all bounded sequences (An) of operators A,~ ~ L(Hn), and the ideal of all sequences (An) ~ ~ tending to zero in the norm. The goal this section is to prove the following characterization of stably regularizable sequences in ~’.

2.2. ALGEBRAIC

CHARACTERIZATION

93

Theorem2.19 A sequence (A~) E jz is stably regularizable i] and only the coset (An)+ G is Moore-Penroseinvertible in the quotient algebra i.e. i] there is a sequence (Bn) ~ if: such that IIA,~B,~An- A,~II ~ O, IIB,~A,~B,~ - B,~II ~ O, II(A,~B~)* - A,~B,~II -~ O, II(B,~A,~)* - BnA,~II--+ The proof of Theorem2.19 requires some additional statements. The first asserts that every ’almost projection’ has a projection in its neighborhood. Proposition 2.20 Let B be a C*-algebra with identity e, and let a ~ B be a self-adjoint element with Ila - a211 < 1/4 (an ’almost projection’). Then there is a self-adjoint element g ~ t3 such that a + g is a projection and

Ilgll- no, then HAn-A~nll < 1/4, and Proposition 2.20 entails the existence of self-adjoint operators Gn ~ L(Hn) such that An + Gn is a projection and IIGnll limsup II(A~An+ IIn)-IlILCH,,~= II((A~An+ IIn) set ~n := I]A~AnIInl], and fix no such that ]](A~An + IIn)-IIIL(H,)

1/ d for al l n k no

(2.21)

and enno.

(2.22)

For n k no, consider the commutative C*-algebra alg (A~A,~, IIH.). This algebra is *-isomorphic to C(X) for some compact X by the GelfandNaimark theorem, and we let ~ and/5 refer to the continuous functions on X corresponding to A~A,, and Hn, respectively. From (2.21) and the fact that *-isomorphisms are isometries we conclude that 5(x) + ~(x) for all x ~ X, whereas the choice of ~,~ implies that 5(x)ih(x) _< en for x ~ X. The function 15 is a projection and can take the values 0 and 1 only. Thus, if 5(x) < d, then necessarily 15(x) = 1, and if ~(x) > en, necessarily 15(x) = 0. By (2.22), this involves that (e,~, d)~5(X) = the spectrum of A~A,~ resp. the singular values of An show the splitting behaviour discussed at the end of the preceding section, which is equivalent to the stable regularizability of (An). For the reverse implication we start with the splitting property of the spectrum of A~An, i.e. we let d > 0 and en k 0 be numbers with limen = 0 such that a(A~An)C_ [0, en] U [d, oo). (It obviously doesn’t matter whether we assume the splitting property for the spectra of A~A, or for the singular values of An.) Choose no so that en < d for n _> no. Given n k no we represent the commutative C*-algebra alg (A~An, I]~,) as the algebra of all continuous functions on some compact X, and we denote by ~ the function which corresponds to the operator A~,An. Our assumptions guarantee that the

2.2. ALGEBRAIC

CHARACTERIZATION

97

function 15:

X-4C,

1 if a(x)~[O, x~ 0 if a(x) E[d,~)

is continuous on X, and one easily checks that 0 < a(x)15(x)

< Cn for all

x EX

(2.23)

and a(x) + 15(x) > max{1, d} for all

x e X.

(2.24)

For n > no, let Hn ~ alg (A~An, IIH.) be the operator which is associated with 15. Then IIn is a projection, IIA~AnIInll 0 and en > 0 are numbers with limen = 0, and let (l’In) ~ SII(An). Then 1-In is the [O, en]-spectral projection of A~Anfor all sufficiently large n.

2.2.3

Finite sections of Toeplitz operators and their stable regularizability

The first application of the characterization of the stable regularizability of a sequence as the Moore-Penroseinvertibility of the coset of that sequence modulo~ concerns the stable regularizability of the sequence (P~T(a)Pn) of the finite sections of a Toeplitz operator T(a) with continuous generating function. In Section 1.4.2 we introduced the C*-algebra S(C), which is the smallest C*-subalgebra of 5r containing all of these sequences, and we proved that S(C) contains the ideal ~ of the zero sequences and that S(C)/~ is *-isomorphic to a subalgebra of L(l 2) x L(/2), the isomorphism sending the coset (A,~) + 6 to the ordered pair (W(An), IV(An)).. Let (An) be an arbitrary sequence in S(C). This sequence is stably regularizable if and only if the coset (An) ÷ ~ is Moore-Penroseinvertible in 5r/~ (Theorem 2.19), and the latter is equivalent to the Moore-Penrose invertibility of (An) + in S(C)/~ (in verse clo sedness, Cor ollary 2.1 8). Finally, by Theorems1.55 and 1.57 (and inverse closedness again), the coset

CHAPTER 2.

98

REGULARIZATION

(An) + 6 is Moore-Penrose invertible in S(C)/~ if and only if the symbol of this coset, i. e. the pair (W(A,~), IYV(An)), is Moore-Penroseinvertible in L(/2) L(12). So, to gether wi th th e ch aracterization of Moore-Penrose invertible operators given in Theorem2.4 we get: Theorem 2.24 The sequence (An)_ E S(C) is stably regularizable only if both operators W(An) and W(An) are normally solvable. A more explicit 1.53.

if

form of this result can be obtained by invoking Theorem

Theorem 2.25 Let a be a continuous function, K and L be compact operators, and (Gn) be a sequence tending to zero in the norm. The sequence (PnT(a)Pn + PnKPn+ RnLRn+ Gn) is stably regularizable if and only the operators T(a) + K and T(5) + L are normally solvable. The operators T(a) + and T( 5) + L belong to the alge bra T(C)which is the smallest closed subalgebra of L(l ~) containing all Toeplitz operators with continuous generating function and, moreover, the operators of the form T(a) + al ready ex haust T( C) (Theorem 1. 51). Fo r th e no rmal solvability of operators belonging to T(C) one has the following result. Proposition 2.26 Let a be a continuous function on ~ and K ~ K(/2). Then the operator T(a) + K is normally solvable if and only if one of the following alternatives holds:

(i) a(t) # o for all t V, (ii) a(t) = 0 for all t e 7~, and g has finite rank. Proof. From (1.21) we know that a Toeplitz operator T(a) is compact if and only if a -- 0. Thus, the mapping T(C) -~ C(T), T(a) + g ~-~

(2.25)

is correctly defined, and with (1.16) and (1.17) and the compactness the Hankel operators H(a) and H(5) (Lemma1.33) one easily checks (2.25) is a *-homomorphism. Let now T(a) ÷ K ~ 7-(C) be normally solvable. Then T(a) + is Moore-Penrose invertible, and (T(a) + + belongs to T(C) again (in verse closedness, Corollary 2.18). Hence, (T(a) + + = T(b) + with b ~ C(~’) and R ~ K(l:), and applying the homomorphism (2.25) to each of the identities (T(a) + K)(T(b) + R)(T(a) + K) (T(b) + R)(T(a) + g)(T(b) + R)

2.2.

ALGEBRAIC CHARACTERIZATION

99

and ((T(a) + K)(T(b) + R))* = (T(a) +

((T(b) + n)(T(a) + (T(b)n)(T(a) one arrives at the Moore-Penroseinvertibility of the function a in C(~). By Theorem2.15(b) this implies that either a has no zero on ~, or vanishes at every point. If, conversely, a(t) ~ for ev ery t E ~, the n T(a) + Kis a Fredholm operator (Theorem 1.30) and, hence, normally solvable, whereas in case a = 0 the operator T(a) + K = is normally sol vable if andonlyif K has finite rank. ¯ So we obtain a third equivalent formulation of Theorem2.24: Theorem 2.27 Let a be a continuous function, K and L be compact operators, and (Gn) be a sequence tending to zero in the norm. The sequence (PnT(a)Pn + PnKPn+ RnLRn+ Gn) is stably regularizable i] and only either (i) a(t) # O.for all t E ~£ (ii) a =_ O, and K and L have finite rank. Wewould like to emphasize two points. The first one concerns the way of proving Theorems2.24-2.27. Observe that the only things we utilized were the characterization of the stably regularizable sequence (An) via the Moore-Penrose invertibility of the coset (A,~) + ~ in ~’/~ (which is true for arbitrary bounded sequences (An)), then an inverse closedness argument which allows us to consider this Moore-Penrose invertibility problem in any C*-subalgebra of ~/~ which contains (An) + ~, and finally an exact description of a certain C*-subalgebra of ~’/~ which contains the coset (P,~T(a)Pn) (andwhichis the res ult of a de tai le d analysi s of the stability problem for the sequences (PnT(a)Pn)). Thus, the ’right’ answer to the usual stability problem for sufficiently manysequences will almost automatically also solve the problem of stable regularizability for these sequences. In Chapter 4, we will examinea lot of (partially muchmore involved) usual stability problems for concrete approximation methods, and we will solve them ’in the right manner’, so that the above analysis also applies. The second remark concerns the stability with respect to small perturbations of stably regularizable approximation sequences. Since stable regularizability is equivalent to Moore-Penroseinvertibility, and MoorePenrose invertibility fails to be stable with respect to small perturbations, one cannot expect stability under small perturbations for stably regularizable sequences. But, in contrast to Moore-Penrosestable sequences, stably regularizible sequences are stable at least under perturbations tending to

100

CHAPTI~R

2.

REGULARIZATION

zero as n tends to infinity. Moreover,in the only interesting case of Theorem 2.24 (when W(An) and l~(An) are Fredholm operators), stably regularizable sequences behave stably with respect to small perturbations (since Fredholmness is a property which is stable under small perturbations, and this property carries over to the corresponding approximation sequences via symbol calculus). Wewill pick up this observation in Chapter 6.

2.2.4

Convergence

of

generalized

condition

numbers

The (generalized) condition number of a Moore-Penrose invertible element of a C*-algebra is defined as conda := Ilall Ila+ll. In this section we are going to examine the convergence of the condition numbers cond An,~ for stably regularizable approximation methods (An). Wewill make use the somewhat more general frame introduced in Section 1.6.1, i.e. we let Cn (n = 0, 1,...) denote unital C*-algebras and consider their product v respective their restricted product 6 which is an ideal in 9r. Also, given a strongly monotonically increasing sequence ~/ : N --~ N, let Rn stand for the restriction operator (an) (a n(n)) an d de fine An:= Rn() for every subalgebra A of ~’. Motivated by Theorem 2.19, we call a sequence (an) E ~ stably regularizable if the coset (an) + 6 is Moore-Penroseinvertible in ~-/6. Theorem 2.28 z. Let A be a fractal C*-subalgebra of :7 (a) A sequence (an) ~ A is stably regularizable if and only if it possesses an infinite stably regularizable subsequence. (b) Let (an) ~ -A be a stably regularizable sequence, and let (bn) ~ jz sequence such that (bn) + 6 is the Moore-Penroseinverse o] (an) + 6. the limit lim,~ ]]anll Hbn]lexists and is equal to ]](an) + 61111(bn)+ 611Proof. (a) If (a,~) ~ A is stably regularizable then, clearly, every finite subsequence of (an) is also stably regularizable. Let, conversely, ~? be a monotonically increasing sequence such that (an(,~)) = Rn(an) is a stably regularizable sequence. By definition, this is equivalent to the Moore-Penroseinvertibility of Rn(a=) + 6n in th e quotient algebra J:n/6n. The inverse closedness property with respect to Moore-Penrose invertibility (Corollary 2.18) entails that Rn(an) + 6n is already Moore-Penrose invertible in (-An + 67)/6n, which, by the third isomorphy theorem, implies the Moore-Penroseinvertibility of Rn(an) + An f3 6n in An/(An ~3 67). Finally, from Corollary 1.68 we infer that .A7 f3 67 = (.A t3 6)7, which involves the Moore-Penroseinvertibility of the coset Rn(a,~) + (.A f3 6)n in An/(A N 6)7. Thus, there is a sequence (b,~) ~ A as well as sequences

2.2. ALGEBRAIC (g(~l)),...,

101

CHARACTERIZATION

(g(n4)) e A n 6 such

(2.26)

By hypothesis, the canonical homomorphism~r from A onto A/(A ~ ~) is fractal, i.e. there is a homomorphism% such that 7r = ~r~ Rv. Applying ~rv to the identities (2.26) gives the Moore-Penroseinvertibility of the coset zr(an) with the Moore-Penrose inverse ~r(an) + = ~r(bn). Hence, (an) stably regularizable. (b) Let (a,~) + ~ be Moore-Penrose invertible in ~’/~, and (bn) ((an) + + with a sequence (b n) E v. Repeating the arguments of p art (a) of this proof, we see that (an) + A N ~ is Moore-Penroseinvertible A/(‘4 N ~). Let (b~) E .4 be a sequence such that (b~) + .4 ~ G is Moore-Penrose inverse of (an) + A ~ G in A/(A N ~). From Theorem 1.71 we know that

and from the uniqueness of the Moore-Penrose inverse in ~’/~ we conclude that [[bn - b~l[ --~ 0. Hence,

which finishes the proof. To illustrate the efficiency of this theorem for a concrete approximation method we consider once more the finite section method for Toeplitz operators, i.e. we specify 5r and ~ accordingly and choose .4 = S(C). This algebra is fractal as we have seen in Corollary 1.70. Theorem 2.29 Let (An) ~ S(C) be st ably re gularizable se ~ > 0 is sufficiently small, then the limit lim condA~ = lim IIAn,~IIIIA~+,~II exists, and it is equal to

max{llW(An)ll, II (A)II}" max{llW(An)+ll,

quence. I]

102

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Proof. As IIAn - An,s]l ~ 0 (by definition) and 6 C_ $(C) (by Theorem 1.53) we see that the sequence (An,s) belongs to the fractal algebra S(C) and, hence, by Theorem 2.28, lim condAn,s = lim IIAn,~ll IIA+~,~II = II(An,s) In Section 1.4.2 we saw that H(A~,~) + ~H = max{HW(A-,~)~, H~(A~,~)~),

Since IIA~,~- A.II ~ 0, it is ~(A~,~) = ~(An), and applying of the identities A~,eA~,~A.,~

immediate that W(An,~) ~ W(A~) the *-homomorphisms W and ~ to

+ ~ + = A~,~, A~,~An,~A.,¢

=A ~,~,

(An,~A~,~)* = An,eA~,~, (A~,~An,¢)* = A~,~ An,~ (which is justified since (An,e) belongs to 6(C)) we W(A~,~) = W(A~,~) + = W(A~) + + and ~(A~,~)

= ~(A~)

which yield8 the assertion. For a further specification, we denote the e-regularizations of the Toeplitz matrices P~T(a)Pa by Tn,~(a). Corollary 2.30 Let a be a continuous ]unction on ~ with a(t) ~ 0 ]or all t ~ ~. I] e > 0 is su~ciently small, then the limit lira condTn,e(a) = lira IIT~,¢(~)II

Proof. If 0 ~ a($) then the sequence (PnT(a)Pn) is stably regularizable by Theorem2.27, and thus the existence of the limit ~ well as the equality lim condTn,e(a) with ~(t) = a(1/t)

m~ (llT(~)ll, II T(a)ll}" ma x(llT(~)+ll, are consequences of Theorem 2.29. The equalities

IIT(~)II : IIT(a)ll~ndIIT(~)+II = IIT(a)+t f~om theiae~titiesT(~) I foUo~ CT(U)C ~ CT(a)*C and

+ = C(T(a)*)+C= C(T(~)+)*~, T(a)+ = (CT(~)*C) where C is the operator (xu) ~ (~) of conjugation

2.2. ALGEBRAIC 2.2.5

Difficulties

CHARACTERIZATION with

Moore-Penrose

103 stability

Weconclude this chapter by a few - partially already mentioned - examples to illustrate somepeculiarities of Moore-Penroseinvertible elements of C*algebras and of Moore-Penrose stable approximation sequences. Example 2.31 A continuous function f : [0, 1] -~ C is invertible (in C[0, 1]) if and only if f(t) is invertible (in C) for every t E [0, 1]. In that sense, invertibility in C[0, 1] (and in every commutativeC*-algebra) is local property. On the other hand, all values of the function f(t) = are Moore-Penrose invertible in C, whereas the function ] fails to be MoorePenrose invertible in C[0, 1]. Thus, even in the commutativesetting, MoorePenrose invertibility is not a local property. ¯ Example 2.32 Consider the n × n diagonal matrices An := diag(0,1,1,...,1),

Bn := (1/n,l,1,...,1).

The sequence (An) is Moore-Penrose stable, whereas (Bn) fails to be so. Since both sequences belong to the same coset modulo G, we conclude that Moore-Penrosestability of a sequence (An) is not an invariant o] the coset (An) + 6. Example2.33 For r E (0, 1), let T(ar) be the Toeplitz operator with erating function at(t) = -1 -r. Obviously, all fini te sect ions PnT(a,.)Pn are invertible, and T(ar) and T(dr) are Fredholm operators. But T(ar) not invertible (the sequence (1,r, r2,...) ~ 12 lies in the kernel of T(ar)). So, Polski’s theorem implies that the sequence (ll(PnT(ar)Pn)-IPnll) is unbounded and, hence, (P,~T(ar)Pn) is not a Moore-Penrose stable sequence. On the other hand, if ao(t) := -~, t hen neither T(ao) nor PnT(ao)P,~ are invertible (the sequence(1, 0, 0,...) ~ 12 resp. the vectors (1, 0,..., Cn lie in the kernels of T(ao) resp. PnT(ao)Pa), but both T(ao) and P~T(ao)Pn are Moore-Penrose invertible with T(a~~) and PnT(a~i)Pn their Moore-Penrose inverses, respectively. Thus we have (PnT(ao)Pn)+ ~ T(ao)+ strongly, i.e. the sequence (P,~T(ao)Pn) is Moore-Penrose stable. What results is that, in every neighbourhood of the Moore-Penrose stable sequence (PnT(ao)Pn), there is a sequence of the form (P~T(a~)Pn) with r > 0 which fails to be Moore-Penrose stable. Thus, Moore-Penrose stability is not stable with respect to small perturbations. ¯ These examples indicate that, concerning Moore-Penrosestability, one cannot expect results of the same generality as for stable regularizability or usual stability. Only in Section 6.1.4 we will formulate some sufficient conditions for Moore-Penrosestability.

104

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REGULARIZATION

Notes and references Basic facts on generalized inversion of matrices and linear operators can be found in the monographs Mitra/Rao [108] and Nashed [111] (from where we also took the motto of the chapter). For generalized invertible elements in C*-algebras and Banach algebras we refer to the papers Harte/Mbekhta [82], [83] and Roch/Silbermann[150], respectively. For the idea of regularization of ill-posed problems see H~mmerlin/Hoffmann[81] and Hofmann [87], for example. All the enclosed material on the spectral theory of selfadjoint operators on Hilbert space we took from the excellent monograph Gohberg/Goldberg/Kaashoek [65]. The idea of considering stably regularizable approximation sequences instead of Moore-Penrose stable sequences goes (as far as we know) back to the 1974 paper Moore/Nashed [109], where norm-convergent approximations of Fredholm integral equations of second kind are studied. In [163], one of the authors raised the analogous problem for the finite section method for Toeplitz operators, and he succeeded in deriving Theorem 2.24. In the forthcoming papers [147], [148] and [149], the approach of [163] has been extended to further classes of approximation methods (finite section methodand polynomial collocation for singular integral operators, for instance), and the connection between the stable regularizability of a sequence and the Moore-Penroseinvertibility of the coset of that sequence in the algebra ~/G as presented in Section 2.2.2 was established. The results of Section 2.2.4 are new.

Chapter 3

Approximation

of spectra Having discussed linear equations and least squares, we now direct our attention to the third major problem area in matrix computations, the algebraic eigenvalue problem. Gene H. Golub, Charles F. Van Loan

The main theme of this chapter is the determination of the cluster or limiting sets of the eigenvalues of the approximation operators, thus answering the question whether the spectrum of a given operator A can be determined approximately by computing the eigenvalues of certain approximations A,~ of A. Wewill see that this is in general possible if A and the A~ are selfadjoint, whereasthis idea can fail drastically in case A is a non-self-adjoint operator. Wewill not spend much time with the self-adjoint case here since this will be the subject of Chapter 7. Rather we want to examine two alternative approaches to the approximation of spectra, namely via pseudospectra and via numerical ranges. Both pseudospectra and numerical ranges can be viewed as approximants of the usual spectra, and both exhibit a muchbetter asymptotic behaviour than the latter.

3.1

Set

sequences

Westart with recalling some basic and elementary facts on set functions. As a general reference we recommendSection 28 in Hausdorff’s classical monograph [84]. 105

106

CHAPTER 3.

APPROXIMATION

OF SPECTRA

3.1.1 Limiting sets of set functions A set function is a mapping which is defined on a metric space and takes values in the set of all subsets of the complex plane C. Especially, a set sequence is a set function which is defined on the natural numbers. If (An) is an approximation sequence, then the mapping n ~+ a(An), which assigns with every n the spectrum of An, is a set sequence in this sense. Definition 3.1 (a) Let (M,~),~=I be a set sequence. The partial limiting set or limes superior lim sup Mn (resp. the uniform limiting set or limes inferior lim inf Mn) of the sequence (M,~) consists of all points m E C which are a partial limit (resp. the limit) of a sequence (ran) of points mn~ (b) Let M be a set function defined on a metric space S. The function is upper semi-continuous at so ~ S if, for every open set U C C containing M(so), there is a neighborhood V of so such that M(s) C_ U for each s Observe that the partial limiting set lim sup Mnis non-emptyif infinitely many of the M~ are non-empty and if tOnM~ is bounded, whereas the uniform limiting set can be empty even under these restrictions as the trivial example M~= {(-1) ~} shows. Proposition 3.2 Both the partial and the uniform limiting sequence are closed.

set of a set

Proof. Let us check this.for the uniform limiting set. Suppose z* belongs to clos liminfMn. Then, for every positive integer k, there are points zk ~ lim inf Mn such that Iz*-zkl < 1/k. Choose sequences (m(nk))~=l with r~t(n k) e M~and lim~-~ m(nk) = Zk, and fix numbers N1 < N2 < N3 < ... such that

I-~) - z~l < 1/k for a~l n _> g~.

Define a sequence (m~) by choosing m,~ ~ Mn arbitrarily if n < N1 and by setting m~ := m(nk) in case Nk ~ n < Nk+~. Then, for all n ~ [N~, N~+I), Iz* - m,~l O, and let L be the polynomial L(A) = bo + b~A + ... + bm~m wi¢h coe~cients in B. Let the coe~cient bm be invertible. Assume there is an open subset U of the complex plane such that L(A) is inve~ible and ~](L(A))-~]] ~ M ]or all ~ ~ U. Then ]](L(A))-’]] < M ]or all ~ e U. Proo£ Wesubdivide the proof into several steps; the third one being identical with Daniluk’s original proof of Theorem3.32. Step 1. Suppose there exists a A0 ~ U such that L(A0) = M. For the shifted polynomial ~, Q(~)

:= L(~+~o)

= a0 +a~+...+a~

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CHAPTER 3.

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OF SPECTRA

one easily checks that Q(A)is invertible, H(Q(A))-ill -~ Mfor all A E U-A0 (= the algebraic difference), and that II(Q(0))-IH = M. Moreover, a,~ = b,~ and a0 = Q(0) are invertible. Further, let (~ refer to the monic polynomial defined by Q(A) O,(A)am. Clearly, (~ (A) is inv ertible if only if so is Q(A). Step 2. Here we recall some facts concerning a special representation called linearization of the inverse of a monic operator polynomial. For details see [152], Chapter 2, Theorem2.5.2. For the desired representation, we introduce the following vectors of length m and matrices of order m x m with entries in the algebra B: Xo := (e

Y :=

0 0 ...

0),

, T :=

I := diag (e e e ...

e),

0 0

0 0

e 0

-.. -..

--a a~n1 --ala~n 1 --a2a~n ~ ....

,

1am_la~n

i.e. T is the companionmatrix of the monic polynomial (~. Then, for all which are not in the spectrum of 8, ~)(~)-1 = Xo(~I - T)-IY. Since Q(A)-1 = a~(A)-1, this yields the representation Q(A)-~ = X(AI - T)-IY

(3.18)

with X = a~nlXo = (a~n 1 0 0 "- 0), which holds for all A such that Q(A)is invertible. Step 3. Beginning with this step, we think of B as an algebra of linear bounded operators acting on some Hilbert space H, which is possible due to the GNSconstruction (Theorem 1.48 (a)). Thus, T is actually an operator acting on the orthogonal sum of m copies of H. Since Q(0) is invertible, the operator T is invertible; hence, for all A the disk [AI _< r with sufficiently small radius r, (T

- AI) -1

: E "~JT-J-I’ j=o

whence X(T - AI)-~Y

= E AJXT-J-~Y" j=O

3.3.

PSEUDOSPECTRA AND THEIR LIMITING

131

SETS

Thus, for all f E H and IAI _< r,

IIQ(A)-~fll~ = IIX(T-AI)-~Y fll ~ = ~ AJ~k (XT-J-~Y f, XT-k-~y f). j,k>_o Integrating this identity with respect to ,k against the circle IAI = r yields

1 fl~ IIQ(A)-~fllzldAI = ~-~rU~IIXT-~-xYflIU. 27rr i= r

j>o

Because IIQ(~)-lfll~ MllYllby hypothesis, this yields for all j _> 1 the estimate

IIXT-IYfll ~ ÷ r2~llST-J-~Yfll2 0. Since, by assumption, IIQ(0)-~II II XT-~YII = M,there is fe in H with norm 1 such that

(3.19) an

ilXT-~Yf~ll~ > Me _ ¢2, which together with estimate (3.19) shows that M2 _ ~2 +

r~llXT-~-~yf~llZ < 2M

or, equivalently, IIXT-J-~Yf~II

< er -~ for all

j > 1.

Step 4. For brevity, set cj = -aia~n~. The operator Co is invertible, and it is easy to check that the inverse of the companionmatrix T is a companion matrix again:

i

0 e

... ...

0 0

0 0

0

...

e

0

.

Computingstep by step the last columns of the operator matrices T -2, T -~ -~, + c~lc~T T-I(T -2 + c~lcl T-l) T= 2, T-3 + c’~lcl -3 -2 -~ , T A- c~c~T + c~c2T

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CHAPTER 3.

APPROXIMATION

OF SPECTRA

we obtain

1(i

~ cg 0 0

respectively, what finally shows that the last columnsof the matrices T -m + c~c~T -m+l -~ + ,... + c~Cm_~T -m-~ -m -2 T + c~lc~T + ... + c~cm_~T are as follows:

Consequently, X(T-’~-~ + c~clT -’~ +...

+ c~ICm_IT-2)Y = a~nlc~ 2 ~. --~ a~la,~a~

(3.21) Step 5. Identity (3.21) implies that Ila~a.~a~f~ll

= IIX(T -m-~ + c~oT -m +...

+

and since Xc~cj -- a~c~cja,~X for all j, we further

= II(XT-m-~y

+ a~n~c~clamXT-~Y

conclude that

+...

O, lim sup a(~) (n,~) = lim inf (e) (L,~) =(e) (P) Proof. Let L stand for the polynomial L(~) := ((Tn(fo)) Due to the fractality

+ ~) + ((Tn(fl))

+ ~)~ +... + ((Tn(fm))

of the algebra S(C), and by Theorems3.36 and 3.38,

lira sup a(e) (Ln) = lira inf a(~) (Ln) = a(~)g For the image of L under the homomorphismsWand IYVone easily finds W(L(A)) = T(fo) + T(fl)A +... + ~ = P(A), I~(L(A)) = T()~) T( fl)A +. .. + T( fm)A"~ =: which, together with Corollary 1.58 and the inverse closedness of S(C)/G in )r/6, yields lim sup a(~) (L,~) = lim inf (~) (L,~) =(~) (P)kl a(~)(/5). Finally, from identity (1.32) we infer that "~ D(A) = CT(fo)*C + CT(f~)*C)~ +... +CT(fm)*CA = C(T(]o) T(fl),~ +. .. + T(.f,~)~")*C = CP(~)*C and, hence, a (~) (P) = (~) ( /5).

134

CHAPTER 3.

APPROXIMATION

3.4 Numerical ranges and their

OF SPECTRA

limiting

sets

The (several kinds of) numerical ranges provide further examples of (upper) spectral approximants which share their good asymptotic behavior for all approximation sequences with the e-pseudospectra. 3.4.1

Spatial

and

algebraic

numerical

ranges

Let H be a (complex) Hilbert space with inner product (., .), and let a linear bounded operator on H. The spatial numerical range SNH(A) of A is the set SNH(A):= ( (Ax, x) : x e H, Ilxll = Thus, SNH(A)consists of all Raileigh quotients (Ax, x)/(x, where x ~ 0. In what follows we want to consider the asymptotic behaviour of the spatial numerical ranges SNc~ (An) of a sequence (An) of approximation operators, and our goal is to identify the limiting set lim SNc~(A,~) with a subset of C which depends on the coset (An) + 6 in 9r/6 only. Therefore we need an analogue of the spatial numerical range for elements of a C*algebra. So let B be a C*-algebra with identity element e. The algebraic numerical range ANu(b) of an element b E B is the set ANu(b) := {~o(b): ~o e B*, I1~11 = 1 and ~o(e) = The special linear functionals ~ on 13 figuring in this definition are called the states of .13. While the spatial numerical range is the more familiar object, the algebraic numerical ranges are distinguished by the more pleasant properties. So we start with a brief account of the latter. For all proofs and further details we refer to the monographs[29] and [76]. The set of all states of a C’algebra B is called the state space of B. The state space is a convex and - with respect to the *-weak topology which we will discuss briefly in Section 4.1.1 - compact subspace of the dual space B* of B. Consequently, one has Proposition 3.40 Let 13 be a unital C*-algebra and b ~ 13. Then AN~(b) is a convex and compact subset of the complex plane. If C is a C*-subalgebraof 13 which contains the identity, then the restriction mapping maps the state space of 13 onto the state space of C (the surjectivity a consequence of the Hahn-Banach theorem). What results is

being

3.4.

NUMERICAL RANGES AND THEIR LIMITING

SETS

135

Proposition 3.41 Let B be a unital C*-algebra and C be C*-subalgebra of B containing the identity. Then, for every c E C, AN~(c) = ANc(c). Thus, C*-algebras are ’inverse closed’ with respect to numerical ranges, and we will often write AN(b) in place of AN~(b). Further one has the following result in which conv Mrefers to the convexhull of the set MC_ C. Proposition 3.42 If13 is a unital C*-algebra and b ~ 13, then conva(b) AN(b). If b is moreover normal, then conv a(b) = AN(b). Therefore one can consider algebraic numerical ranges as majorizations of spectra. Moreover, it turns out that manyof the properties of algebraic numerical ranges and spectra are quite similar, but as a rule, the results for numerical ranges are a bit stronger. Here are a few further examples. Proposition 3.43 Let 13 be a unital C*-algebra. (a) The mapping b ~+ AN(b) is upper semi-continuous

b

(as the mapping

is).

(b) The numerical radius r(b) := rad AN(b)satisfies the inequalities ~llbll r(b) 0, there is a vector (Xk) ¯ ~Hk with norm 1 such that

Let 1~ C_ N denote the set of all k with Xk ~ O, choose arbitrary elements Yk ¯ Hk with [[Yk[[ = 1 for k ¢ l~, and set x~ := { xk/[[x~[[ ifif Yk Then

k=l

k~M

138

CHAPTER 3.

APPROXIMATION

OF SPECTRA

and, consequently, m

- ~ IIzkll=(alex[.,x[.}H~ le=l

Because IIxkll= >_0 andF~_IIIx~ll= -- II(z~)ll~,~-- 1, this

showsthat m can be approximated by convex linear combinations of points (alezle, xle) U~SNH~(ale) as closely as desired. Hence, and by Theorem3.45, me clos conv t3le SNH~(ale) C_ clos cony Uk ANL(H~)(ak), and employing inverse closedness once more we find mE clos conyt_Jle ANc~(a~).

(3.24)

Since tAkANc~(ak) is bounded (the radius of this set is not greater than [[(ak) [[~-), and since clos conv M= cony clos Mfor every bounded subset Mof the complex plane, (3.24) is just the assertion. Proof of Theorem 3.46. We know from Proposition

3.43(d)

that

AY:r/g((a,~) + 6) = VI(9,,)Eg AN.r((a,~) whence, in combination with Theorem 3.48, AN.r/6((an) + 6) = f~(~)eg convclos t2n ANc,, (an + gn).

(3.25)

For every k E N, choose an m~ ~ ANc,, (an) and define a sequence (G(nk)) ~ 6 by g(nk)={o--an+token if if nn>k. k ANc,,(a,~). Applying Lemma3.47 to the sets Mk:: clos

~n>_k

ANc,, (a,~) gives

AN.r/g((an) + 6) conv ¢q~o=~ clos I.. Jn>_k ANc , (an ), and the set on the right hand side coincides with conv lim sup ANc,, by Proposition 3.5, which proves one half of the theorem.

3.4.

NUMERICAL RANGES AND THEIR LIMITING

SETS

139

For the second half, let m E limsup ANc~ (an). Then there exist a subsequence (nk) C_ N tending to infinity as k -+ eo as well as points mk ~ ANc,,~ (a~) which converge to m as k --~ oo. Further choose states Ck of Cn~with Ck (an~) in k. Let ~ stand for the linear subspace of ~" consisting of all sequences (bn) := o~(an)-b ~(en) -twhere en is the identity of C~, a,/~ ~ ~2, and (gn) runs through the ideal 6. If (bn) ~ ~, then the limit limk~ooCk(bn~)exists. Indeed, it is clearly sufficient to check the existence of that limit for the generating sequences of £, for which one has Ck (a~) = m~ -~ m, Ck (e~) = 1 -~ 1

Thus, there is a linear functional ¢ on ~2 defined by ¢((bn)) = lim Ck(bn~) k--~oo

which mapsthe sequences (an), (en) and (gn) into m, 1 and 0, respectively, and which is continuous with norm 1: I¢((b~))l -- IlimCk(bn~)l < suplCk(bn~)I g supllb,~l k

k

Using the Hahn-Banachtheorem, one can extend ¢ to a linear functional with norm 1 on all of 9v, and we denote this extension by ¢ again. Since II¢ll = ¢((en)) ---- 1, this functional is a state r. Further, the ideal lies in the kernel of ¢ by construction, so one can define a functional ¢ on 5r/6 by

¢ : c, (cn) + Clearly, ¢((e,~) + 6) = 1 and ¢((an) + 6) = m, and ¢ is continuous norm 1. Thus, ¢ is a state of ~-/6 which implies that m and, consequently, lim sup ANc,~(an) C_ AN.r/g( (an) Finally, algebraic numerical ranges are convex as we knowfrom Proposition 3.40. Thus, cony limsup ANcn(a,~) C_ AN.r/g( (a,~) 6) which verifies the second ,half of ~he theorem.

¯

140

3.4.3

CHAPTER 3.

The case of fractal

APPROXIMATION

OF SPECTRA

sequences

Our final goal in this section is the consequences of the fractality of the sequence (an) for the asymptotic behaviour of the partial and uniform limiting sets of the numerical ranges of an. Theorem 3.49 Let fit be a fractal C*-subalgebra of ~ which contains the identity of ~. Then, for every (an) ¯ lim sup ANcn Proof. The inclusion lim inf ANc. (an) C_ lim sup ANc~ (an) is obvious. For the reverse inclusion observe that the numerical ranges ANc~(an) are convex, and that the uniform limiting set of convex sets is convex again. Thus, the inclusion lim sup ANc, (an) C_ lim inf ANcn(an) holds if and only if conv lim sup ANc~(an) C_ lim inf ANon(an). So, by taking into account Theorem 3.46, what we have to prove is that AN~:/g((a,~) +~) C_ liminfANc~(an) for every (an) ¯ fit. From the inverse closedness (Proposition 3.41) we conclude that Ag~:/~((an) + ~) = AN(.,t+~)/~((an) and the third isomorphy theorem for C*-algebras (Theorem 1.47) further implies that AN.r/~((an) + ~) = ANA/(An~)((a,~) + Let m ¯ ANA/(.an~)((a,~) + A ~ and let ¢ be a sta te of A/(A ~6) such that m = ¢((an)+fit~) = ¢(~r(an)) where we use the notations of 1.6.1. Since ~ is fractal, one has m = ¢(~,Rv(an) ) for every monotonically increasing sequence ~]. It is obvious that ¢ o ~r, is a state on fitv = Rufit. Hence, rn ¯ AN.a, (Ru(an)) whence, by Theorem 3.48, m ¯ convclos ~n ANc,(.)(a,~(n)) for every ~?.

(3.26)

Nowassume there exists an m ¯ AN.r/g((a.n)+~) which does not belong to the uniform limiting set of the ANc~(an). Then there is a strongly monotonically increasing sequencey* C_ N such that dist (m, ANc,. (~) (a,~. (,~))) d > 0 for all n.

3.4.

NUMERICAL RANGES AND THEIR LIMITING

SETS

141

Due to the compactness and convexity of ANc,.(~ (an.(n)) (Proposition 3.40), there exist points mn. (n) ANc ,.(n~ (an * (n)such that [m- mn.(n)l --- dist (m, ANc,.(~)(a~.(,~))), and these points m~*(n) are unique for every n. From Proposition 3.43(b) we further knowthat all ~n*(n) lie in the disk around the origin with radius r := sup I]an[[ -- [[(an)[[y, hence, there is at least one cluster point m* the sequence (ran. (n))neNNow,given ¢ > 0, consider those ~*(n) for which the m~*(n) belong to the e-neighborhood U of m*. These 7" (n) single out an (infinite) subsequence ye of 7*. A little thought reveals that for this subsequence dist (m, conv clos ~J,~(n) ANc,~(,)(and(n))) _> d/2

(3.27)

if only e is small enough. Since (3.26) holds for every sequence (particularly for ~), (3.27) contradicts (3.26). Uniform limiting sets of convex sets are convex again, hence, from Theorem 3.46, Proposition 3.4 and the theorem just proved we get: Corollary 3.50 Let A be a fractal C*-subalgebra of J: which contains the identity. Then, for every sequence (an) ~ lim sup ANc~(an) = lim inf ANc~(an) = ANy/~((an) and limradANc~ (an)

=

radAN~:/~((an) +

To illlustrate these results, we consider once again the finite section method for Toeplitz operators, i.e. we let .4 = 8(C) with accordingly chosen algebras ~" and 6. Theorem 3.51 If (An) ~ $(C), lim inf ANc~×~ (A~) li ra in f SNc~ (An) = lira sup ANc,×~ (A~) li m sup SNc, (A n) = Ag:~/g((An)

+ ~) = ANL(~)×L(~)((W(An),

(*2 cony (ANL(~:)(W(An)) tA ANL(~:)(IV(A~))) (**) =conv (clos SNt: (W(An)) cl os SN~: (I V(An))). Proof. The identities in the first two lines are consequences of Corollary 3.50 and Theorem 3.45. The identity (.) can be checked by repeating the

142

CHAPTER 3.

APPROXIMATION

OF SPECTRA

arguments of the proof of Theorem 3.48, where we verified an analogous result for sequences instead of pairs of operators, and by taking into account the convexity of algebraic numerical ranges. The identity (**) follows from Theorem 3.45. ¯ Let us emphasize that the real importance of Theorems 3.49 and 3.51 lies in the fact that they allow us to determine the limiting sets of numerical ranges for any element of a fractal algebra .4. In the case of the pure finite section method, these limiting sets can be identified mucheasier. Theorem 3.52 Let H be a Hilbert space, (P,~) a sequence of orthogonal projections on H which converge strongly to the identity operator, and A E L(H). Then lim sup SNIm Pn (P,~APn) = lim inf SNIm pn (P,~APn) = clos SNH(A). Proof. Let rnn ~ SN~mp~(P,~AP,~), and choose a vector x~ ~ ImP~ with Ilxnll = 1 such that mn= (PnAPnxn,x,~). Then, clearly, mn= (APnxn, P,~xn) and IIPnxnll = 1, whence mn E SNH(A) and, consequently, li~n sup SNImp~ (PnAP~,) C_ clos SNH(A). Let, conversely, m ~ SNH(A), and let x ~ H be a vector such that Ilxll = 1 and m= (Ax, x). Then P~x--~ x and IIP,~xll --~ []xll = 1 as n --~ cx3, which in particular shows that Pnx ~ 0 for all sufficiently large n. For these n, set xn := Pnx/]IPnxlI. Then x n -~ x as n -~ o¢, and (P~AP~x~,’ ’ x~) (Axn,xn)-+(Ax, = ’

x)=m as

Hence, rn ~ liminf SN~mp~(P~AP~). Specifying this result to the finite section method for arbitrary Toeplitz operators we obtain Corollary 3.53 Let a ~ L~(~), and let Pn refer to the orthogonal projection from 12 onto the subspace of all (Xk)k=O with Xk = 0 for > n. Then lim sup SNc, (P,~T(a)Pn) = lim inf SNc, (P~T(a)P,~) clos SNt= (T(a)). Taking into account Proposition 3.42 and Theorem3.45 and recalling that T(a) is just the compression of the Laurent operator L(a) to the subspace 12 = /2(Z+) of/2(Z), one can moreover identify closSNt:(T(a)) ANLq:) (T(a)) as the convex hull of the essential range of the function a ~ L~(~’). The details are left to the reader.

3.4.

NUMERICAL RANGES AND THEIR LIMITING

SETS

143

Notes and references Section 3.1 - 3.2: Most of the material presented in these two sections is well known.[30], [130] or [154] ca serve as basic references. Weare grateful to Torsten Ehrhardt for bringing Proposition 3.6 and Theorem3.7 to our attention; both results can be found in [84]. The characterization of the limiting sets of spectra of normal sequences and of the set of the singular values is taken from [146]. Theorem3.19 is new; it might yield another explaination of the figures in Section 3.3 and in [15], [27] and [129] showing the asymptotics of the pseudospectra of Toeplitz operators. A proof of Lin’s celebrated theorem can be found in [104], Chapter 19. Section 3.3: As far as we know, Landau[99], [100] was the first to introduce and to study the behavior of pseudoeigenvalues and pseudospectra in the context of Toeplitz and Wiener-Hopf operators. The recent popularity of pseudospectra of Toeplitz operators has its roots in the the papers Reichel and Trefethen [129] and BSttcher [15]. The alternative description of pseudospectra observed in Theorem3.27 belongs to our colleagues Tilo Finck and Torsten Ehrhardt and is first published in [147]. The characterization of the e-invertibility of a linear bounded operator via its e-kernel and e-range is perhaps new and published here for the first time. Theorem3.32 is due to Daniluk; its proof and some commentson the history of the problem can be found .in [15]. The only thing what is new in Theorem3.31 is its formulation; the main steps of its proof are taken from [15], where the special case of the limiting sets of pseudospectra of finite sections of Toeplitz operators is considered. The fact that the latter theoremcarries over to the general case without essential changes points out once more the pioneering role of the finite section method for Toeplitz operators. All the results concerning the pseudospectra of operator polynomials can be found in [140]. Section 3.4: The results of Section 3.4.1 are classic. The monographs [29] and [76] provide excellent introductions into this field. The HausdorffToeplitz theorem is in [45] and [80]. The material presented in Sections 3.4.2 and 3.4.3 is taken from [138] and [141].

Chapter 4

Stability concrete methods

analysis for approximation

Today,a student cannot get very far in the C*-algebraliterature withoutbeing somewhat familiar withthe lexicon of examplesthat now dot the landscape. KennethR. Davidson Roughlyspeaking, the mainresult of the previous three chapters is that, for an arbitrary approximation sequence (A,~), the basic problems mentioned in the introduction are equivalent to certain problems for the coset (An)+~ of the sequence (An) in the algebra ~" of all bounded sequences factored by the ideal 6 of the zero sequences. Specifically, the stability of (An) corresponds to the invertibility (An) + 6, the stable regularizability of (An) to the Moore-Penroseinvertibility of (An)+~, and the limiting set of the spectra (pseudospectra, numerical range) of the An is related with the spectrum (pseudospectrum, numerical range) of (An) + G. Moreover, both the invertibility, Moore-Penroseinvertibility and the spectrum (pseudospectrum, numerical range) of (An) remain invariant when passing from ~’/~ to a certain C*-subalgebra of ~-/~ which contains (An) ÷ G and the identity element. Hence, one can freely choose a convenient C*-subalgebra of ~’/G in which the above mentioned problems for (An) + 6 will be considered. So what one needs is a precise description of C*-subalgebras of 5r/~ 145

CHAPTER4.

146

STABILITY ANALYSIS

which contain the approximation sequences one is actually interested in. As we have seen in Section 1.4.4, the desired detailed knowledge about a certain subalgebra .4 of ~’/6 can be attained by investigat.ing the invertibility problem for every coset (An) + ~ in .4, and exactly this will be subject of the present chapter. That is, we will consider several C*-subalgebras of 5r/~ which are generated by some ’interesting’ approximation sequences, and we will answer the stability problem not only for the ’interesting’, but for arbitrary elements of these algebras, which will then enable us to get also information about stable regularizability or spectral asymptotics. Thereby, our main emphasis will be on introducing and motivating some technical ingredients such as local principles and lifting theorems which apply to the analysis of several concrete subalgebras of the basic algebra ~’/~.

4.1 Local principles Local principles can be viewed as far-reaching generalizations of partitionof-linity-techniques. Westart with one of the simplest local principles (which we have already encountered several times): the Gelfand theory commutative C*-algebras. There are several ways to generalize this theory to the non-commutative setting (which is important for us because nonnormal approximation methods generate non-commutative subalgebras of ~’), and one of these generalizations is the local principle by Allan and Douglas, which fits perfectly to our purposes. Finally, in Order to illustrate the application of this local principle, we will derive the Fredholmtheory for Toeplitz operators with piecewise continuous generating function. 4.1.1

Commutative

C*-algebras

Recall from Section 1.4.1 that every commutative and unital C*-algebra 91 is *-isomorphic to the algebra C(X) of all continuous complex-valued functions on some compact space X. Weare going to outline the proof of this result. Let 9/ be a commutative C*-algebra with identity element e ~ 0. A character of 91 is a non-zero homomorphism from 91 into the algebra C of complex numbers. Further we call an ideal ~ of 91 maximal if ~ ~ 91 and if there is no ideal 3 of 91 being different from J~ and 91 such that J~ C_ ~ C_ 91. Proposition 4.1 Let 91 be a commutative C*-algebra with e ~ O. (a) I] a E 91 and W is a character of 91, then W(a) ~ a(a). (b) Every character o] 91 is unital, continuous with norm1, and symmetric.

4.1.

LOCALPRINCIPLES

147

The proof is elementary. One starts with showing that W(e) -= 1, which implies (a). From a) one c oncludes t hat I W(a)l ~_ p(a) ~ _ I lall, hence, W is continuous with norm 1. For the symmetryone has to take into account that self-adjoint elements of C*-algebras have real spectra. ¯ Proposition 4.2 Every proper ideal of a C*-algebra 92 lies in a maximal ideal. The proof relies on a standard application of Zorn’s lemma. The next result states that there is a one-to-one correspondence between the characters and the maximalideals of 92. Proposition 4.3 Let 92 be a commutative C*-algebra with e # O. Then the mappingW ~ Ker W is a bijection ]rom the set o] the characters onto the set o] the maximalideals o] 91. Proof. If Wis a character then, by the third isomorphy theorem, 92/KerW ----

ImW= C,

hence KerWis maximal. Let, conversely, ~ be a maximal ideal of 92. A little thought shows that ~ must be closed (otherwise the closure of J~ would give an ideal which lies between ~ and 92). Weclaim that the quotient algebra 92/~ is isomorphic to C. Indeed, let a + J~ be a non-zero element of 92/~, and consider the subset 92 ¯ a + ~ in 92. The set 91. a + ~ is an ideal in 92 which contains ~, but is strictly larger than ~. Since ~ is maximal, this involves that 92. a + ~ = 92. Hence, there are elements b E 92 and k E J~ such that ba ÷ k = e which implies that every non-zero element a + ~ of 92/~ is invertible. Further, the closedness of ~ entails that 92/J~ is a C*-algebra again and, thus, every element a + ~ of P2/~ has a non-empty spectrum, i.e. there is a A ~ C such that a- Ae+~ is not invertible. Since 0+~ is the only non-invertible element of 92/J~, this involves that a + ~ = Ae + J~, i.e. every element of 92/J~ is a multiple of the identity. Consequently, 92/~ ~ C (which is also known as the Gelfand-Mazur theorem), and the canonical homomorphism from 92 onto 92/~ induces a character of .4 having J~ as its kernel. ¯ Thus, characters and maximal ideals can be identified, and we will make use of this identification throughout what follows. Let M(92)denote the set of all maximalideals of the C*-algebra ~. Since maximalideals correspond to characters, and characters are special linear functionals, one can think of M(92).as a subset of the dual space 92* of 92. Wewill use this observation in order to define a topology on

148

CHAPTER 4.

There are several natural topologies on the one which is most important in our context is convergence or the *-weak topology. This is the which all mappings 92* -~ C, ] ~-~ ](a) with Equivalently, a neighborhoodbase of a point f ¯ U~,~2..... ~,~(f) - (g ¯ 91": If(ai)-g(ai)l

STABILITY

ANALYSIS

dual space 91" of 91. That the topology of pointwise weakest topology on 91" for a ¯ 91 become continuous. 91" is provided by the sets < ~ for i =

where k runs through the positive integers, the ai through 91, and ~ through the positive reals. The importance of this topology is a result of the following theorem. Theorem 4.4 (Banach-Alaoglu) The unit ball o] 91" is compact with respect to the *-weak topology. For a proof see [127], TheoremIV.21. Moreover, the *-weak topology is Hausdorff: Given distinct points ] and g of 91", choose a E 91 such that f(a) ~ g(a) and set ¢ :---- If(a) - g(a)]/3. Then the sets Ua,e(f) = {h P2*: If (a) - h( U~,~(g) = {h ¯ P2*: Ig(a) - h(a)l are open neighborhoods of f and g, respectively, which are disjoint. The maximal ideal space of the commutative and unital C*-algebra 91 is defined as the set M(91) provided with the topology which is induced the *-weaktopology of the dual 91" of 91. It is easy to check that the *-weak limit of characters is a character again, thus, M(91) is a closed subset the unit ball of 91", and the Banach-Alaoglu theorem implies: Corollary 4.5 M(91) is a compact Hausdorff space. Weclaim that M(P2) is just the compact X which figures in the GelfandNaimark theorem for the commutative C*-algebra 91. Wehave to associate with every a ¯ 91 a continuous function on M(91), which can be easily done: Given a ¯ 91 define a function Ga on M(91) by (Ga)(]) :-- f(a). It is evident from the definition of the *-weak topology that Ga is a continuous function on M(91) (and, conversely, one can showthat the topology on M(91) defined above is the weakest topology which makes every function Ga with a ¯ 91 continuous). The function Ga is called the Gel]and trans]orm of 91, and the mapping G : 91 -+ C(M(91)), a ~ is t he Gel]and tran s]ormation. One can now restate Theorem1.48 (b) as follows.

4.1.

LOCAL PRINCIPLES

149

Theorem 4.6 (Gelfand-Naimark) The Gelfand translormation is a *- isomorphism ~rom 91 onto Proof outline. Proposition 4.1 implies that G is a *-homomorphismand that (Ga)(f) C_ a(a) for every a. Let, conversely, h E a(a). Then there is an f e M(91) such that (Ga)(f) = h. Indeed, since a - he is not invertible, 91. (a - he) is a proper ideal of 91 whichlies in a certain maximalideal of (Proposition 4.2). If f is the character associated with this maximalideal then, evidently, ](a - he) = 0 resp. f(a) = h. Consequently, sup I(Ga)(f) I = sup Ihl = p(a),

feM(92)

)~ea(a)

and since p(a) = ]lall due to the commutativity of 91, we conclude that G is an isometry from 91 onto a closed subalgebra of C(M(91)). Further, this subalgebra separates the points of M(92), it contains the function f ~-~ and, together with a function f, the complex-conjugate f of f belongs to this subalgebra, too. Thus, the subalgebra coincides with all of by the Stone-Weierstraf~ theorem ([127], Theorem IV.10). Example 4.7 Let X be a compact Hausdorffspace clearly, the mapping

and 91 = C(X). Then,

is a character of C(X) for every fixed x ~ X. Conversely, every character of C(X) is of this form. Observe that for every proper closed ideal I of C(X) there is a point Xo ~ X at which all functions in I vanish. If I is maximal, then xo is unique, and 5xo is the character a~sociated with I. One can further show that the mapping X ~ M(C(X)),

x ~-~

(4.1)

is not only bijective but even continuous (i.e. a homeomorphism),and if X and M(C(X)) are identified via (4.1) then G is nothing but the identity mapping of C(X). 4.1.2

The local

principle

by Allan

and

Douglas

Nowwe turn over to non-commutative C*-algebras. The center of an algebra 92 is the set of all elements a ~ 92 which commutemultiplicatively with each other element of 92. Clearly, the center of a C*-algebra is a C*-algebra again, and the center contains the identity element in case 91 is unital. If the algebra 91 is commutative then its center coincides with

150

CHAPTER4.

STABILITY ANALYSIS

the algebra itself, whereas in case 91 = L(H) for a Hilbert space H, the center of 91 consists of the scalar multiples of the identity operator only. Thus, the center may be considered as a measure of non-commutativity of an algebra. Let now 91 be a C*-algebra with identity, and let ~ be a C*-subalgebra of the center of 91 which contains the identity. Then ~ is a commutative C*-algebra and, hence, *-isomorphic to C(X) with X = M(~) referring the maximalideal space of ~. For every maximalideal x of ~, let Ix denote the smallest closed ideal of 91 which contains x, i.e. le~ Ix stand for the closure in 91 of the set of all elements ~’~=1a~c~ where n ¯ Z+, a~ ¯ 91, and ci ¯ x ¯ M(E). Theorem 4.8 (Local principle by Allan/Douglas) Let 91, ~, M(E) and be as above. Then I~ is a proper ideal o] 91 ]or every x ¯ M(E), and the ]ollowing assertions are equivalent for every a ¯ 91: (i). a is invertible in 91. (ii) a ÷ I~ is invertible in the quotient algebra 91/Ix for every x ¯ M(~). For a proof’see [1], [26] (Sections 1.32 - 1.43), [50], or [77] (Section 1.4.4). ¯

Let us first see what this local principle says in the two simple extremal situations where 91 -- C(X) or 91 = L(H). For more interesting applications we refer to the following subsection for an application in operator theory and to the remaining sections of this chapter for an analysis of some concrete algebras of approximation sequences by means of the local principle. Example 4.9 Let X be a compact Hausdorff space and 91 = C(X). As we have already remarked, the maximalideal space of 91 can be identified with X, and the Gelfand transform G is the identical mapping then. Thus, given x ¯ X ---- M(C(X)), one has I~ = (f ¯ C(X) : f(x) 0} = x, and the quotient algebra 91/I~ is clearly *-isomorphic to the complex field C, the isomorphism sending the coset f ÷ Ix into f(x). The local principle says that a function f ¯ C(X) is invertible in C(X) if and only if f(x) is invertible in C for every x ¯ X, i.e. if f has no zeros on X. ¯ Example 4.10 Let H be a Hilbert space and 91 = L(H): Then the center of 91 is equal to CI (Schur’s lemma), which obviously implies that the maximal ideal space of the center consists of the zero ideal {0} only and that the Gelfand transform from CI onto C({0}) can be identified with the identical mapping. Consequently, there is only one ideal Io, which is just the zero ideal of A, and the local principle reduces itself to the triviality that a ¯ 91 is invertible if and only if a + {0} is invertible in 91/{0}. ¯

4.1.

LOCAL PRINCIPLES

151

Before coming to more interesting examples, let us summarize some prerequisities for the practical applicability of the local principle. (A) The local principle applies to invertibility problems in C*-algebras. So, one can only ’localize’ problems which are equivalent to invertibility problems (which suggests to try to localize the stability problem). (B) The algebra in which the local principle will be applied has to possess a sufficiently rich center (which excludes manyalgebras such as L(H) from a direct localization), but observe that also the algebra non-trivial center does not fit very well to the localization procedure since: (C) There must be a subalgebra in the center for which both its maximal ideal space as well as the local algebras modulo the ideals Ix can determined explicitely (which seems to be quite hard for Observe that this third point is often essentially simplified by the circumstance that the local principle allows us to localize over subalgebras of the center the description of which is sometimes much more easy than that of the complete center (see the application of the local principle in the following subsection). 4.1.3

Fredholmness of Toeplitz wise continuous generating

operators function

with

piece-

Thought both as an application of the local principle and as a preparation for the study of the finite section method, we are now going to examine the Fredholm properties of Toeplitz operators with piecewise continuous generating function. A function a on the unit circle T is said to be piecewise continuous if it possesses one-sided finite limits a(t ÷ 0) and a(t - 0) at every point t E and if a(t + O) = a(t) for all t E T. If one considers piecewise continuous functions not as functions but as elements of L°° (T) (as it is sufficient in the present section) then the latter condition can be ignored. One can show that a piecewise continuous function can possess an at most countable numberof discontinuities. The set of all piecewise continuous functions on ~ will be denoted by PC(T); this class is a C*-algebra under pointwise operations and the supremumnorm. For some concrete applications which involve Toeplitz operators with piecewise continuous generating function see Section 4.2.3. The Fredholm criterion for Toeplitz operators with piecewise continuous generating function is a surprising generalization of the corresponding criterion for the case of continuous functions. Let a ~ PC(T). This function mapsthe unit circle T into an, in general non-connected, set of curves, provided with a natural orientation (see Figure 4.1).

CHAPTER 4.

152

STABILITY

ANALYSIS

a(V) ~’~’~a(t~-

O)

a(t3+o) a

+o) a(t2 + O)

~ Figure 4.1: The oriented curve a This system of curves can be made to one closed oriented curve a ~ by joining a(t-0) to a(t + 0) by a straight line for every point t of discontinuity of a, and by naturally extending the orientation from a(’l~) onto all of a~. Theorem 4.11 (Widom, Gohberg, Krupnik) Let a E PC(Z). Then the Toeplitz operator T(a) is Fredholmon 12 if and only if 0 ~ ~. In t his c ase, ind T(a) = -wind a~ . Together with Coburn’s theorem (Theorem 1.29), this yields the following invertibility criterion: Theorem 4.12 Let a ~ PC(T). Then the Toeplitz operator T(a) is vertible on 12 if and only if it is Fredholmand wind a ~ = 0. Wewill not only outline the proof of Theorem4.11 but moreover derive a Fredholmcriterion for arbitrary operators belonging to the smallest closed subalgebra T(PC) of L(l 2) which contains all Toeplitz operators with piecewise continuous generating function. The structure of the algebra T(PC) is more involved than that of ~he algebra T(C) described in Theorem1.51. This is mainly due to the fact that the Hankel operator H(a) is no longer compact for general a ~ PC. Our starting point is a theorem by Calkin showing that Fredholmness of an operator is equivalent to an invertibility problem. Theorem 4.13 (Calkin) Let X be a Banach space. An operator A ~ L(X) is Fredholmi] and only i] its coset A + K ( X ) modulothe compactoperators is invertible in the Calkin algebra L(X)/K(X).

4.1.

153

LOCAL PRINCIPLES

The proof can be found in many textbooks and monographs; for instance see [72], Chapter 4, Theorem7.1. ¯ So what we have to deal with is an invertibility problem in the Calkin algebra L(12)/K(l~). A direct application of the local principle to this problem fails since the center of L(12)/K(l 2) is trivial. But T(PC) is a C*-algebra (obvious) which contains the ideal K(/2) (Theorem 1.51); so one can the quotient algebra T(PC)/K(12), which is a C*-subalgebra of the Calkin algebra and, thus, inverse closed in L(12)/K(l~). The following proposition states that the center of this algebra is not trivial, and so it offers the possibility of applying the local principle to examinethe invertibility of the cosets T(a) + K(12). Proposition 4.14 The set C of all cosets T(f) g(/2) with f e C(~£) is a C*-subalgebra of the center o] T(PC)/K(I2) which is *-isomorphic to C(q£), the isomorphism being given by T(f) g( /2) ~ Proof. The identity T(af) = T(a)T(f) + g(a)g(]), functions a, f E L¢~(~?)(see (1.16)), implies T(a)T(f)

- T(f)T(a)

holding for arbitrary

= H(f)H(5)

If f is continuous then H(f) and H(]) are compact by Lemma1.33 and thus the commutator T(a)T(f)-T(f)T(a) is compact for every a E L~(’~). In particular this shows that C is contained in the center of T(PC)/K(12). Moreover we see that T(af) - T(a)T(f) is a compact operator for every a ~ L~(~?) and f ~ C(~’), from which it easily follows that C is even subalgebra of the center of T(PC)/K(!~) and that the mapping ~r : C(’~) -~ T(PC)/K(Ie), f ~ T(f) ~) is a *-homomorphism.Weclaim that the kernel of ~r is trivial. from (1.21) we infer that IIT(a)ll

= liT(a)

Indeed,

+ K(~)II a e L~(V),

and IIT(a)ll = Ilalloo by the Brown-Halmostheorem (Theorem 1.28). Hence, if T(a) K(2) is thezerocoset , then a = 0 , showing that the k ernel of ~r is indeed trivial. Thus, ~r is a *-isomorphism, and the first isomorphy theorem whence yields the closedness of C in T(PC)/K(12). The maximalideals of C(T) are in bijection with the points of "l~ via x {f e C(~£) : f(x) = 0} (see Example 4.7). Since C(~’) and C are isomorphic, the maximal ideal space of C is also homeomorphic with ~, and the maximalideal of C which corresponds to x E "1~ is {T(f) + g(/2), f e C(’II’) f(x) 0}.

(4.2)

CHAPTER4.

154

STABILITY ANALYSIS

In accordance with the local principle, let ~Tx denote the smallest closed ideal of T(PC)/K(l2) which contains the maximal ideal (4.2) of C. Abbreviate the quotient algebra (T(PC)/K(12))/~7~ to T~ and denote the canonical homomorphismfrom T(PC) onto T~ by ~. Then the local principle in combination with Calkin’s theorem states that an operator A E T(PC) is Fredholmif and only if the ’local’ cosets rx (A) are invertible in ~ for every x E ~I’. So what we have to deal with is invertibility problems in the algebras T~ which arise from the algebra T(PC) by twice factorizations. Wewill see nowhowthis (at the first glance quite heavy) procedure of double factorization simplifies things essentially: Indeed, the outcome of the following considerations will be an identification of each of the ’local’ algebras T~ with a very familiar object - the algebra C[0, 1] of the continuous functions on the interval [0, 1]! Let us agree upon calling a C*-algebra 91 with identity element e to be singly generated if there is an element a ~ P2 such that the collection of all polynomials coe + Cla + c2a 2 ~- ... + Cr ar with r ~ Z+ and c~ ~ C is dense in 91 and upon calling a a generator of 91 in this case. Further, given x C "~, let X~ refer to the piecewise constant function on "F with jumps at x and -x which is 1 on the arc from x to -x (with respect to the common orientation of ~’) and which is 0 on the arc joining -x to x. Proposition 4.15 Every algebra T~ is singly generated, and 7~(T(x~) ) a generator of Proof. It is sufficient to verify that, if a is a piecewise continuousfunction, then the coset ~(T(a)) is a linear combination of ~z(I) and ~z(T(xx) ). Denote the one-sided limits of a at x ~ "if’ by a(x ÷ 0) and a(x - 0), set a~ := a(x + O)X~ -a(x- 0)(1 -X~), and consider the function b := aThis function is continuous at x and vanishes there (Figure 4.2). a

Figure 4.2: Localization of a piecewise continuous function Let f C C(~’) be a function with f(x) = 1. Then (1-f)(x)

= 0, hence,

4.1.

155

LOCAL PRINCIPLES

the coset T(1- f)+ K(/2) = I- T(f)÷ K(l2) lies in the local ideal Jx, and taking into account the compactness of T(f)T(b) - T(fb) -- -H(f)H(~) one gets ~rx (T(b)) = ~r~ (I. T(b)) = ~r~(T(f)T(b)) whence, by Theorem 1.28,

If the support of f is chosen sufficiently small, then the normIlfbll~ can be made less than any prescribed ~ > 0. Thus, II~r~(T(b))[I 0,which implies ~rz(T(a)) = a(x 0)rz(T(x~)) + a(x - 0)(~rx(I) -

~r~(T(xz))),

hence, the coset r~(T(x~)) ) indeed generates Tz.

(4.3) ¯

In particular we see that T~ is a commutative C*-algebra. For singly generated commutative C*-algebras one can specify the Gelfand-Naimark theorem as follows: Theorem4.16 Let 92 be a singly generated unital C*-algebra, and let a 6 92 be a generator of 92. Then the maximalideal space of 92 is homeomorphic to the spectrum a(a) of a, and the Gelfand transform 92 --~ C(a(a)) a to the identical mappingon a(a). For a proof see the standard textbooks on Banach and C*-algebras.

¯

So the only thing that remains to do is to compute the spectrum of the coset ~r~(T(x~)) in T~. To this end, we recall a result by Hartman and Wintner, which identifies the spectrum as well as the essential spectrum (= the set of all A e C for which T(a) - is notFredholm) of self -adjoint Toeplitz operators. Theorem 4.17 (Hartman/Wintner) Let a E L°°(~) be a real-valued function. Then both the spectrum and the essential spectrum of T(a) coincide with the interval [ess inf a(t), ess sup a(t)]. A proof is in [26], Section 2.36.

¯

Proposition 4.18 The spectrum of ~r~(T(x~)) in 7-~ is the interval [0, 1]. Proof. Assumex = 1 without loss of generality, and write X in place of X1. The essential spectrum of T(X) is the interval [0, 1] due to the Hartman/Wintner theorem, and so the local principle implies that [0, 1] = a~_(pc)/~:(,2) (T(x) K(/2)) = Uue’r a~-~ (~r~,(T(x))).

156

CHAPTER 4.

STABILITY

ANALYSIS

If y E ~7 \ {-1, 1}, then ~y(T(x)) is either ~(0) or ~(I) by (4.3) hence, a(~r~(T(x)) ) is either {0} or {1}. Thus, (4.4) involves (0, 1) C_ a(Trl(T(X)) ) L]

a(~r_l(T(x))) C_[0, 11,

and since spectra are closed, this shows that [0, 1] = ~(~1 (T(~))) U o(~-I(T(x)))-

(4.5)

Weclaim that a(rl (T(x))) = a(w_~ (T(X))), whence via (4.5) the assertion follows. Define operators J and C from 12 onto l ~ by J: (xk)k=0 ~ ((--1)

Xk)k=0 and C: o ~ (x~) a= C is anti-linear, and J: -- C2 ~ I. Thus, the

The operator J is linear, mappings V : A ~ JAJ and W : A ~ CA*C

define an isomorphism and an anti-isomorphism (which means a linear mapping W satisfying W(AB) = W(B)W(A) for all A, B) of L(l 2) onto L(/2), respectively. In particular, if a ~ L¢~(~’), V(T(a))

= T(a) and W(T(a))

= T(a),

(4.6)

where ~(t) a(-t) and 5( t) = a(1/t). Since ~ and ~ ar e pi ecewise continuous again if a ~ PC, the identities (4.6) show that both V and Wmap the algebra T(PC) onto itself. It is further evident that V and Wmap the ideal K(1~) of T(PC) onto itself, which implies that both the mappings A + K(12) ~ V(A) + ~) and A + K( 2) ~ W(A) + K(1~) are correctly defined and that they provide us with an isomorphism and an anti-isomorphism from T(PC)/K(I~) onto itself. Wedenote these mappings by V and W again. Let now f and g be continuous functions on ~" with ](1) = 0 and g(-1) = 0. Then (4.6) further yields V(T(f)

+ g(12))

= T(]) ~) and W(T(g) + g(12) = T(~) + K(

where ] and ~ are continuous functions satisfying ](-1) = 0 and t~(-1) So we finally conclude that the mappings ~r~(A) ~ r_~(V(A))

and ~r_l(A)

~ ~-I(W(A))

are a (correctly defined) isomorphism from T1 onto 7-1 and a (correctly defined) anti-isomorphism of T-~ onto T-~, respectively.

4.1.

LOCALPRINCIPLES

157

These mappings send ~rl(T(x)) into ~r_l(T(~)) = ~r_l(T(1 - X)) =-I(T(1 = X)) into ~r-l(T(~ - )~)) ~r-I(T(X)), and si nce both is omorphisms and anti-isomorphisms preserve spectra, we obtain at, (~rl(T(x))) 1 (~r_l (T(1 - X)) ) = at_, (~r-I which proves our claim.

¯

It is nowclear how to derive the Fredholm criterion in Theorem4.11: The Toeplitz operator T(a) is Fredholm if and only if the coset 2) T(a) + K(l is invertible in the Calkin algebra (Calkin’s theorem). This happens if and only if the local coset ~r~(T(a)) is invertible for every x E T (the local principle), and ~r~ (T(a)) is invertible if and only if the function t ~, a(x + O)t + a(x 0)(1 -

(4.7)

is invertible in C[O,1], i.e. if and only if it has no zeros in the interval [0, 1]. Moreover, the same arguments show that, for arbitrary piecewise con-

tinuous functions a,j, theoperator E=I only if none of the functions !

T(a,j)is redholm if and

J

[0, 1] -~ C, t ~ ~ H(a~j(x + O)t + a~(x 0) (1 - t)

(4.8)

i=l j=l

with x running through ~, has a zero between 0 and 1. Let us emphasize another remarkable aspect: Wealready observed that every local algebra T~ is singly generated and, hence, commutative. This implies that the ’global’ algebra T(PC)/K(I2) is commutative, too. To get this, we need the following completion of the local principle. Proposition 4.19 Let the notations be as in Theorem ~.8. Then, ]or every a E ~2, Ila[l~ -=~sM(~) max Ila+ Proof. The proof is easy: Let, for a moment, II refer to the product of the local C*-algebras P2/I~ with x 6 M(~). The local principle states that the mapping P2 -~ II which assigns with every a ~ ~2 the ’function’ (a + Ix)~m(¢) is a symbol mapping in the sense of Definition 1.56 and, consequently, an isometry by Theorem1.57. . Thus, if A + K(/2) and B + K(/2) are arbitrary cosets T(PC)/K(12), then [[AB - BA + g(12)ll = max[[~r~(AB - BA)I[ = 0

158

CHAPTER4.

STABILITY ANALYSIS

whence the compactness of AB - BA for arbitrary A, B E T(PC) follows. The (as we now know) commutative C*-algebra T(PC)/K(l 2) is subject to the Gelfand theory, and having in mind our above derivation, it is not hard to identify the maximalideal space of this algebra. It turns out that this maximal ideal space is homeomorphicto the cylinder ~" × [0, 1] (but provided with a topology which is not the standard Euclidean one, see Section 5.1.4, Example4 for details) and that the Gelfand transform of the coset T(a) K(/2) is thefunction ~’x[0,1]-~C,

(x,t)~a(x+O)t+a(x-O)(1-t)

which arises by ’glueing’ together the functions (4.7). Observe that the application of the ’non-commutative’ local principle to this commutative algebra essentially simplifies the determination of the maximalideal space ~" × [0, 1]; the main reason is that ~" x [0, 1] is a Cartesian product, and localization allows to determine each of its factors ~" and [0, 1] separately.

4.2

Finite sections of Toeplitz operators generated by a piecewise continuous function

In this section we are going to extend the results of Sections 1.3.3 and 1.4.2 - 1.4.4 to Toeplitz operators with piecewise continuous generating function. The basic idea is to use the local principle introduced in the previous section for the stability analysis of the finite section method. This natural idea fails to work immediately since the related C*-algebra has a trivial center. So we will start with a further ingredient - the so called lifting theorem - which (in some instances) allows to render a C*-algebra accessible to the application of local principles.

4.2.1 The lifting

theorem

Let, as in Section 1.4.2, ~" refer to the C*-algebra of all boundedsequences (A,~) with An E ~×n, and write ~ for th e id eal of ~- consisting of all sequences (Gn) tending to zero in the norm. Further, we let ,~(PC) denote the smallest closed subalgebra of ~- which contains all sequences (PnT(a)P,~) where now a runs through the piecewise continuous functions. Clearly, S(PC) is a C*-algebra. It is further evident that the strong limits s-limAnPn and s-limR,~AnPn as n -~ oo (where Rn is the reflexion operator introduced in Section 1.4.2) exist for every sequence (An) S(PC). Wedenote the se lim its by W(A~) and I~V(A,~), respectively. Thus, W and IYd may be considered

4.2.

FINITE SECTIONS-

PC FUNCTIONS

159

as *-homomorphisms from S(PC) into L(/2) (and even into T(PC) as one easily checks). Our goal in this section is to prove the following generalization of Theorem1.54. Theorem 4.20 Let (An) E S(PC)_. The sequence (An) is stable only i] both operators W(An) and W(An) are invertible. In particular, if An = PnT(a)Pn (= the nth finite section of T(a)), then Theorem4.20 states that the finite section method (PnT(a)Pn) is stable if and only if the operators W(PnT(a)Pn) = T(a) and ITV(PnT(a)Pn) with a(t) a(1/t) ar e in vertible. Si nce wefur ther kno w that T(a) and T(5) are invertible only simultaneously (which is a consequenceof identity (1.32)) we arrive at the following corollary. Corollary 4.21 Let a ~ PC. Then the finite section method (PnT(a)Pn) is stable if and only if the Toeplitz operator T(a) is invertible. To prove Theorem4.20, it is natural to start with reformulating the assertion into an invertibility problem in a C*-algebra, which is possible due to Kozak’s theorem: A sequence (An) S( PC) is sta ble if andonly if th e coset (An) ÷ is invertible in thequotient alge bra 9r/G, which on i ts hand is equivalent to the invertibility of (An) + in S(PC)/G (re call tha t G i an ideal of ,~(PC) by Theorem1.53). So we are left with an invertibility problem in The experience gathered in Section 4.1.3 (where we localized the quotient algebra T(PC)/K(l 2) via the cosets T(f) + K(l 2) with continuous f) suggests to try to localize the algebra S(PC)/G via the cosets (PnT(f)Pn) wher e f is a conti nuous funct ion. We ha ve alrea dy remarkedthat this idea fails (one can even show that the center of S(PC)/G consists of the multiples of the identity element only), but let us nevertheless look what we would need in order to localize in the desired manner. Widom’sformula (Lemma1.52) implies that PnT(a)Pn" PnT(f)Pn - PnT(f)Pn" PnT(a)Pn = Pn(H(.f)g({z) - g(a)g(]))Pn + Rn(H(])g(a) for arbitrary functions a,f e L~(~£). If one of these functions, say f, is continuous then this expression is of the form PnKPn+ RnLRnwith compact operators K and L. So we could localize if we were able to factor out from the algebra $(PC)/G all cosets of the form (PnKPn+RnLRn) with K and L compact. This factorization would require that these cosets are contained in some ideal of $(PC)/G and, moreover, one would also need that invertibility modulo this larger ideal has something to do with the invertibility moduloG, i.e. with stability.

160

CHAPTER4.

STABILITY ANALYSIS

To make this idea precise, we will introduce a C*-subalgebra 9~Wcof 5 such that 2) and (Gn) e (A) Gw := {(PnKPn + RnLRn + an) with K, L E K(l is a closed ideal of ~-w. (B) the invertibility of the coset (An)+ Gw3rw/6 W is - u nder cer tain additional conditions which have to be specified - equivalent to the invertibility of (An) (C) the algebra $(PC) we are interested

in is contained in ~-w.

As a guide how to construct the algebra ~w can serve the algebra ~-c with its ideal ~c introduced in Section 1.2.4. Recall that, in order to make to an ideal, we had to diminish the algebra ~ by requiring the existence of certain strong limits. In analogy, we let ~-w stand for the collection of all sequences (An) ~ J: for which there exist two operators W(An) and l~d(An) in L(I 2) such that AnPn --~ W(An), A~Pn --~ W(An)*, RnAnRn -~ IfV(An),

RnA~Rn --+ IfV(An)*

in the sense of the strong convergence. Then one has the following counterpart to Theorems 1.18 and 1.19. Theorem 4.22 (a) :~w is a C*-subalgebra o] J: which ~ontains the identity clement o] ~. (b) The mappings W : :W - ~ L(12), ( An) ~s- lim,~-~ooA,~Pn, an d IT ~rw _~ L(12), (An) ~ s-limn-,~oRnAnRn, are *-homomorphisms. (c) 6Wis a closed ideal o] :~w. Proof. The proof is quite similar to those of Theorems1.18 and 1.19; so we will indicate a few points only. The inclusion ~w C ~w was shown in the proof of Theorem1.53, where we moreover verified that W(PnKP,~+R,~LRn+Gn)

= K, ITV(PnKPn+RnLRn+Gn)

(4 .9

whenever K and L are compact and HG,~[[ tends to zero. The identities (4.9) together with the estimates

IIW(An)l] 0, k < 0 and N > 0, one has E an(Xn(eiS) + iyn(eiS))

-iks ds

\n=l

i ei~

(4.27) The first term on the right hand side of (4.27) is less than

M being a constant independent of ~. The second term on the right hand side of (4.27) can be estimated from above by 2~ times

172

CHAPTER 4.

STABILITY

ANALYSIS

For N large enough, this expression becomes less than ~. Hence if k < 0, then the kth Fourier coefficient of ~ an(xn ÷ iyn) has absolute value less than (2M + 1)~ for every ~ > 0; therefore it vanishes. Thus, an(xn ÷ iy,~) E ~ and a= ~ a, ~z,~ ~ C ÷ H°°. Employing th e de composition

n----1

n----1

n=l

in an analogous manner, one finds that a ~ C ÷ H°°, i.e.

a ~ QC.

¯

Wewill now see how this result can be used for the construction of noncontinuous but quasicontinuous functions. Set rn = 1 - e-’~2 and consider the functions F~ on ~ given by In (1 - r~z) 2n ’

Fn(z) In (1 - rnz ) _ In (1 - r~)

where In refers to the continuous branch of the logarithm which is defined on C \ (-~, 0] by In z := In [z[ + i argz with the argument of z chosen (-w, ~r). The function Fn has its only pole at 1/r,~; hence it is analytic in the open unit disk and bounded on T, i.e. F,~ ~ H~. Set X,~ :-- ReF,~ and Yn := Im Fn. Then, clearly, [IYnl]~ -< 2 In (1 - rn) - 2

(4.28)

and X,~(eis) In[1 - rheim[ _ In (1 + r2n- 2r,~ coss) In (1 - rn) 2 In (1 - rn) Because (1

- rn) 2 =

1n + r 2_2r,~_ ~), whichverifies (iii). Finally, (i) is satisfied since x,~ + iy,~ is just a rotation of the H°C-function

4.3.2 Stability of the finite section method Let ~" and ~ be specified as in Section 4.2.1, and let $(QC) denote the smallest closed subalgebra of ~" which contains all sequences (PnT(a)Pn) with a a quasicontinuous function. Clearly, S(QC) is a C*-algebra, and the strong limits s-limn-~o~AnPn and s-limn_~RnAnR~exist for every sequence (An) S(QC) (here Rnagain ref ers to then × n re fl ection matr ix). Wedenote these limits by W(An) and IV(An), respectively. Our goal the following generalization of Theorems1.32 and 1.54. Theorem4.33 (a) Let (AN) ~ $(QC). Then (An) is stable if and only both operators W(An) and IV(An) are invertible. (b) Let a ~ QC. The finite section method (PnT(a)Pn) is stable if and if the operator T(a) is invertible. Proof. Let us start with assertion (b). A look at the proof of Theorem 1.32 reveals that we can derive this assertion in the very same way, once we have convinced ourselves that the following implications are valid for arbitrary quasicontinuous functions a: T(a) invertible =~ a invertible =~ T(a-1) invertible.

(4.32)

174

CHAPTER4.

STABILITY ANALYSIS

In Section 1.3.3, we verified these implications for continuous functions a by having recourse to the invertibility criterion Theorem1.31, which we do not have at our disposal in the quasicontinuous setting. For the first implication in (4.32), let Un(with n E Z) denote shil Ct operator on the Hilbert space /2(Z) of the two-sided squared summable sequences, Un : (Xk)keZ ~-} (Yk)keZ, Yk : Xk-n, and let/4 stand for the collection of all operators A E L(/2(Z)) for which the strong limits S(A) := s-lim

U_,~AU,~ and S(A)* := s-lim

U-nA*Un

exist. It is elementary to check that L/is a C*-subalgebra of L(12(Z)) and that S is a *-homomorphismfrom/4 into L(12(Z)) the range of which is the C*-algebra of all Laurent operators L(b) with b ~ L°~(T). Further, if we think of 12 as a subspace of/2(Z) and, thus, identify every operator B e L(12) with the operator on/2(Z) acting as B on 12 and as the zero operator on (/2)±, then we easily get that T(a) e lg and S(T(a))

=

for arbitrary a ~ L~(~). So, if T(a) is invertible for some a e QC, i.e. if there is an operator B ~ L(l 2) with T(a)B = BT(a) then B ~ L/ (inverse closedness), to equality (4.33) gives L(a)S(B)

and application

(4.33) of the homomorphism

-- S(B)L(a)

i.e. the invertibility of the Laurent operator L(a). Nowrecall that L(a) can be identified with (or is unitarily equivalent to) the operator of multiplication by a (see the proof of Theorem1.27) to get the invertibility of in L°°(~") and, consequently, in QC. For the second implication in (4.32), consider the identity I = T(aa -1) = T(a)T(a -1) -=1) + H(a)H(5 or, equivalently, T(a)-’

= T(a -~) + T(a)-~H(a)H(5-’),

which implies that the operator T(a-1) is of the form ’invertible plus compact’. Hence, T(a-1) is a Fredholm Toeplitz operator with index zero and

4.3.

FINITE

SECTIONS-

QUASICONTINUOUS FUNCTIONS

175

is therefore, by Coburn’s theorem, invertible. This finishes the proof of assertion (b). Nowone can prove in complete analogy with Theorem 1.53 that S(QC) = {(PnT(a)P~ + PnKPn + RnLRn + wher e a ¯ K and L are compact, (Gn) ¯ and, hence, twice applying the perturbation Theorem 1.54 we arrive at assertion (a). 4.3.3

Some other

classes

QC,

theorem as in the proof of

of oscillating

functions

Comparing Theorem 4.20 with Theorem 4.33 one might ask whether there is a more general result containing both theorems as particular cases. And indeed, such a result exists. Let PQCstand for the smallest closed subalgebra of L°~(T) which contains both the algebra PC of the piecewise continuous functions and the algebra QCof the quasicontinuous functions. The functions in PQCare called piecewise quasicontinuous. Theorem 4.34 (a) Let (An) ¯ S(PQC). Then (An) is stable if and if the operators W(An)and I~V(An) are invertible. (b) Let a ¯ PQC.The finite section method (PnT(a)P,~) is stable only if the operator T(a) is invertible. Sketch of the proof. The set ~w introduced in Section 4.2.1 is also a closed ideal of S(PQC). So, by the lifting theorem (Theorem 4.23), sequence (An) S(PQC) is sta ble if andonly if th e o pera tors W(An) and ITV(An) are invertible, and if the coset (An) + Gwis invertible the quotient algebra S(PQC)/~W. It remains to verify that invertibility (even Fredholmness will be enough) of W(A~) implies the invertibility of (An) ÷ Gw. This can be done by showing that C := ((P,~T(b)Pn)

+Gw, b¯

is a C*-subalgebra in the center of $(PQC)/GW which is *-isomorphic to QCitself, and by applying the local principle in order to localize over the maximal ideal space of QC. The proof that Fredholmness of W(An) involves local invertibility of the coset (An) ÷ Gwat every point ~ of M(QC) is the hard part of the proof of Theorem 4.34 and requires some subtle knowledge about M(QC). For details we refer to [160], [161] or to the monograph [26], Section 7.33. ¯ So we see again that the lifting

theorem in combination with a very fine

176

CHAPTER 4.

STABILITY

ANALYSIS

localization (where the points of T are replaced by points of M(QC))provides an adequate tool for tackling stability problems. After this success one might ask whether the assertion of Theorem4.34 holds for arbitrary functions in L°° (~i’). Fortunately (since otherwise all we had done up to were for nothing) this is not the case! S. R. Treil succeeded in constructing an L~-function a such that T(a) is invertible, but the finite section method (PnT(a)Pn) is not stable. Treil’s construction can be found in [170] and [26], Theorem7.92, and see also [19] and [27] for other examples. Weconclude this section by considering a non-symmetric (but nevertheless quite elementary) problem: the finite section method for Toeplitz °°. operators with generating functions in C + H Theorem 4.35 Let a E C + H~. The finite section method (PnT(a)Pn) is stable i] and only if the Toeplitz operator T(a) is invertible. Weprepare the proof of Theorem4.35 by an elementary but useful result. Let Rn again denote the n × n reflection matrices, write V and V-1 for the forward and backward shift operators on 12 introduced in (1.19), and set Vn := Vn and V_,~ := (V_I) ~ for n _> 1. Further, let T/:: (which stands for Toeplitz-like) denote the set of all operators A E L(l 2) for which the following eight strong limits exist: T(A) 5b(A) H(A) /~(A)

:= := := :=

s-lim s-lim s-lim s-lim

V_nAV~, R~AR,~, V_nAR~, RnAVn,

T(A)* ~(A)* H(A)* /~(A)*

:= := := :=

s-lim s-lim s-lim s-lim

V-~A*Vn, R,~A*R,~, (4.34) V_~A*R,~, RnA*V~.

Theorem4.36 (a) T£~ is a C*-subalgebra of L(/2). (b) Let A, B ~ T£. Then T(AB) = T(A)T(B) ~(AB) = ~(A)~(B)

+ H(A)[I(B), + [-I(A)H(B),

H(AB) = H(A)~(B)

+ T(A)H(B),:

[I(AB)

+ ~(A)I:I(B).

= H(A)T(B)

(c) Let a ~ L~(~). Then T(a) ~ 7-1~, T(T(a)) = T(a), ~(T(a)) -- T((t),

H(T(a)) = H(a),

Proof. Let us check, for example, the first identity in (b). For A, B ~ 7-£: one has V-,~ABVn = V-~A(R,~R,~

+ VnV_n)BVn

= (V-~AR,~)(RnBVn) + (V_,~AV,~)(V_,~BV,~),

4.4.

POLYNOMIAL COLLOCATION

177

and the assertion follows by letting n go to infinity. The other identities can be obtained similarly. Consequently, T£ is an algebra. The proofs of the symmetryand the closedness of T£: are straightforward. For (c) one easily verifies that V-nT(a)Vn = T(a), RnT(a)Rn = PnT(~)Pn, V-nT(a)Rn = H(a)P,~ and R,~T(a)V,~ = PnH(~), and the assertion again follows by taking the strong limit as n -+ oo. ¯ °° and suppose T(a) to be invertProof of Theorem 4.35. Let a E C+H ible. Since T(a) belongs to the C*-algebra T£:, and C*-algebras are inverse closed, we conclude that B := T(a) -1 belongs to Tt:, too. In particular, the eight limits (4.34) exist for B in place of Write a as h + c where h E H°° and c ~ C. Then PnT(h) = PnT(h)Pn for all n, which is an effect caused by the triangular form of T(h), and from T(a)B = weobt ain P,~(T(h) + T (c ))BP, = P and Pn(T(h)

+ T(c))PnBPn = Pn - PnT(c)VnV-nBPn.

The second term on the right hand side is equal to R,H(~)V-nBRn and since H(~) is compact and (V_,BR,)* converges to H(B)* strongly as n --~ c~ by Theorem4.36, we conclude via Lemma1.5 (b) that P~(T(h)

+ T(c))P~BPn

= Pn + R~LRn

L = -H(a)H(B)* being compact and (Gn) tending to zero in the norm. Nowthe arguments used in the proof of the lifting theorem apply to obtain the invertibility of the coset ( P,~T( a)P,) + 6 = ( Pn(T( h ) + T(c) in 5r/~ from the right hand side. The invertibility of this coset from the left hand side is a consequence of this since for the P,~T(a)P,, being squared matrices, the one-sided invertiblity implies two-sided invertibility. ¯ °°) is not a symmetric algebra, the Let us finally observe that, since $(C+H results of Chapters 2 and 3 do not automatically hold for the finite section method for Toeplitz operators T(a) with a ~ C + H¢~. Nevertheless, they hold, and for a quite general approach to this problem we refer to [166].

4.4

Polynomial collocation methods for singular integral operators with piecewise continuous coefficients

Our next candidate for examining stability is a collocation method for singular integral operators where the approximate solution is sought in the form of a trigonometric polynomial.

CHAPTER4.

178 4.4.1

Singular

integral

STABILITY ANALYSIS

operators

Westart with recalling somebasic facts on singular integral operators with piecewise continuous coefficients on the unit circle ~, i.e. on operators of the form aI + bS where I is the identity, a and b are piecewise continuous functions on ~ (more precisely: operators of multiplication by piecewise continuous functions), and S is the operator of singular integration against 1 ]~ u(s) ds, t E

(su)(t)

(4.35)

As already remarked, the integral in (4.35) exists as a Cauchy principal value if only u is smoothenough (say, HSlder continuous), and, in this case, [[Su[[2 5 [[u[[2

(4.36)

with [[-[[~ referring to the L2 (~)-norm. The HSlder continuous functions are dense in L2(~); so (4.36) guarantees that S can be extended to a bounded operator on MI of L~(~). Since, moreover

for all bounded functions a on 7, we can think of aI + bS ~ a bounded 2(~). operator on L Let ek(e ~) := e~k*/~ for k S Z. Then {ek}ke~ forms an orthonormal basis of L2 (~), and a straightforw~d calculation shows that Sek

: ~ ¢k if k ~ 0 -e~ if k < 0.

Thus, the matrix representation of S with respect to this basis is the twosided infinite diagonal matrix diag (..., -1, -1, 1, 1,...), which in particular implies that S is an isometry on L2(~), that 2 =I and S*= S, and that P :=

1

~(I+S)

and

Q :=

(I-S)

are complementaryorthogonal projections. With these ~rojections, one can write the operator aI + bS also ~ cP + d~ where c = a + b and d = a - b, which is often of advantage. ~or the special singular integrM operator cP + Q one h~ the identity

eP + = (PeP + where I + QcPis an invertible operator whose inverse is I - QcP. ~aking into account ~hat the matrix representation of the operator of multiplication by c with respect to the b~is {e~} is given by the (two-sided infinite)

4.4.

179

POLYNOMIAL COLLOCATION

(compare the proof of Theorem1.27) one gets ~ Laurent matrix ( ci_j)i,j=_o~ the following matrix representation of PcP÷Q: fo

CO

C-I

C--2

Cl

CO

C-I

C2

Cl

CO

°oo

Thus, (4.37) closely relates the singular integral operator cP ÷ Q with the Toeplitz operator T(c) acting on 12, and since I + QcPis invertible, we see that cP ÷ Q is invertible (or Fredholm of index k) if and only if T(c) is invertible (or Fredholm of index k). Further, the same arguments those used in the proof of Theorem4.33 yield that, if c,d E L°°(~) and cP ÷ dQ is Fredholm, then both c and d are invertible in L~(T). Thus, writing cP ÷ dQ as d(~ P ÷ Q) in this case, one can derive the Fredholmand invertibility properties of the singular integral operator cP ÷ dQ completely from those of the related Toeplitz operator T(c/d). For example, one can get the following theorem of Coburn type in this way. Theorem 4.37 Let c, d ~ L°°(T). The singular integral operator cP ÷ is invertible on L2 (~) i] and only i] it is Fredholmwith index Furthermore, if c/d is a piecewise continuous function, then it is an easy exercise to translate the Fredholm criterion and index formulae for the Toeplitz operator T(c/d) obtained in Theorem4.11 into corresponding criteria for the singular integral operator cP ÷ dQ. What we are going to do now is a little bit more: Wewill derive a Fredholm criterion which applies to an arbitrary operator belonging to the smallest closed subalgebra I(PC) of L(L2(~)) which contains all singular integral operators cP ÷ dQ with c and d piecewise continuous. Clearly, :~(PC) is a C*-algebra. Proposition 4.38 :~(PC) contains the ideal K(L2(~’)) the compact operators on L2(~’). Proof. Translate everything into operators (ek}. Then the operators of multiplication z ~ z -1 go over into the shift operators (xk) respectively, and the image of the projection

on /2(Z) by using the basis by the functions z ~-~ z and ~+ (Xk-~) and (Xk) ~-~ (Xk+l), P is the orthogonal projection

180

CHAPTER 4.

STABILITY ANALYSIS

from 12(Z) onto/2(Z+). Nowthe assertion follows in a similar way as of the inclusion K(/2) _C T(C) in Theorem 1.51 by verifying that Z(PC) contains a projection with rank 1 and that every compact operator on can be approximated by linear combinations of shifts of that projection. ¯ So one can form the quotient algebra Z(PC)/K(L2), and this algebra can be localized via Allan/Douglas, as the following result demonstrates. Proposition 4.39 (a) If f is continuous, then fS - SfI is compact. (b) The set :={fI + K(L2) : f E C(~’is a C* -subalgebra in the center of Z(PC)/K(L 2) which is *-isomorphic to C(V), with the isomorphism being given by fI + K(L2) ~+ f . Proof. (a) Because fS-

SfI

= f(2P-

I) (2 P- I) fI =

2( fP- Pf I) =

2( QfP- Pf

it suffices to show that PfQ and Q.fP are compact. The matrix representation of PfQ with respect to the basis {ek} is

¯

-.

0 0

"’.

0

0

f3 0

so the compactness of PfQ follows from the compactness of H(.f) by Lemma1.33. (b) The set C belongs to the center of Z(PC)/K(L2) by assertion (a). The proof of the isomorphy can be done as that in Proposition 4.14 by making use of the equality (1.20) with the projections Qn in (1.20) specified Qn:

ZXkek kEZ

yielding that II.fI

÷ K(L2)II

~-~

Z Xkek-t-ZXkek, k~_--n-1

k~_n

IIT(I)II = If f[l~, ev en fo r ar bitrary f ~

The maximal ideal space of C is homeomorphic to ~ via the mapping ~ ~ x ~ (fI+ K(L~) : f ~ C(~), f(x)

0}~ M(C).

(4.38)

4.4.

POLYNOMIAL COLLOCATION

181

In accordance with the local principle, let I~ denote the smallest closed ideal of Z(PC)/K(L~) which contains the maximal ideal (4.38) of C, and abbreviate the local algebra (I(PC)/K(L2))/I~ by/7~ and the canonical homomorphismfrom Z(PC) onto Z~ by 7r,. Further, write c(x 5= 0) for the one-sided limits of the piecewise continuous function c at x, and let X~ refer to the special piecewise constant function introduced in Section 4.1.3. As in the proof of Proposition 4.15 of that section one can showthat, for arbitrary c, d E PC, 7r~(cP + dQ) = r~(c)Tr~(P) + 7r~(d)Tr~(Q) = (c(x + O)Tr,(x,I) + c(x 0)(Tr, (I ) - 7r,(x,I)). 7r ~(P) + (4.39) + (d(x + O)Tr,(x,I ) + d(x - O)(r~(I) - 7r~(x,I)). (r~(I) revealing that the local algebra Z, is generated by the two cosets 7r~ (P) and 7rx(XxI) and by the identity coset 7r,(I). The cosets 7r,(P) rc,( X~:I) do not commute(see Proposition 4.41 below), thus, 27, is not a commutative algebra and, thus, not subject to the Gelfand-Naimark theorem for commutative C*-algebras. Moreover, doubly generated C*-algebras can be of a quite involved structure in general, and there is no universal approach to study them. Fortunately, the generators of/7, have some additional properties which render the algebra Z, accessible: they are projections, which is a trivial consequence of the fact that P and XI are projection operators. For C*algebras generated by two (not necessarily commuting)projections and the identity element, one has the following characterization, where we restrict ourselves to a special case being sufficient for our purposes. Theorem 4.40 (Halmos’ two projections theorem) Let 92 be a C*-algebra with identity element, and let p,q ~ 92 be projections (i.e. Self-adjoint idempotent elements) such that a~a(pqp) = [0, 1]. Then the smallest closed subalgebra of 92 which contains p, q and e, is *-isomorphic to the C*-algebra of all continuous 2 × 2 matrix functions on [0, 1] which are diagonal at 0 and 1. The isomorphism can be chosen in such a way that it sends e,p and q into the functions t~

0

1

,

t~-~

0 0 ’

V/~-t)

’

respectively. For a proof see [80], and several generalizations can be found in [61], [73] and [74]. ¯ So all we have to do in order to apply the two projections theorem to the

182

CHAPTER4.

STABILITY ANALYSIS

algebra Zx with p and q corresponding to the cosets ~r~(P) and r,(X,I) to determine the spectrum of the coset ~r,(Px~P ). Proposition

is

4.41 a~ (~r~(Px~P)) = [0,1] for every x ¯ ~£.

This can be verified in the very same manner as Proposition 4.18 for the spectrum of ~r~ (T(x~)). Thus, Halmos’ theorem applies to the description of the local algebras and in combination with the local principle it yields the following Fredholm criterion for operators in Z(PC): Theorem 4.42 (a) There is a *-isomorphism ~ from Z(PC)/K(L 2) onto a C*-algebraof 2 × 2 matrix functions living on the cylinder T × [0, 1]. This isomorphism maps the cosets I + K(L2), P + K(L2) and cI + K(L2) to the functions (x,t)

~-+ 0 1 ’ (1 0) (x,t)~(l

0 and (x,t

)~

(c(x+O)t+c(x-O)(l-t)

(c(x + O) - c(x t) c(x + (c(x o)(1 - +O)-c(x.-O))~ t) + c(x -

respectively. (b) An operator A ¯ :~(PC) is Fredholm if and only if the function ~(A K(L2)) is invertible, i.e. if det (~(A K(L2))(x,t)) # O fo r ev ery (x "F × [0, 1]. If A = cP + Q with c ¯ PC, then ~(A + K(L2)) is the function (x,t)~_

(c(x+O)-c(x (c(x+0)t+c(x-0)(1-t)

0))V/~-t)

which is invertible if and only if 0 ¢ [c(x - 0), c(x 0)] fo r ev ery x This is of course the same criterion we would have obtained when employing the identity (4.37) and the Fredholmcriterion for Toeplitz operators. Analogously, the determinant of the function associated with cP + dQ is (x, t) ~ d(x - O) c(x + O)t + c(x - O) 0)(1- t), and this function does not vanish on "1~ × [0, 1] if and only if 0 ~ [d(x - O) c(x + 0), c(x - O) 0)] for every

4.4.

POLYNOMIAL COLLOCATION

4.4.2 Stability

183

of the polynomial collocation method

Let R(~) stand for the set of the Riemannintegrable functions on the unit circle. Provided with pointwise operations and the supremumnorm, R(q~) becomes a (commutative) C*-algebra (observe that the elements of R(~?) are really functions in the commonsense, and not cosets of functions such as the elements of By Ha we denote the class of all trigonometric polynomials Un(Z) ~=-n CkZ~ on ~, and we set zj := exp (2~rij/(2n 1)). Gi ven a function f E R(~), there exists one and only one polynomial Ln] ~ IIn such that (Lnf)(zj) = f(zj) for every j ~ {-n, ..., n}. The function Lnf is called the Lagrange interpolation polynomial of f, and the operator Ln (for which obviously L~ = Ln) is the Lagrange interpolation projector. Besides this (non-orthogonal) projection, we introduce the orthogonal projection from L2(’~) onto 1-In which associates with the function g ~ L2(~) polynomial (Png)(z)

~

gkz ~ where gk = -~l fo2~g(eiZ)e-~nz dx.

One can show that

IlPnf - fll2 --> 0 ’) as

n -+ oo for every f 6 L2(qI

and

IlL.f - $112-~ 0 as n -+ oo for everyf e R(T).

(4.40)

Consider the singular integral equation on T, (aI + bS)u = f,

(4.41)

with Riemann-integrable right hand side f and piecewise continuous coefficients a and b which, clearly, also belong to R(q~). For the approximate solution of (4.41) by the collocation method, we seek polynomials un E 1-In by solving the linear (2n + 1) × (2n + 1) - system a(zj)un(zj)

+ b(zj)(Sun)(zj)

= f(zj),

j ~

which can be written equivalently in the form Ln(aI + bS)Pn un = L,~f, and our objective is to examine the stability bS)Pn).

of the sequence (Ln(aI

184

CHAPTER 4.

STABILITY

ANALYSIS

Introduce the C*-algebra ~ of all bounded sequences (An) of operators An ¯ L(II,~) as well as the ideal ~ of all sequences (Gn) ¯ z tending to zero in the norm. Further define the reflection operators Rn by R,~ : L2(~’)

~ H,~,

ECkZk~C_lZ-~+...+C_nZ-l+c~zO+...+COZn,

and write $-w for the subset of ~ consisting of all sequences (A,~) for which the strong limits W(A~) := s-lim I?¢’(A,~)

A~Pn, W(An)* := s-lim

A~P~,

:= s-lim RnAnRn, I~V(An)* := s-lim RnA~R~

exist. Finally, let ~w refer to the collection of all sequences in 9~ of the form (PnKPn + RnLRn + Gn) with K, L ¯ K(L2) and (Gn) ¯ and let ~(PC) denote the smallest closed subalgebra of ~" which contains all sequences of the form (Ln(aI + bS)P,~) with a, b ¯ PC as well as all sequences belonging to Gw. The stability result for sequences in ]~(PC) reads as follows: Theorem 4.43 ](:(PC) is a subalgebra of jzw, and a sequence (An) ]E( PC)is stable i] and only i] the operators W( An) and I~V ( An) are invertible. The first step of proving Theorem4.43 is again a lifting theorem which can be derived as the corresponding Theorem 4.23. Theorem4.44 (a) :pw is a C*-subalgebra o~ z, and ~w i s a cl osed id eal

w. of :r

(b) A sequence (An) :wis st able if an d only the o perators W(An)and I~V(An) are invertible and i] the coset (An) Wis invertible in Jzw/6w. The next result makes the first precise.

assertion of the basic Theorem4.43 more

Proposition 4.45 ~(PC) is a C*-subalgebra of ~:w, and the *- homomorphisms W and ~V act on the generating sequences o] ]C(PC) as ]ollows: W(L,~(aI + bS)P,~) = aI + bS, I~V(Ln(aI + bS)Pn) = where 5(t) := a(1/t) again, W(PnKPn + R,~LRn + G~) = K, ITV(PnKP~ + RnLR~ + G~)

4.4..

185

POLYNOMIAL COLLOCATION

The proof of this result is much more involved than its counterpart for the finite section method of Toeplitz operators. Weonly remark a certain ’semi-commutativity’ of the projections Ln with multiplication operators and of the orthogonal projections P,~ with the singular integral, LnaI = LnaLn,

SP~ =P~SP~,

(4.42)

two relations for the adjoint sequences, (LnaP,~)*

= Ln~Pn, (PnSPn)*

= PnSPn,

(4.43)

and a relation for ’reflected’ sequences, R,~(L,~(aI + bS)P~)R~ = L,~(5I + ~S)P~. Details can be found in [90].

(4.44) ¯

The lifting theorem reduces the stability problem for sequences in 1C(PC) essentially to an invertibility problem in j:w/~w and thus, since IC(PC) is a C*-algebra, to an invertibility problem in 1C(PC)/~W (recall that ~w C_ IC(PC) by definition). What we are going to show next is that, for every sequence (A,~) IC(PC), th e Fr edholmness ofW(A~) alr eady implies the invertibility of the coset (An) + Gw. This will be done via localizing, and localization is indeed possible due to the following proposition. Proposition 4.46 (a) If f is continuous on T, then the coset (LnfPn) W. ~w belongs to the center of I~(PC)/G (b) The set C of all cosets (LnfPn) W with f E C(T)is a C*-subalgebra of the center of IC(PC)/~w which is *-isomorphic to C(~), the isomorphism being given by (LnfP~) + W ~-~ f . The proof proceeds similarly to the one of Proposition 4.14 and Proposition 4.39. The main difficulty is to verify that (L,~fP,~)(P,~SP,~)

- (P,,SP,~)(LnfP,~)

whenever f ~ C (’ ~)

for which we again refer to [90].

¯

Consequently, given x ~ ~’, let Ix denote the smallest closed ideal of 1C(PC)/6W which contains the maximal ideal {(LnfPn) + Gw: f ~ C(V) and f(x) -- 0} of C, abbreviate the algebra (1C(PC)/Gw)/Ix to/Cx, and write (I)x for the canonical homomorphismfrom 1C( PC) onto ICx.

186

CHAPTER 4.

STABILITY

ANALYSIS

In what follows it is more convenient to write the sequences (Ln (aI bS)Pn) as (Ln(cP + dQ)Pn) with c = a + b and d = a ~ b. For arbitrary functions c, d 6 PC(~2) and f 6 C(~), the estimate I](LnfPn)(L~(cP + dQ)P~) + ~wI] ]]f cl]~ + ]]f dlloo holds (see [90]), which allows us to rewrite the local coset ~(Ln(cP dQ)P,) ¯ ~ (in ((c(x+O)x~ +c(x-O)(1 -X~))P+ (d(x+O)x~ +d(x-O)(1 Due to the semi-commutator property (4.42), this coset coincides with (c(x 0)O~(L,~x,P,) + c( x - 0) O,(Ln(1 - x~)P~))~,(P~PPn) + (d(x + O)~(Lnx, Pn) + d(x 0) ~,(Ln(1 - x~)Pn))~x(P,~QPn), that is, the local algebra/C, is generated by the cosets ¢b,(L~x,P~) and ~ (PnPPn) and by the identity coset a2x(Pn). Further we conclude from (4.42) that ~x(LnxxPn) and ~x(PnPPn) are idempotent cosets, and from (4.43) we infer that both cosets are self-adjoint. What we still need for applying the two projections theorem is provided by the following proposition. Proposition 4.47 a~:, ((~ (PnPP,~L,~xzP,~P,~PP~)) = [0, 11. Proof. The coset ~2,(PnPP, aLnx, PnPnPPn)is self-adjoint, non-negative, and has a norm not greater than 1. Hence, the desired spectrum is contained in the interval [0, 1]. That, conversely, every point between 0 and 1 belongs to that spectrum can be checked by repeating the arguments used in the proof of Proposition 4.26 in order to relate the local spectrum a(~2~ (P,~PPnL~x~:PnP~PPn)) to the spectrum of ~r~ (Px~P) which is equal to [0, 1] as we know from Proposition 4.41. ¯ Having this local spectrum at our disposal, we can apply the two projections theorem and the local principle to get a description of the quotient algebra IC( PC) / w. Theorem 4.48 (a) There is a *-isomorphism ~ from IC(PC)/~ TM onto a C*-algebra of 2 × 2 matrix functions living on the cylinder ~ × [0, 1]. This W isomorphism sends the cosets (p,~)+Gw, (p,~ppn)+Gw and (L,cPn)+G into the functions (x,t)

~t 0 1 (1 0)(x,t)~

(

0 0 ’ and (x,t)

4.4. POLYNOMIAL

COLLOCATION

c(x+ 0)t + c(x- 0)(1- t)

187

%

(c(x + 0)

(c(x + O) - c(x - O))vfi~ - t) 0)(1 - t) + c( x - O)tJ respectively. (b) A coset (A,)+Gw is invertible in IC(PC)/6W if and only if the function ~((A,~) + ~w) is invertible V x[0, 1]. The following result, which can be verified in complete analogy with Theorem 4.27, finishes the proof of Theorem4.43. Theorem 4.49 The C*-algebras Z(PC)/K(L ~) and ~(PC)/~ W are *isomo~hic under the isomo~hism given by ~(PC)/~ W ~ Z(PC)/K(L~), 4.4.3

Collocation

versus

(A~) + ~w ~ W(A~) K(L2). Galerkin

methods

The quite comfortable approach to polynomial collocation methods for singular integral operators presented in Section 4.4.2 is essentially due to the semi-commutator relations L,~aI = LnaLn and SPn = PnSP,~, which allow us to decompose the sequences (L~(aI + bS)P~) into much simpler sequences, which locally behave as projections. For the Galerkin method (P,~(aI+bS)P,~) for singular integral equations one also has a decomposition (P,~(aI + bS)P,~) = (P~aPn) + (P,~bP,~)(P,~SP,~), but this decomposition turns out to be much less useful than its collocation analogue, since the sequences (P,~aP,~) remain quite complicated objects due to the absence of a relation of the type LnaI = L,~aL,~ for the projection P,~ in place of L,~. To illustrate this, we consider the smallest closed subalgebra 7~(PC) of ~ which contains all Galerkin sequences (Pn(aI + bS)P,~) with a, b E PC(Z). This algebra can be treated via lifting theorem and local principle, too. Indeed, the cosets (PnfPn) + ~w with f e C(~) and W as i n Section 4.2.2 form a C*-subalgebra of the center of 7~(PC)/~W, which is *-isomorphic to C(~’)). But the resulting local algebras ~P~ prove be of a quite involved structure. In particular, it is no longer true that the Fredholmness (or even invertibility) of W(An)and I~V(A~)implies the invertibility of every local coset (I)~(A~). Thus, every local invertibility problem for ~z(An) in 7~z yields an additional necessary condition for the stability of (An), and only the union of the ’global’ conditions (invertibility of W(A,~) and IYd(An) in accordance with the lifting theorem) with

188

CHAPTER 4.

STABILITY

ANALYSIS

’local’ invertibility conditions (one for every point of 2i’) gives a necessary and sufficient stability criterion. For example, the condition for the local invertibility of the sequence (Pn(cP + dQ)Pn) at the point x e 2I’ reads as follows: The point 0 has to lie outside the triangle with vertices 1, c(x+O)/d(x÷O)and c(x-O)/d(x-O) (Figure 4.3). For a proof of this and related results we refer the reader [126] and [132] as well as to the monographs[77] (Section 4.1.2) and [123].

c(x+0)

d(x-O)

Figure 4.3: The local stability

4.5

Paired circulants tion methods

condition

and spline

approxima-

A further extensively employedtool for solving singular integral equations (and more general pseudodifferential equations) is provided by spline approximation methods. Weare going to point out how these methods fit into the picture drawn in the previous sections. As in Section 4.4, we are concerned with the singular integral equation Au := (aI ÷ bS)u =

(4.45)

on L2(T). For simplicity we suppose that the coefficients a and b are continuous functions whereas the right hand side f can be an arbitrary L2 (’1~) function for the moment.

4.5.

PAIRED CIRCULANTS

189

Via the parametrization s ~ exp(27ris), we have a one-to-one correspondence between functions w on q~ and 1-periodic functions Wpon l~ given by Wp(S) := w(exp (27ris)). In what follows, we will thouroughly identify functions w with their periodization wp. Given integers d _> 0 and n _> 1, let S~d denote the space of smoothest 1-periodic splines of degree d over the uniform meshZ/n. Thus, in case d > 1, Sna consists of all 1-periodic d-IC functions the restriction of which onto each interval [k/n, (k + 1)/n] is polynomial of degree d, whereas the elements of S~° are just the 1-periodic functions which are constant on each of these intervals. Here are a few concrete spline approximation methods for the singular integral equation (4.45). Galerkin methods. The simplest Galerkin method (with the same functions used both as ansatz and test functions) determines an appr6ximate solution un e Sdn of (4.45) such that (Au,, ~o) = (f, ~o) for all ~ e

(4.46)

where (., .) again refers to the usual scalar product on L2(’~). e-collocation methods. Choose and fix e in [0, 1) if d _> 1 and in (0, if d = 0, and suppose f Riemannintegrable. The e-collocation defines an approximate solution un E sdn of (4.45) such that (Aun)((k + e)/n)

= f((k

for all

k = 0, . .. n - 1 . (4.

47

Quadrature methods. The perhaps simplest quadrature method for solving (4.45) is the so-called methodof discrete vortices which works follows. For k = 0, ...n - 1, let s~’~) := exp(2ri(k 1/ 2)/n) and t~ ’~) := exp (27rik/n) and, for Riemannintegrable f, determine approximate values ~(kn) for u(t~ n)) by solving the linear system (4.48)

where k = 0,...n- 1. If this system possesses a unique solution (~)) dand if d is a positive odd integer, then the interpolating spline u~ E S~ satisfying

= for all k = O,...n - 1 can be thought of as an approximate solution of (4.45).

CHAPTER 4.

190

STABILITY

ANALYSIS

It turns out that, once a suitable basis of S, d is chosen, each of the equations (4.46) - (4.48) can be written as an n × n linear system for unknowncoefficients of un with a system matrix of a very special form: a so-called paired circulant (see below). Also numerous other approximation methods lead to this special structure of the system matrix; cp. [123], Chapter 10. So, in what follows, we will have to examinealgebras generated by sequences of paired circulants. 4.5.1.

Circulants

and paired

circulants

A finite Toeplitz matrix ([a J-k)j,k=0 ~n-1 a-k ~ an-k for k ----

1, ...,

n - 1.

diagonal matrices: if Un and Un :-- n-1/2(e2~rikj/n)~l

U~-1

~

is said to be a circulant if Circulants are unitarily equivalent to U~ refer to the n x n unitary matrices ~n×n

1 o and U~ := n-i/2(e-2~rikj/n)~lo,

then a matrix A E Cnxn is a circulant if and only if there is a diagonal ~. Clearly, in this matrix D = diag((o, ..., ¢n-1) such that A = U~DU~ case, the numbers (k are just the eigenvalues of A, and the vectors ~k -n-1are the corresponding eigenvectors. n-1/2(exp (2m~k/n))j= o Given ~ e [0, 1), set t~~) := exp (2~rik/n) and T(kn) := exp (2~ri(k ~)/n) and, for each boundedfunction p on ~’, let Pn and ~5~ stand for the diagonal matrices pn := diag(p(t(on)),...,

P(t(nn)--~)) and /Sn := diag(p(To(~)), ...,

Further write/~n for the circulant

p(~’n(n)~)).

~. Obviously, Unp~U~

]]/~[] = []p~][ _< ]]p[]~ and ][/~,]] _< I]P[]~.

(4.49)

Let a, b, a, fl be boundedfunctions on the unit circle. Then an n × n paired circulant is a matrix Bn of the form ~. := ~. + {,~.

(4.50)

If (An) is the sequence of the system matrices for one of the methods(4.46) - (4.48), then it turns out that (An) can be written as (Bn) + (Cn) where the matrices B, are given by (4.50) and where the Ca are matrices tending to 0 in the norm. Similar descriptions hold for the system matrices of numerous other approximation methods for the singular integral operator aI + bS. Thereby, as a rule, the diagonal matrices 5~ and ~, correspond to the coefficients a and b of the singular integral operator (and are almost independent of the concrete approximation method), whereas the circulants &nand ~n arise from discretizations of the identity operator I and

4.5.

191

PAIRED CIRCULANTS

of the singular integral S (and reflect heavily the properties of the chosen discretization procedure). In general, the functions ~ and ~ are piecewise continuous, and they are given in form of infinite series (similar to the series in (4.25)) in manyinstances. For their computation we refer again [123], Chapter 10. Thus, we are concerned with the stability of sequences (Bn) where Bn is as in (4.50) and where a and b are continuous and c~ /~ are piecewise continuous functions. 4.5.2

The stability

theorem

As in the previous sections, we put all sequences we are interested in into one C*-algebra and try to analyse this algebra by means of local principles. It is advantegeous in what follows to identify the approximation matrices An with operators acting on the range of a certain projection operator Ln having range dimension n. More precisely, for k -0, ..., n - 1, let X(~n) stand for the characteristic function of the subarc [exp (2~ik/n), exp (2~i(k + 1)/n)) of the unit circle, n), g~’~) := v /dX(k and let Ln denote the orthogonal projection from L2(~l ’) onto the linear span of the functions g~n) with 0 < k < n - 1. These functions form an orthogonal basis of Im L~, and we identify operators on Im Ln with their matrices with respect to this basis. Accordingly, we introduce the C*-algebra ~- of all bounded sequences (An) of operators An E L(Im Ln) (provided with elementwise|y defined operations and the supremum norm). For fixed z E [0, 1), let A stand for the smallest closed subalgebra of ~ which contains all sequences (An) = (Bn) ÷ (Cn) where Bn is as in (4.50) a, b continuous and c~, ~3 piecewise continuous, and where lim IICnll = 0. (Recall that the coefficients of the paired circulants Bn dependon z.) It evident from (4.49) that these sequences belong to the algebra ~. Observe further that, if Bn is as in (4.50), then

= + fin, hence, ,4 is a C*-subalgebra of 9r. Let us emphasize that this algebra also contains a lot of practically relevant approximation sequences which are no paired circulants (but, of course, generated by sequences of paired circulants); the spline qualocation method can serve as an important example. The stability criterion for sequences in .4 will be given again in the form that a sequence is stable if and only if certain strong limits associated ¯ with that sequenceare invertible. It turns out that in the case at hand, this stability criterion involves an infinite family of strong limit homomorphisms which we are going to introduce now. Besides this family, we will define a

192

CHAPTER 4.

STABILITY ANALYSIS

second family of homomorphismswhich is only needed in the proof of the stability theorem. For the construction of the first family, associate with every T E ~" an integer kr,~ E {0, ..., n - 1} depending on n such that T e (exp (2~ri(kr,e + ¢ - 1)/n), exp (2~ri(kr,~ + and set ~ := exp (2~rik~.,~/n). Further define operators n--1

n--1

j=o

j=o

and set T~’,0f]’E~":=[[ vnvn 1

Notice that, for a continuous and a piecewise continuous, Er,1~r~r,1,--1 n an(l~n )

--~ 5n and Er’l~ (Er’’~-’ = "~. with o’(t) a(~-ot).

Moreover, we let Pn refer to the orthogonal projection from L2 (~) onto its closed subspace spanned by the polynomials t [(n - 1)/2] (with [x] denoting the largest integer which is not larger than x). There is an isometry En : Im L,~ --> Im P,~ given by ~--1

[(n--1)/2]

j----0

j=0

~--1

j=[(n--1)/2]q-1

Finally, define E~’2 := E,~T~’~ : Im L~ -~ Im P~. Proposition 4.50 (a) For (An) ~ A and r ~ ~, the strong limit s-lim E~" An( E~’l -1 =: W r,I ( A~ exists, and the mapping W~,I : .4 --r L(L2(~)) is a *-homomorphism. particular, if B,~ is as in (4.50), then W~,~ (Bn) = a(a(r + O)P + - O)Q)+ b(f l( T + O)P+ - O)Q)(4.51 where P = (I + S)/2 and Q = I (b) For (An) ~ A and r ~ q~, the strong limit - ’ =: W~,:( A,~ s-lira E~’2A,~( E~’2)

4.5.

PAIRED CIRCULANTS

193

exists, and the mapping Wr,2 : A --~ L(L2(T)) is a *-hQmomorphism.In particular, if B,~ is as in (4.50), then Wr,2(Bn) = a(T)a~eI I+ b(T)fl~ where arel(t):=

(4.52)

a(1/t).

A proof is in [123]. Nowwe are in the position to formulate the stability criterion for sequences in A. Theorem4.51 A sequence (A~) ~ A is stable if and only if the operators W~,I (An) are invenible for all T ~ ~. In pa~icular, the sequence ( Bn) B~ as in (4.50) is stable if and only if all operators (4.51) a~ invertible. The proof follows the same lines ~ its an~ogues in the previous sections and we will indicate only the main steps and emph~ize some differences. The lifting theorem. Again, we want to attack the stability problem by localization over the unit circle. Wecould perform this localization if ¯ we would know that the sequences (in) with f continuous commutewith all other sequences in the algebra A modulo a certain ideal of .4 which can be lifted. It is clear that (]~) commuteswith all sequences of the same form (hn) with a ¯ C(T). For the circulant sequences (&n) with piecewise continuous, a tricky analysis (for details see [123] again) shows that, if ~ possesses exactly one jump discontinuity, say at ~- ¯ T, then the commutator (]n)(&n) - (&n)(]n) is of the form ((E~’I)-ILnKE~ ’1) + (C,~) with K ¯ K(L2(T)) and (Cn) ¯ Let 3.~ stand for the set of all sequences of this form. Since every piecewise continuous functions can be approximated by a finite sum of piecewise continuous functions having exactly one discontinuity, it is clear that we have to factor out all sets 3.~ with T ¯ T. For this goal, let 3. denote the closure in ~" of the set

and let .40 stand for the smallest closed subalgebra of ~" which contains the algebra .4 and the set 3.. It turns out that 3" is a closed ideal of .40 which can be lifted: Theorem4.52 The set fl is a closed ideal of ‘4o. A sequence (A,~) ¯ .40 is stable (i.e. the coset (An) + ~ is invertible in .4o/~) if and only if operators Wr,I( An) are invertible for every ~" ¯ ~ and if the coset (An) is invertible in the quotient algebra Ao/ fl.

194

CHAPTER 4.

STABILITY

ANALYSIS

This theorem will be proved in a more general setting in the forthcoming section 5.3. Here we only remark that the strong limits Wr,I(A~) and Wr,2(A~) also exist for sequences (An) e ‘7. In particular, for (Jn) Jn -- (E,~’I)-I LnKE~one has Wr,~(J,)=

K if r=w and 0 if T¢W or

j=l j=2.

(4.53)

Localization. It is now clear that, for f continuous, the cosets (in) lie in the center of the algebra Ao/,7. Let C stand for the smallest closed subalgebra of ,40/,7 which contains all of these cosets. Our next goal is to determine the maximal ideal space of the (commutative) C*-algebra First observe that, due to (4.53), every homomorphismW~,I induces *-homomorphism from Ao/,7 into the Calkin algebra L(L2(~))/K(L2(~)) via (A,~) + ,7 ~ Wr,I(An) K(L2(V)), which we again denote by Wr,~. Similarly, every homomorphism Wr,2. generates a *-homomorphismform Ao/,7 into L(L2(’I~)) by (An) + ,7 ~ Wr,2(An), which will be denoted by W~,2 again. In particular, implies that

Proposition 4.50 (b)

+ ,7) = f(r)i.

(4.54)

With this observation it is easy to conclude that the maximalideal space of C is homeomorphic to the unit circle ~ and that, hence, C is *-isomorphic to C(’I~). In accordance with the local principle (Theorem4.8), we associate with every point r e ~ the corresponding local ideal Z~ of Ao/,7. Our next goal is to identify the local algebras From equality (4.54) we conclude that the mapping (Ao/,7)/Zr

"-+ L(L2(V)), ((An) + ,7) + Zr ~ W~,2(A,~)

is well defined. Wedenote this mapping by Wr,2 again. Evidently, Wr,2 is a *-homomorphism from (Ao/‘7)/Z~ onto the C*-subalgebra of L(L~(~)) which consists of all operators of multiplication by a piecewise continuous function. The latter algebra is clearly isomorphic to the algebra PC(’~) of all piecewise continuous functions on "1~, provided with the supremum norm. Proposition

4.53 Wr,2 is a *-isomorphism from (.Ao/Y)/Z, onto PC(T).

4.5. Proof.

PAIRED CIRCULANTS

195

Consider the *-homomorphism

vT: Pc(v) (,4ol3)1z

+ J)

where c~rel(t) := c~(1/t) as before. From Proposition 4.50 we know W~,2Vr is the identity operator on PC(’I~), and we claim that V~W~,2is the identity operator on (Ao/J)/ZT. For this claim, it is sufficient to check whether VrW~,2maps every generating coset of (Ao/,7)/Zr onto itself (recall that V~ W~,2is a homomorphism).Since the sequences (~)(&n) a E C(~’) and ~ PC(~£) generate th e al gebra A, andsinc e ((an)(an) + :~) + Z, = (a(~)(a,) it is, thus, sufficient to check whether VrWr,2((a(r)(&n) + if)

+ Zr) = (a(T)(&n)

for every piecewise continuous function ~. But this equality is again a simple consequence of Proposition 4.50. ¯ Local invertibility. To finish the proof of Theorem 4.51, we have to prove that, if all operators W~,I(An) with T ~ ~I’ are invertible, then the coset (An) + is invertible. If thi s is shown, then the lift ing theorem yields the stability of the sequence (A~). By the local principle and Proposition 4.53, the invertibility of (An)÷ is equivalent to the invertibility of all operators Wr,~(A,~) with r ~ ~. Thus, we are left with the task of showingthat the invertibility of all operators W~,~(An) implies the invertibility of all operators WT,~(An). Actually, we will see that already the Fredholmnessof all operators W~,~ (An) is enough to guarantee the invertibility of all Wr,~(A,~). Wewill verify this implication for sequences (A,~) ~ .4 of the form (A~) = ((dl)~)((d~)n)

((d~) n)((dk),~) + (Jn)

(4.55)

where the aj are continuous, the aj are piecewise continuous, and where (Jn) ~ ~7. These seqences form a dense subalgebra of A0, thus, if the implication is true for these sequences then a simple approximation argument yields its validity for all sequencesin ‘4oLet (An) be as in (4.55), and let T, a E ~’. By Proposition 4.50, W¢,~(A,~)

= (a~a~(T +0) +... (alal

+a~a~(T +

(T -- 0) ÷...

with a certain compact operator K, and

(An)

+...

akO~k(T --

O))Q + K

196

CHAPTER4.

STABILITY ANALYSIS

From Theorem 4.42 we infer that the singular integral operator cP + dQ with continuous coefficients c and d is Fredholmif and only if c(a) ~ and d(a) ~ for al l a E ~. Thus, the Fredholmness of a ll operators W~,I(An) implies that al(a)~l(T±0)W...+ak(a)~k(T±0)

#0

for all a, T E ~. But then, evidently, all functions al(a)(a~)ref with a E ~" are invertible.

.. .ak(a)(a~)ref

Thus, the proof of Theorem4.51 is complete.

Applications. For applications of the stability theorem to concrete approximation sequences (An) ~ one ha s to compute all the asso ciated strong limits W~-,j(An) with T ~ ~" and j ~ {1, 2}. This computation often requires to represent the matrices A,~ as paired circulants which can prove to be n quite serious problem as already mentioned. For the concrete methods (4.46) - (4.48), details of this computation can be found in Chapter 10 of [123], and here are the results presented in a geometric language. In all cases, the approximation method is applied to the singular integral operator aI + bS with continuous coefficients. The sequence (As) of the Galerkin method (4.46) belongs to algebra A where the parameter e is zero. This method is stable if and only if a(T) + #b(T) for all T e ~"and #e [- 1, 1].

(4.56)

¯ The sequence (An) of the e-collocation method (4.47) belongs to algebra A with algebra parameter e. If ~ ~ (0, 1) \ {1/2}, then sequence (An) is stable if and only if condition (4.56) is satisfied. same condition appears in case ~ = 0 and d is odd, and also in case ¢ = 1/2 and d is even. If ~ = 0 and d is even, then the collocation method(4.47) is stable if and only tta(~-) + b(T) for all T ~ V and # E [- 1, 1].

(4.57)

¯ The sequence (A,~) of the quadrature method (4.48) belongs to the algebra A again with parameter s -= 0. This method is stable if and only if condition (4.57) is satisfied.

4.6.

4.6

FINITE

Finite ators

SECTIONS

OF BAND-DOMINATED OPERATORS

sections

197

of band-dominated oper-

Our last example concerns the finite section method for band and band dominated operators. A linear bounded operator A on /2(Z) is band operator if the ijth entry in the matrix representation of A with respect to the standard basis (ej) of/2(Z) vanishes whenever[i-j[ _> r for some fixed r. Normlimits of band operators are called band dominated operators. Clearly, every band operator can be uniquely written as a finite sum ~ aiV~ where the ai are bounded multiplication operators (which are given by a diagonal matrix) and where the V~ are the shift operators on /2(Z) mappingej to ej+i. Conversely, every finite sum of this kind defines a band operator, and this equivalence allows to think of band operators as being built of two "bricks" : the multiplication and the shift operators. Weshall adopt this point of view when introducing the multidimensional analogues of band and band dominated operators. 4.6.1

Multidimensional

band

dominated

operators

Let k be a positive integer which is fixed in this section, and let 12 denote the Hilbert space of all complex-valued functions f on Zk such that

Further, let Vm k, stand for the operator of shifting a function by m E Z which acts on 12 as the unitary operator (V,~f)(l)

:= f(l-m),

k.

In what follows, we will not consider the most general class of multiplication operators on/2; instead of, we will only deal with operators of multiplication by functions which behave sufficiently well at infinity. To be more precise, let Sk-1 and Bk refer to the unit sphere {x ~ ~k : ix ] = 1} and the unit ball {x ~ Rk : Ixl < 1} in ~k, respectively. Further, given a continuous complex-valued function a on ll~ k and a positive real t, denote by at the function at :.S k-~ "-+ C, rI ~q. a(t~l). Wesay that a belongs to the class C(l~k) if the functions at converge uniformly on S~-~ as t tends to infinity. With pointwise operations and the supremumnorm, the set C(IRk) becomes a commutative C*-algebra. If

198

CHAPTER 4.

STABILITY

ANALYSIS

a belongs to C(~k) then the limit of the functions a, will be denoted by a~°. Clearly, a°~ is a continuous function on the unit sphere S~-1, and aC~(~/) = lim a(t~l) for every r/¯ Sk-l: t---~oo

The non-trivial multiplicative functionals on C(I~k) are either of the form x* : a ~-~ a(x) with some x ¯ l~ ~ or ~/* : a ~+ a°°(~/) with some r/ Sk-~. So the maximal ideal space of C(I~~) can be thought of as a union of ~k with the ’infinite sphere’, or as the compactification of ~k by a sphere (the notation Ii~ k is chosen to indicate this compactification). The Gelfand (weak*) topology makes this maximal ideal space homeomorphic to the ball Bk with a natural bijection between ll~ k and Bk \S k-1 and with identifying the sphere of the infinite points with the boundary Sk-~ of Bk. In particular, for each point r/E Sk-~, kthere is a sequence (h,~) C_ ll~ (tending to infinity in the Euclidean topology) which tends to the functional ~/* in the Gelfand topology. It is not hard to see that the h,~ can be already chosen amongthe points with integer coordinates. Given a function a ¯ C(~k) we let 5 refer to the restriction of a onto k. Z Again, the collection C(Zk) of all functions & with a ¯ C([¢~) forms a commutative C*-algebra under pointwise operations and the supremum norm(now over Z ~), and the non-trivial multiplicative functionals on C(Z are precisely those of the form x* with x ¯ Zk and r/* with ~/ ¯ k-~. S Clearly, every function a E C(Z~) induces a bounded linear operator on 12 via (a f)(1) := a(l)f(l) which we call operator of multiplication by a and denote by aI. The band operators on 12 we will be concerned with in what follows are just the operators of the form ~’~l aIV~ where at ¯ C(Z~) and where the summationis over a finite subset of Zk. Further we consider the closure/3 of the set of all band operators in L(/2), the elements of which we call band dominated operators. Both the sets of the band and the band dominated operators are algebras, and B is moreover a C*-subalgebra of L(12). 4.6.2

Fredholmness

of

band

dominated

operators

Our next goal is a Fredholm criterion for band dominated operators. Given ~ ¯ Sk-~, we choose a sequence (hm) C_ k which c onverges t o r /* i n t he Gelfand topology of the maximal ideal space of C(Zk). Proposition 4.54 I] A is a band dominated operator, then the strong limit (4.58) exists, and this limit is independent of the choice o] the sequence (hm).

4.6.

FINITE

SECTIONS

OF BAND-DOMINATED OPERATORS

199

Proof. It is sufficient to verify the existence of the strong limits (4.58) for shift operators A = Vt and for multiplication operators A = aI with a E C(7/,k). For shift operators one evidently has V-hmVtVhm= Vt. Thus, the sequence in (4.58) converges in that case to Vt even in the norm. So let A = aI be a multiplication operator. For a moment,denote by ahm the function 1 ~-a(l + hm), which belongs to C(Zk) again. Obviously V-h~ aIVh.~ = ah.~I, and if f is an element of l 2 then (ah.~f)(l) = a(l + hm)f(1). Nowsuppose that f vanishes outside a certain finite subset of Zk. Since the sequence (l + hm) also converges to r/as m ~ c~, one has

lim ahf =

=

The finitely supported functions are dense in 12, so the Banach-Steinhaus theorem yields the assertion for arbitrary multiplication operators. ¯ Abbreviate the strong limit (4.58) (which depends on A and r/ only) A,. Thus, if A is the band operator ~t a~Vt, then A~ = ~t a~(r/)Vt, e. An is simply a linear combination of shift operators. Moregeneral, the mapping A ~-~ An is a *-homomorphismfrom the C*-algebra B of all band dominated operators onto the smallest closed subalgebra of L(l ~) which contains all shift operators V,~. In particular, all limit operators An are shift invariant: V-mA, V,~ = A,. Theorem 4.55 A band dominated operator A is Fredholm if and only if the limit operators An are invertible for all ~1 ~ Sk-1. Sketch of the proof. One starts with showing that the algebra B contains the set K(l ~) of the compact operators as its ideal, and that the image C~(Zk) of C(Zk) in the quotient algebra B/K(I2) belongs to the center of that algebra. To this end, one has to take into account that, whenever a belongs to C(Zk) and am is the function I ~-r a(l+m), then the function aa,~ has limit zero at infinity and, hence, induces a compactmultiplication operator. So one can localize B/K(12) over C~(Zk) via Allan’s local principle. The maximal ideal space of C~(Zk) is homeomorphicto the sphere k-~ .S It remains to observe that, for every band dominated operator A, the limit operators Ao only depend on the coset A + K(/2) (thus, Fredholmness of implies invertibility of An)’ and that the cosets A + K(l~) and An ~) + K(l coincide locally at ~/~ Sk-1 (thus, if every o i s i nvertible, t hen A+ K(/2) is invertible by Allan’s principle).

CHAPTER 4. STABILITY ANALYSIS

200 4.6.3

Finite

sections

of band

dominated

operators

Wecontinue with the finite section method for band dominated operators. To simplify notations, we assume k -- 2 throughout this subsection. Let fl be a compactand convex polygon in ll~ 2 with vertices in Z2, and suppose that (0, 0) is an inner point of ft. The boundary of f~ will denoted by 0~. Further, let X~ stand for the characteristic function of ~, set Xm~(x) := X~ (x/m) for positive integers m, and denote the restriction of Xm~to Z2 by )~m~. For every band dominated operator A, we consider its finite sections Am := ~m~A~mf~I thought of as acting on the space Im ()~maI). What we are interested in again stability criteria for the sequence (Am)m>~. Let Ul, ..., Uk denote the vertices of ~, and set uk+l :---- ul and u0 :---Uk. For j ---- 1, ..., k, let Ij stand for the open segmentjoining uj to uj+~, write Hj for the half plane which is bounded by the straight line through uj and u~+l and contains the origin (0, 0), and set Kj := H~._~ NH~(with Ho :---- Hk). Further, define H~ and KS as the algebraic differences H~. - uj and K~ - uy, respectively. The characteristic functions Of the sets Hj and Kj will be abbreviated to XHj and XK~, and we set

xm.j :=

and Xmg~ (X) := XK~(XI’~)

for every positive integer mand write )~mH~and )~mK~for the restrictions of )~mH~ and XmKi to Z2, respectively. Finally, let C(I~2) stand for the C*-algebra of all functions which are continuous on ~2 and possess a limit at infinity, and write again ] for the restriction of the function f ~ C(]~2 ) to Z2 and f,~ for the function x ~ f(x/m). Nowintroduce the smallest closed *-subalgebra A of the C*-algebra 5r of all bounded sequences (Am) of linear bounded operators on/2(Z2), which contains (A) all constant sequences (A) with A (B) all sequences (f(mH~I)m>_~with j = 1, ...,

k,

(C) all sequences (Gm) with [[am[[ -+ 0, (D) all sequences (]mI) 2). with f e C(~ The sequences under (C) form a closed ideal ~ of A and, clearly, a sequence (Am) ~ .4 is stable if and only if the coset (Am) + is invertible in the quotient algebra A/G. Notice further that Xmais just the product of

4.6.

FINITE,

SECTIONS OF BAND-DOMINATED OPERATORS 201

all functions XmH~;hence, the sequences (~m~Af~m~I)m>lwith A band dominated belong to .4, too. The derivation of the desired stability criterion is based on the following two simple observations. Proposition 4.56 If f e C(I~2), then the coset (]mI) + 6 belongs to the center of Proof. The sequences (]mI) commute with every sequence of multiplication operators. So it remains to check whether

for every constant sequence (Vt) of shift operators. Since

with g(x) = f(~) -f(~A), this is a consequence of the uniform continuity 2of. the function f on I~ ¯ For the next result we need some more notations. Let x ¯ 0~. Then there is a unique point ~ = r~(x) in the sphere of the infinite points such that mx -~ rl(x) in the Gelfand topology as m ~ oo (under the natural identification of the infinite sphere with the unit sphere S1 one simply has ~(x) = x/llxll ). Further, we associate with every x ¯ 0f~ and every positive integer m a point Xm ¯ Z2 as follows: If x is the vertex uj of f~, then Xm := muj. In case x lies on some open interval Ij, we choose A0 ¯ (0, 1) such that x = Aouj + (1 - A0)uj+~, and then define Xm := [mAo]uj + (m - [mA0])Uj+l. Here [y] refers to the integer part of the real y. In any case, the point Xm belongs to the boundary of mHj, and the sequence (Xm)m>lconverges to O(x) in the Gelfand topology. With these remarks, the proof of the following proposition, establishing the existence of certain strong limits, is straightforward. Proposition 4.57 (a) If (Am) ¯ A, then the strong limits s-lim Am =: W(A,,~) and s-lim A~ exist. In particular, one has for the generating sequences (A) W(A) = A for band dominated operators (B)

W(~m.~I)

:

fo r j = 1, ..., k.

(c) W(G) = o Ior

202

CHAPTER 4.

(D) W(]mI) = f(O)I

STABILITY

ANALYSIS

for f ¯ C(~2).

(b) Let x ¯ Off and (Am) ¯ ,4. Then the strong limits s-lim

V-z.~AmVzm =: Wz(Am) and s-lim

V_zmA*mV~m

exist. In particular, one has for the generating sequences of .4 (A) W,(A) = An(z) for band dominated operators (B) for j = 1,...,

k,

2~1

Wz (YimHj I)

{

I 0

if xeOHj if x ¯ interior of Hj if x ¯ exterior of Hi.

(c) wz(am)= o/or (cm) (D) Wz(]mI) = f(x)X for f ¯ C(~2). Obviously, the mappings W and Wz are *-homomorphisms from the C*algebra .4 into L(12). Let further 79 stand for the smallest closed subalgebra of .4 which contains all sequences (f~m~AYl,~I + (1 - Y~mu)I)m>l with A band dominated. The desired stability criterion reads as follows. Theorem4.58 A sequence (Am) ¯ 79 is stable if and only if the operators W(Am) and W~(Am)are invertible for every Proof. Wewill consider sequences in 79 as elements of .4 and work in this larger algebra. From Proposition 4.56 we know that the coset (]mI) belongs to the center of A/G whenever f ¯ C(I~:). It is moreover not hard to check that the set C of all of these cosets forms a C*-subalgebra of the center of A/G, which is *-isomorphic to the algebra C(l~ 2) and, consequently, has a maximal ideal space which is homeomorphic to the compactification ~2 of IR~ by the single point oo. So we can make use of Allan’s local principle in order to localize A/Gover its central subalgebra C. The outcome of this localization is local algebras .4z with canonical homomorphisms(hz : .4 -~ .4z for every x ¯ l~ ~. Allan’s local principle then states that a sequence (A,~) ¯ .4 is stable if and only if the local representatives ff~ (Am) are invertible in Az for every x ¯ ]~2. Let now (A,~) be a sequence in 79. It is evident from the definition that this sequence can be written in the form (y~,~eB,~f~,~nI + a(1

4.6.

FINITE

SECTIONS

OF BAND-DOMINATED OPERATORS

203

fimn)I)m>i with a complex number a and with a sequence (Bin) which can be approximated as closely as desired by sum~of products of sequences (~,~Aijf6~I) with Aij band dominated. Nowobserve that, for x belonging to the interior of fl, one has

whereas for exterior x or x = oo, O~(~:r~nI)= ’I~,(O). This yields

=

+ - f m )I)

for interior and ’~(Am) = (~(f~muB.~f~.mI

a( 1 - ;~ mn)I) = ¢~(~I)

for exterior x. Wewill see in a momentthat the invertibility of any of the cosets ¢~(A,~) with x E 0~ implies a ~ 0 and, hence, the invertibility ¯ ~(A.~) for all x in the exterior of f~. Further it is clear that stability (Am) implies the invertibility of the strong limit operator W(Am)of that sequence and that, conversely, the invertibility of W(Am)guarantees the invertibility of the cosets (~x(Am) for all x in the interior of f~. Consequently, a sequence (Am) in 79 is stable if and only if the operator W(A.~) is invertible, and if the local cosets ~(Am)are invertible for all x in the boundary of fL In the next step we are going to showthat, for x E 0f~, the coset is invertible if and only if the operator W~(Am) is invertible. FromProposition 4.57 (b) we infer that the operators W~(Am)only depend on the coset ¢~(Am). Thus, we can think of W~as homomorphisms acting the local algebras A~, and this clearly shows that the operators W~(A~) are invertible for invertible cosets (~x(Am). Let, conversely, the operator W~(Am) be invertible. Then the sequence (V~,~ W~(Am)V-~,,)~>~is stable, and this sequence belongs to the algebra A as one easily checks by considering the generating sequences of that algebra in place of (A,~). Thus, the coset q~ (V~ W~(A~)V_~,~) is invertible in A~. The assertion follows from the equality (~(Vx,,Wz(Am)V-z,.) = Oz(Am), which holds for all sequences (A.~) ~ A, and which is again easy to verify for the generating sequences of A.

204

CHAPTER 4.

STABILITY

ANALYSIS

Finally we observe that, if x E Ij for some j and Am= a(1 - fim~)I, then

=

+ c41-

Hence, ~ cannot be zero for invertible W~(Am).

¯

An analogous result holds for k > 2, if the finite sections of the band operators are generated by the characteristic function of a convex polytope with vertices in Zk and with 0 in its interior.

Notes and references Section 4.1: The simplest local principle is the classical Gelfand theory, which reveals that invertibility in commutativeBanachalgebras is local in nature (see, e.g. [154], 11.8 - 11.9). A first step in establishing generalizations of the Gelfand theory to the context of not necessarily commutative algebras was done by Simonenko [167]. He both realized the local nature of the Fredholmness of convolution and related operators and, at the same time, created a powerful machinery (his local principle) for tackling successfully a whole series of problems. Simonenko’s local principle was generalized by Kozak to arbitrary Banach algebras. Another modification of Simonenko’sprinciple, which is distinguished for its simplicity on the one hand and for its wide range of applicability on the other, was proposed by Gohbergand Krupnik ([72], Section 5.1). For a general look at all these local principles we refer to [22]. In the present textbook we prefer the local principle by Allan and Douglas due to its elegance and its appropriateness to our purposes. It was stated by Douglas [50] for the C*-algebra case and by Allan [1] in the general Banach algebra setting. Douglas was also the first to realize the importance of this local principle for problems in concrete operator theory, such as the Fredholmness of Toeplitz and singular integral operators with piecewise continuous coefficients [50, 51, 52]. Also the idea of combining local principles with a two projections theorem as in Section 4.4.1 in order to derive a criterion for the l~redholmness for singular integral operators is Douglas’. For further applications of local principles both in operator theory and numerical analysis see the monographs[26], [50], [72], [77] and [123], for instance. Section 4.2: There is a long history of the investigation of the finite section method for Toeplitz operators with several classes of generating functions; for details we must refer to the relevant monographssuch as [26], [77] or

4.6.

FINITE

SECTIONS

OF BAND-DOMINATED OPERATORS 205

[123]. Both the main results of Section 4.2 as well as the approach to the stability of the finite section methodvia lifting theoremand local principle is due to one of the authors [159, 162]. It was the very success of this approach which stimulated and determined the development in the piece of numerical analysis which deals with stability problems for several kinds of projection methods for complicated operators up to now. Here are, in addition to the approximation methods considered in Sections 4.3 and 4.4, a few more examples where this approach proved to be successful. The given references are by no meanscomplete; their only goal is to provide the interested reader with a starting point for further reading. ¯ Spline and wavelet projection methods(Galerkin, collocation, qualocation) and methods based on composedquadrature rules for singular integral operators with piecewise continuous coefficients on composed curves (circles, lines, intervals, more involved curves with intersections) ([77], [120], [123], [155]). ¯ Same methods for singular integral operators with conjugation and for double layer potential operators on curves ([46], [47], [48]). ¯ Same methods for Mellin convolution operators and Wiener-Hopf operators on the half line ([57], [77], [120], [123], [137]). ¯ Same methods for singular integral operators on spheres and other manifolds without boundary ([88]) and on the half plane ([59, 60]). ¯ Finite section methodfor Toeplitz plus Hankel operators, for singular integral operators with Carlemanshift ([145]). ¯ Finite section method as well as other Galerkin-Petrov methods for Toeplitz operators on the Bergmanspace ([28]). ¯ Finite section method with respect to weighted Chebyshev polynomials for singular integral operators on intervals ([89]). Section 4.3~ The results on the finite section methodfor Toeplitz operators with quasicontinuous generating functions and with generating functions in H~ ÷ C are quite obvious generalizations of the corresponding results in the continuous setting considered in Chapter 1. Theorem4.32 as well as the subsequent explicit construction of a quasicontinuous function are taken from [55], and Theorem4.36 is from [143]. The finite section method for Toeplitz operators with piecewise quasicontinuous generating function was first studied in [160, 161]; see also the monograph[26]. A far reaching generalization (operator-valued piecewise quasicontinuous coefficients, sums of products of Toeplitz operators) can be found in [56].

206

CHAPTER 4.

STABILITY ANALYSIS

Section 4.4: The results pertaining to the collocation method are due to Junghanns and Silbermann [90]; the approach via the two projections theorem is taken from [148]. Section 4.5: The discovery that the application of discretization procedures to singular integral equations leads in manycases ’to approximation matrices in form of paired circulants goes essentially back to PrSssdorf and Rathsfeld, [119]. In the same paper they also derived a stability criterion for sequences of paired circulants as in (4.50). The Banach algebra approach to study this stability problem is taken from [78]. In both papers [78] and [119], a more general class of sequences of paired circulants is considered which also includes approximation methods for singular integral operators with piecewise continuous coefficients. See also Chapter 10 in [123]. Section 4.6: These results are taken from the paper [124] where also more general classes of band dominated operators are studied. Especially, there is a Fredholmcriterion for arbitrary band dominated operators (with multiplication operators in l ~ rather than in classes of continuous functions) which is formulated in terms of so-called limit operators and which generalizes Theorem4.55 essentially. Addedin proo]:. In [125], there is derived a stability criterion for the finite section method applied to an arbitrary band-dominated operator on /2(Zk). This criterion is based on the following observation: If A is banddominated on 12 (Zk), then the sequence of the finite section approximations of A can be identified with a band-dominatedoperator acting on 12 (Zk+l), and this sequence is stable if and only if the associated operator is Fredholm. So the results concerning Fredholmness of band-dominated operators immediately apply to give a stability criterion for the finite section method for band-dominated operators. Since the algebra generated by the finite section sequences of all band-dominated operators is far away from being fractal, we hope and expect that the ~pproach of [125] c~n also serve as a model for dealing with the stability of other non-fractal approximation sequences.

Chapter 5

Representation

theory

A C*-algebra is either extremely well behaved (type I) or totally misbehaved (antiliminary). ... Thus there is a natural temptation to concentrate on type I C*-algebras and forget about the rest. As long as the theory is applied to group representations this point of view is quite fruitful, because a large number of interesting groups (among them all compact groups) give rise to C*-algebras of type I. For the applications in theoretical physics, however, the situation is not so easy. As a matter of fact all the relevant algebras are antiliminary. G. K. Pedersen

Roughly speaking, all we did in the previous chapter was the following: we put some interesting approximation sequences into a C*-algebra .4 and tried to find a family {Wt}tET of homomorphismsfrom this sequence algebra .4 into algebras of operators on a Hilbert space having the property that a sequence (An) E .4 is stable if and only if all associated operators Wt(An) are invertible. Two points are of importance and should be emphasized;the first is that stability is equivalent to the invertibility o] ’something’, whereas the second is that the ’something’ is operators on a Hilbert space. This second point is of importance since operators have kernels (which have a dimension), they have ranges (which are sometimes closed), there is a spectral theorem, a Fredholmtheory, and a lot of further 207

208

CHAPTER 5.

REPRESENTATION

THEORY

ingredients which are not immediately available for elements of a general C*-algebra, and which make it much easier to work with operators rather than with elements of an algebra. It is just this latter point whichstands in the center of the present chapter: Wewill consider *-homomorphismsfrom a C*-algebra into the algebra of all linear boundedoperators on a Hilbert space, so-called representations of the algebra. In Section 5.1 we summarize the needed prerequisites from representation theory and determine all irreducible representations of some of the algebras we already met in Chapters 1 and 4 (thus, convincing the reader that all we have done up to nowis nothing but representation theory of concrete C*-algebras). Then (Section 5.2) we are going to single out a special but very comfortable class of C*-algebras which encloses all the concrete algebras considered before. In the concluding Section 5.3, we examine the connections between representation theory on the one hand and one of our main instruments in Chapter 4, the lifting theorems, on the other hand. Only at the end of Section 5.3 we turn back to the first point mentioned above: the search for conditions which guarantee that the invertibility of all operators Wt(An)is sufficient for the stability of (A~).

5.1 Representations Westart with a brief introduction of some basic notions of representation theory, such as unitary equivalence and irreducibility of representations, and primitive ideals. Moredetailed expositions as well as the proofs of the cited results can be found in almost every book on C*-algebras. In particular we recommend ARVESON [3], Bt~ATTELI,ROBINSON [32], DIXMIER[49] (which serves as our main reference here), FELL, DOP~AN [58], KADISON, RINGROSE [91], KHELEMSKI! [94], MURPHY [110] and PEDERSEN [114]. 5.1.1

The

spectrum

of

a C*-algebra

A representation of a C*-algebra P.l is a pair (H, ~r) constituted by a Hilbert space H and a *-homomorphism r from P2 into L(H). The representation (H, ~r) is faithful if the kernel of the homomorphism ~r consists of the zero element only. If (H, r) is a faithful representation 92, then 92 is *-isomorphic to a C*-subalgebra of L(H). The famous GNStheory (Theorem 1.48) states that every C*-algebra possesses a faithful representation. Let 92 be a C*-algebra and (H, r) a representation of 92. A subspace

5.1. REPRESENTATIONS

209

of H is invariant for ~r if ~r(a)KC_K for all

aE91.

Thus, if K is a closed subspace of H, and PK denotes the orthogonal projection from H onto K, then invariance of K for r just means that Pg~r(a)Pg = ~r(a)PK for all a e 91.

(5.1)

A closed subspace K of H is invariant for ~r if and only if Pgr(a) = ~r(a)PK for all a e 91.

(5.2)

Indeed, (5.1) implies (5.2): Pg~r(a) = (Tc(a*)PK)* = (PKTc(a*)PK)* = Plc~c(a)Pg On the other hand, (5.1) follows from (5.2) by multiplication PK. The zero space {0} and the space H itself are invariant for every representation. A non-zero representation (H, re) of a C*-algebra 9/is irreducible if {0} and H are the only closed subspaces of H which are invariant for ~r. If 91 is a C*-algebra with identity e then every irreducible representation (H, ~r) of 91 mapse into the identity operator on Remark. It would be more correct to refer to irreducible representations as topologically irreducible representations, in contrast to algebraically irreducible representations, for which {0} and H are the only (not necessarily closed) invariant subspaces. Both notions coincide in the case of C*-algebras and *-homomorphisms(see [49], 2.8.4), but observe that this is no longer true for arbitrary Banach algebras. ,, The following theorem summarizessome characterizations

of irreducibility.

Theorem5.1 Let (H, ~r) be a representation of a C*-algebra 91. The following assertions are equivalent: (a) (H, ~r) is irreducible. (b) If B e L(H) commutes with every operator ~r(a), a e 91, then scalar multiple of the identity operator. (c) The set {~r(a)x : a ~ 91} is dense in H for every non-zero vector x ~ H. (d) {~r(a)x : a ~ 91} = H for every non-zero vector x ~ H. The equivalence of (a) and (b) is knownSchur’s lem ma, andasse rtions (c) and (d) are often rephrased as follows: Every non-zero vector in H is topologically respective algebraically cyclic for 7r. For a proof see [49],2.3.1.

210

CHAPTER 5.

REPRESENTATION

THEORY

It is not at all clear whether a given C*-algebra possesses irreducible representations. The following theorem showes that every C*-algebra possesses, in some sense, sufficiently manyirreducible representations. Theorem 5.2 Let 91 be a C*-algebra and a E 91. Then there exists irreducible representation( g, ~r) o] 91 such that Ilall =II~r(a)ll. For a proof, see [32], 2.3.23, and [49], 2.7.3.

an .

Tworepresentations (H1, ~rl) and (H2, ~r2) of a C*-algebra unit arily equivalent if there is a unitary operator U from H1 onto H2 such that -~ for all a E 91. ~r~(a) = U~r~(a)U Equivalence of representations is an equivalence relation in the set of all representations. If two representations of a given algebra are equivalent, and if one of them is irreducible, then so also is the other. The set of all equivalence classes of irreducible representations of a C*-algebra 92 is called the spectrum (or the structure space) of 91, and we will denote it by Spec 92. In what follows we denote the equivalence class of Spec 92 containing the representation (H, ~r) simply by (H, ~r). 5.1.2

Primitive

ideals

The kernels of irreducible representations of a C*-algebra 92 are called the primitive ideals of 92. Wedenote the set of all primitive ideals of 91 by Prim 91. If two irreducible representations of a given algebra are unitarily equivalent, then their kernels coincide (but the converse is false in general). there is a natural mapping Spec 91 -~ Prim 91, (H, ~r) ~ Ker

(5.3)

which is onto, but not one-to-one in general. A C*-algebra is simple, if its only closed ideals are the zero ideal and the algebra itself. Examplesof simple C*-algebras are the algebras of the complex k × k matrices and the algebra of the compact operators on a Hilbert space. If 91 is a simple C*-algebra, then Prim 91 consists of the zero ideal only. On the other hand, there are examples of simple C*algebras (e. g. the irrational rotation algebras, see [44], Chapter VI) which possess a great number of mutually non-equivalent irreducible representations. Thus, Spec 91 reflects the structure of 92 with higher precision than Prim 91, whereas the latter space is better accessible. Roughly speaking, the spectrum Spec 91 is accessible only in the rare cases where it coincides

5.1.

211

REPRESENTATIONS

with Prim 91, i.e. where the mapping(5.3) is a bijection. In what follows we will only have to deal with these ’good’ situations. Here are some elementary properties of primitive ideals. Theorem 5.3 Let 91 be a C*-algebra. Then (a) the intersection of all primitive ideals of 91 is {0), (b) every proper closed ideal 3 of 92 is equal to the intersection of all primitive ideals of 91 whichcontain 3. Assertion (a) is an immediate consequence of Theorem5.2, and the simple proof of (b) is in [49], 2.9.7. The space Prim 91 carries a natural topology the definition of which is based on the following observation (see [49], 2.11.4). Proposition 5.4 Every primitive ideal of a C*-algebra 92 is prime, i.e. if 3 E Prim 91, and if 31 and 32 are closed ideals of 91 with 3~32 C_ 3, then Given a subset Mof Prim 91, we define its kernel resp. its hull by ker M:=

N3EM~

resp. hull M := {3 E Prim 91 : ker MC_ 3}.

Theorem 5.5 Let 9_1 be a C*-algebra. The mapping M ~ hull M which is defined on the subsets of Prim 91, satisfies Kuratovski’s axioms o] closure, i.e. (i) hull 0 = (ii) MC_ hull (iii) hull (hull M)= hull M. (iv) hull M1ID hull M2= hull (M1 U M2). The simple proof is an easy exercise (compare [49], 3.1.1). verification of axiom (iv) is based on Proposition 5.4.

Note that the

Thus, the sets hull Mwith MC_ Prim 91 are the closed sets of a certain topology on Prim 91, the so-called hull-kernel or Jacobson topology. The properties of the topological space Prim 91 are less convenient than those one is accustomed from the maximal ideal space of a commutative C*algebra. In particular, Prim 91 is no longer Hausdorff, but it is still a Tospace, i.e. for any two distinct points of the space there is a neighborhood of one of the points which does not contain the other. Moreover, if 91 is unital, then Prim 91 is compact(see [49], 3.1.3 and 3.1.8). The closed one-elementic sets in Prim 91 can be characterized as follows ([49], 3.1.4):

212

CHAPTER 5.

REPRESENTATION

THEORY

Proposition 5.6 For 3 E Prim 91, the singleton {3} is closed if and only if 3 is maximalin the set of primitive ideals. The pre-image of the hull-kernel topology under the mapping(5.3) defines a natural topology on the spectrum of 91. For details we refer once more to [49], Chapter 3.

5.1.3 The spectrum of an ideal and of a quotient Let 91 be a C*-algebra. Every closed ideal 3 of 91 involves a decomposition of the spaces Spec 91 and Prim 91. Indeed, consider Spec~P.l := {(H, ~-) E Spec P2 : ~r(3) = Spec391 := {(H, r) ~ Spec 91: r(3) # Obviously, Spec391 U Spec391= Spec 91, Spec391~3 Spec~91= 0, and Spec391 and Spec~91are closed and open subsets of Spec 91, respectively. Theorem5.7 Let 91 be a C*-algebra and 3 be a closed ideal o] 91. (a) For every (H, ~) ~ Spec3P2, let ~r /3 denote the quotient homomorphism a + 3 ~ ~r(a) from 91/3 into L(H). The mapping 7r ~-+ ~r/3 is a homeomorphism from Spec~91 onto Spec (91/3). (b) For every (H, ~) Spec391, let 7r 13 denote th e re striction of The mapping r ~-~ ~1~ is a homeomorphismfrom Spec~91 onto Spec3. See [49], 2.11.2 and 3.2.1 for a proof. In an analogous manner, we introduce Prim~91:= {I ~ Prim 91 : 3 C_ I}, Prim~91:= {I E Prim 91 : 3 q[ I}. As above, Prim~91 U Prim~91 = Prim 91, Prim~P2 N Prim391 = 0, and the sets Prim391 and Prim391 are closed and open in Prim 91, respectively. Theorem 5.8 Let 91 be a C*-algebra and 3 be a closed ideal of (a) I ~ I/3 is a homeomorphismfrom Prim~91 onto Prim (91/3). (b) I ~-~ I ~3 3 is a homeomorphismfrom Prim391 onto Prim 3. A proof is in [49], 2.11.5 and 3.2.1. Thus, by the preceding two theorems, there are canonical mappings Spec (91/3)

~ Spec 91 ~---

Spec

Prim (9.1/3)

--~ Prim 91 ~- Prim

5.1.

REPRESENTATIONS

213

Moreover, by Theorem5.7, the set Spec 3 can be viewed as an open subset of Spec 91. The converse statement is also true: Theorem 5.9 Let 91 be a C*-algebra. The mapping 3 ~-~ Spec 3 is a bijection from the set of the closed ideals of 91 onto the set of the open subsets of Spec 91. Moreover, 31 C_ 32 ¢==V Spec 31 C_ Spec 32. See [49], 3.2.2 for a proof of this result, and [49], 3.2.3 for its following corollary. Theorem5.10 Let 91 be a C*-algebra and 31, 32 closed ideals o]91. Then Spec (31 +32) = Spec 31USpec 32 and Spec (31N32) = Spec 31NSpec 32. In particular, 31 f~ 32 = {0} if and only if Spec 5.1.4

Representations

of

some concrete

The following examplesare intented as illustrations oped in the preceding sections.

algebras of the concepts devel-

Example 1: Commutative C*-algebras Every character (i.e. every non-zero multiplicative functional) of a commutative C*-algebra is an irreducible representation of this algebra and, conversely, every irreducible representation is unitarily equivalent to a character. Moreover, the correspondence between characters and equivalence classes of irreducible representations is bijective. Hence, the primitive ideals of a commutative C*-algebra are just its maximalideals, and one can mutually identify the spaces Spec 91, Prim 91, and the maximalideal space of 91. Underthis identification, the hull-kernel topology on Prim 91 coincides with the Gelfand topology on the maximal ideal space discussed in Section 4.1.1 (compare [58], Chapter VII, Section 3.2 and Corollary 5.11). The coincidence of the two topologies is a typical C*-effect: For commutative Banach algebras one can only show that the hull-kernel topology is contained in the Gelfand topology, and there are examples already in the class of commutativeBanach*-algebras where these topologies are distinct (for details see [58], Chapter VII, Proposition 3.11 and Example3.12). Example 2: An algebra

of matrix functions

Let 91 stand for the C*-algebra of all continuous 2 × 2 complex matrix functions on the interval [0, 1] whosevalue at 0 is a diagonal matrix. Every

214

CHAPTER 5.

REPRESENTATION

THEORY

point t E (0, 1] gives rise to a two-dimensional representation of 92 via ’evaluation’, 7rt: 91-~L(C2), f~f(t), and there are two one-dimensional representations associated with t -- 0: a: 92-+L(C),

f~f11(0)

and f~:

92-~L(C),

f~f22(0).

These are (up to unitary equivalence) all irreducible representations of 92. Hence, Spec 92 is homeomorphicto Prim 92 and can be identified with the union (c~,/~} U (0, 1] with two separate points a and/~ in place of 0. The restriction of the hull-kernel topology onto (0, 1] is the usual (Euclidean) topology, whereas a neighborhood base of c~ (resp. f~) is given by the sets (~} t~ (0, e) (resp. (/~} U (0, e)), e running through (0, 1). Thus, neighborhoods of c~ and /~ have a non-empty intersection, which reveals that the hull-kernel topology on Prim 92 is not Hausdorff. ¯ Example 3: Algebras

of compact operators

Let H be a Hilbert space and 92 = K(H) be the C*-algebra of all compact linear operators on H. Then every irreducible representation of 92 is equivalent to the identical mapping of K(H) into L(H). Hence, both sets Spec 92 and Prim 92 are singletons, and {0} is the only primitive ideal of 91. For details see [49], 4.1.5. As a consequence, one has ([49], 4.1.8): Theorem 5.11 Let G and H be Hilbert spaces, and let 7r be a *- isomorphism from K(G) onto K(H). Then there is a unitary operator V from onto G which defines ~r in the sense that ~r(K) = Y*gY ]or all Moreover, V is unique up to multiplication

g ¯ K(G). by a unimodular number.

A C*-algebra 92 is called elementary if there is a Hilbert space H such that 92 is *-isomorphic to K(H). Every elementary algebra is simple because Prim K(H) consists of the zero ideal only and since every closed ideal ~i of 91 is the intersection of all primitive ideals containing 3 (Theorem5.3). Let now H be an infinite-dimensional Hilbert space and consider the C*algebra 91 = CI+K(H) of all operators AI+Kwith complex A and compact K. If ~r is an irreducible representation of 92 then either K(H) C_ Ker ~r or K(H) Ker ~r . In the first case, the quotient representation ~r/K(H) exists. It is an irreducible representation of the quotient algebra 92/K(H) ~- C~ by Theorem 5.7 and hence unitarily equivalent to the representation a: 91--+C,

AI+K~-~A.

5.1.

215

REPRESENTATIONS

In the second case, again by Theorem5.7, the restriction of r onto K(H) is an irreducible representation of K(H) which is, as just mentioned, unitarily equivalent to the identical mapping from K(H) into L(H). Since the mapping rr ~-~ 71"[K(H) is a bijection from SpecK(H)~onto SpecK(H) (Theorem5.7 once more), we conclude that 7r is unitarily equivalent to the identical representation ~ : ~-+ L(H), AI + K ~ AI + Thus, Spec 9.1 as well as Prim 91 are doubletons, and {0} = Ker ~ and K(H) = Ker a are the primitive ideals of 9~. The hull-kernel topology on Prim 93 is not the (obvious) discrete one. Indeed, while the closure of {K(H)}is the point {K(H)}itself, hull (K(H)) = {3 E Prim 91: K(H) C_ 3} -- {K(H)}, one has hull ({0}) = {~ e Prim ~: {0} C 3} -- {{0}, K(H)} = Prim ~[. Thus, the closure of the point {~} is all of Spec 92 or, in other words, the constant sequence (~)~=0 has the two limits ~ and a. ¯ Example 4: Algebras of Toeplitz

operators

Let again T(C) refer to the smallest closed subalgebra of L(l 2) which contains all Toeplitz operators with continuous generating function. Every operator in T(C) can be uniquely written as T(a) + where a ~ C(T) an K is compact (Theorem 1.51); furthermore, T(C) is a C*-algebra which contains the ideal K(/2) of the compactoperators, and the quotient algebra T(C)/K(I2) is *- isomorphic to C(T) (Section 1.4). If zr is an irreducible representation of T(C), then either K(/2) C_ Ker ~ or K(/2) (~ Ker ~r. As in the preceding example, we conclude in the first case that the quotient representation ~r/K(l2) is an irreducible representation of the algebra T(C)/K(l 2) ~- C(~) and hence zr is unitarily equivalent to an representation of the form ~ : T(C) -~ C, T(a) + K ~ with some t E T due to Example 1. In the second case, ~r is unitarily equivalent to the identical representation ~ : T(C) -+ L(/2), T(a) + g ~-+ T(a)

216

CHAPTER 5.

REPRESENTATION

THEORY

The mappings ~ and (it with t E ~ exhaust (up to unitary equivalence) the irreducible representations of the Toeplitz algebra T(C), and we can think of both Spec T(C) and Prim T(C) as the union ~7(~ {~}. The restriction of the hull-kernel topology onto the component ~7 coincides with the familiar Euclidean topology, whereas the closure of the point {~} is all of Spec T(C). In other words, the constant sequence (~)~=0 converges to each of the infinitely manypoints in the spectrum of T(C). It is convenient to think of Spec T(C) as the closed disk ~ U l~ whose boundary points t are identified with the representations (it and whoseinterior ll) is thought of the identical representation ~ (Figure 5.1).

Figure 5.1: Spectrum of T(C) The appearance of one point of the spectrum which lies dense in the whole spectrum, as observed in the latter two examples, is typical for socalled primitive algebras, i.e. for algebras having {0} as its primitive ideal (compare[49], 3.9.1). Proposition 5.12 I] 91 is a primitive C*-algebra, then its spectrum contains one point which is dense in Spec 91. In a similar way, one can describe Spec T(PC) ~- Prim T(PC) for the algebra T(PC) generated by the Toeplitz operators with piecewise continuous generating function. Besides the identical representation 5, all other irreducible representations are one-dimensional, and the one-dimensional representations are in one-to-one correspondence with the points of the cylinder Z = ~I" x [0, 1]. The restriction of the hull-kernel topology to Z (or, what is the same, the Gelfand topology on the maximal ideal space of the commutative C*-algebra T(PC)/K(12)) is not the standard (Euclidean) topology: An open neighborhood base of the point (t, x) where t E ~" and x E (0, 1) is formed by the sets {t}

× (x-6,

x+6)

with

0<e<min{x,l-x}.

5.1.

REPRESENTATIONS

217

The sets ([t,

te ~)x (1-~,l])U((t,

~) x[ 0,1-~]) wi

th 0< ~o wi th/_i co

P~K~P~ P,~K12P,~))

0

))+~

~o

mpact),

+~ withKijcompact},

with~compact}.

{((0 One can show (by having recourse to the results of Ex~ple 6, for instance) that ~ is congNnedin ~(C) and that ~_~, ~ ~d ~ are subsets of the quotient algebra ~ := P(C)/0 and therefore are closed ideals of this algebra. Moreover,the ideal ~0 is *-isomorphic to the ideal of the compactoperators on l~(E), the isomorphism being & : (RnKRn)

+ G

P~K21P~ K~

PnK~P~ K~

~>o

’

whereas both ~_~ and ~ are *-isomorphic to K(l~(g+)) via the isomorphisms S_~ : S1 :

0

0

~o

0 W~L~W~

~o respectively. The mappings S0 resp. S-1 and S~ can be extended to homomorphismsof ~ into L(/:(Z)) resp. L(/2(Z+)) S0 : (An) +~ ~ s-lim S-1

: (An)+~

s- lim(

AnRn, Wn0 ) An( Wn

0

~

"

It is an e~y matter to check that these homomorphismsare irreducible representations of N. Put ~ := ~_~ + ~ + ~. This is a closed ideal of ~, ~d ~/~ is a commutative C*-algebra which is *-isomorphic to C(X), where

222

CHAPTER 5.

REPRESENTATION THEORY

X is the union of two circles (~_~and ~1, say) without commonpoints. The one-dimensional representations of P.I associated with the points of "1~1 and T-1 act on the generating cosets of the algebra 92 as t ¯ ~ ~- ~F1 : (Rn(L(a)P + L(b)Q)R~) + ~ t ¯ ~ ~- T_I: (R~(L(a)P + L(b)Q)Rn) + ~ Thus, we have found (up to unitary equivalence) all irreducible representations of 91, and the spaces Spec 91 ~ Prim 91 can be identified with ~’_1 L) ~’~ L) (S_;, So, $1}. The restriction of the hull-kernel topology "F_~ LJ~’l coincides with the Euclidean one, whereasthe closures of the points {S_~}, {So} and {$1} are {S-l} U~-I, (So} U~]~-I U’]I~l and {S~} U’]~l, respectively. It is advantageousto think of Spec 91 as the surface of a cylinder provided with its top and bottom (Figure 5.5).

..........

{So} {SI} ~-I

Figure 5.5: Spectrum of P(C)/~ In contrast to the finite section methodfor Toeplitz operators, the representation theory for the finite section methodfor singular integrals with piecewise continuous coefficients is essentially more involved than in the continuous case.

5.2 Postliminal

algebras

This section is devoted to the introduction of some special classes of C*algebras: the liminal and postliminal algebras, which owna lot of pleasant properties that makethem accessible to investigation (in fact, we will see that all the concrete algebras met before belong to one of these classes) and their counterpart, the antiliminal algebras. Mainly for historical reasons, the terminology used in this field is not unique. So, liminal algebras are sometimes called CCR-algebras (with CCRreferring to completely continuous representations), the notations GCR-algebrasand type I C*-algebras

5.2.

POSTLIMINAL

ALGEBRAS

223

are synonymous for postliminal algebras, and NGCR-algebrasis another name for antiliminal algebras. Then we continue with a subclass of the liminal algebras, the dual algebras, which are closely related to the lifting theorems and, thus, to the structure of algebras of approximation sequences. Again we restrict our exposition to some basic facts and refer to the monographs mentioned at the beginning of the previous section for details and proofs. Weconclude by an example of an algebra which arises from the finite section method for Wiener-Hopfoperators and is not postliminal. 5.2.1

Liminal

and postliminal

algebras

A C*-algebra 91 is liminal if, for every irreducible representation (H, r) 91 and for every element a E 91, the operator u(a) is compact on H. The algebra 91 is postliminal if, for every irreducible representation (H, ~) 91, the range 7~(91) contains a non-zero compact operator on H. Finally, C*-algebrais antiliminal if its only liminal ideal is the zero ideal (a liminal ideal is an ideal which is at the same time a liminal algebra). The algebra L(H) of all bounded linear operators on an infinite - dimensional Hilbert space H is neither post- nor antiliminal, whereas the corresponding Calkin algebra L(H)/K(H) is antiliminal (see [49], 4.7.22). Also the sequence algebra ~- consisting of all bounded sequences (An) with A,~ E L(Cn) is neither post- nor antiliminal (compare[49], 4.7.6. and [93], Lemma7.5). On the other hand, all algebras considered in Examples 1 7 of Section 5.1.4 are postliminal, or even liminal (Examples 1 and 2 and the algebra K(H) of the compact operators in Example 3), as one easily checks. In what follows we will concentrate our attention on the postliminal algebras. Concerning the antiliminal ones, we only mention the following result, which states that every C*-algebra consists of a (good) postliminal and a (bad) antiliminal part (see Section 5.2.3 for an example). Theorem5.13 Let 91 be a C*-algebra. Then 91 possesses a largest postliminal ideal ~, and the quotient algebra 91/~ is antiliminal. A proof is in [49], 4.3.6. Weproceed with equivalent characterizations liminal and postliminal algebras. Theorem5.14 (a) A C*-algebra 91 is liminal if and only if ~(91) = for every non-trivial irreducible representation (H, ~) of 91. (b) A C*-algebra 91 is postliminal i~ and only if ~r(91) ~_ K(H) for non-trivial irreducible representation (H, ~r) of 91.

of

224

CHAPTER 5.

REPRESENTATION

THEORY

The proof is based on the fact that every irreducible representation of the ideal K(H) is equivalent to the identical representation. It can be found in detail in [49], 4.2.3 and 4.3.7. For other equivalent descriptions of postliminal algebras we refer to [114], Theorem6.8.7. Also the criterion of the following result proves to be necessary and sufficient for the postliminality of P2 if 92 is supposedto be a separable algebra. Theorem 5.15 Let 92 be a postliminal C*-algebra and let (H1, ~rl) and (H2, ~r2) be irreducible representations of 92 with the same kernel. Then these representations are unitarily equivalent. In other words: If P./is postliminal, then the mapping Spec P2 -~ Prim 92, (H, ~r) ~t Ker is bijective and the spaces Spec 92 and Prim 92 can therefore be identified. For a proof see [49], 4.3.7. Weturn over to ideals and quotients of postliminal algebras. Theorem5.16 (a) Let 92 be a postliminal C*-algebra. Then every C*subalgebra and every quotient C*-algebra of 92 are postliminal. Conversely, if 3 is a closed ideal of 92 and if both 3 and 92/3 are postliminal, then the algebra92 is postliminal itself. (b) Let P2 be a liminal C*-algebra. Then every C*-subalgebra and every quotient C*-algebraof P.I are liminal. See [49], 4.2.4, 4.3.4 and 4.3.5 for a proof. Notice that the converse of assertion (b) is false: If 92 is the Toeplitz algebra T(C) considered in Example 4 in 5.1.4, then both the ideal K(l2) of 92 as well as the quotient algebra 92/K(12) ~ C(~) are liminal, but the identical mappingof ~ is irreducible, and so P2 cannot be liminal itself. Checking the Examples 4 - 7 once more one can in each case observe the appearance of a natural ideal (the compact operators or the ideal 3) which is liminal and which obviously plays a particular role. Here are some results on liminMideals of postliminal algebras. Theorem5.17 (a) Every (not necessarily postliminal) C*-algebra ~1 possesses a largest liminal ideal. This ideal coincides with the set of all a having the property that ~r(a) is compact for every irreducible representation (H, rr) of 92. (b) Let 3 be the largest liminal ideal of a postliminal C*-algebra P2. Then Spec 3 is dense in Spec 9d. Moreover, every kernel of an irreducible representation of 3 is a minimal primitive ideal of 92.

5.2.

225

POSTLIMINAL ALGEBRAS

A proof is in [49], 4.2.6 and 4.7.8. To have an example, consider the algebra 92 = S(C)/6 of the finite section method for Toeplitz operators with continuous generating functions (Example 6 in 5.1.4), and let J~ stand for the largest liminal ideal of 92. We knowfrom Corollary 1.58 that the algebra 92 is *-isomorphic to the algebra of all ordered pairs (W(An), IfV(An)) with (An) S(C), and fr om Section 5.1.4 we recall that W/6and l~d/6 are (up to equivalence) the only infinitedimensional representations of 92. Thus, a coset (An) + G 6 92 belongs ~ if and only if both operators W(A,~) and I~V(An) are compact (the compactness of ~r(An) for finite-dimensional representations ~r involves no extra condition). From the afore-mentioned isomorphy we further conclude that the only cosets with this property are (P, KP,~ + W,,LW,~)n>o + ~ with K, L compact, thus, ~ C_ ~. Conversely, the ideal ~ is liminal, which implies that ~ is the largest liminal ideal of 92. Further, the spectrum of ~ is the set {W/G,I/~V/G} (with the discrete topology), and the closure of this doubleton in Spec 92 indeed yields all of Spec 91 (Figure 5.6). Another peculiarity

{Wlg} cl°suref_xf_>//~. ..........

~

Figure 5.6: The closure of Spec ~ is Spec 92 of this example(and also of Example7) is that the quotient algebra 92]~ not only postliminal, but even liminal (and even commutative). This fact reflects a further general property of postliminal algebras. A composition series of a C*-algebra is a family {J~}0_ 0 and 0 for x < 0, then W(a) = x+C(a)IL=(~t+) = x+FaF-~IL~(~t+). Here are some examples of convolution and Wiener-Hopf operators: Example 1. Let k ~ L~ (ll~),

and set

a(x) := v~(F-Ik)(x)

=

eiZtf(t)

The function a is continuous on I~ and vanishes at infinity (i. e. lim a(x) = as x -~ +~ and x -+ -oc), and the convolution theorem (Theorem IX.3 in [127]) says that C(a)f = FaF-~f = x/~F(g-lk)(F-~f) that is,

(C(a)f)(~) = t)I( t) dr, z ~

=

5.2.

231

POSTLIMINAL ALGEBRAS

for every function f E L2(I~). Thus, C(a) is a classical convolution operator, and (W(a)f)(x)

= k(x-

t)f(t)dt,

is a classical Wiener-Hopfoperator in this case.

¯

Example2. For ~ ~ If(, let U~ denote the shift operator U~ : L2(l~) -~ L2(R), (V,f)(x)

= f(x

An elementary calculation yields U~ = C(e,)

:= i~, e

where e,(x)

and the associated Wiener-Hopf operator Va := W(ea) acts on L2(~+) by (Yaf)(x)

(f(x

-~) if x_>max(0, c~}, if 0_< x < ma~(0, ~}.

Example 3. The operator of singular integration 1 /_~ f(t)

against

dt, xell~,

(the integral existing as a Cauchyprincipal value for good, e. g., compactly supported and Hhlder continuous functions f) is the convolution operator Sa=C(-sgn)

where

sgnx=

-1 1

if x0.

For a proof see [54], Lemma1.35.

¯

Only incidentally, we mention that Wiener-Hopf operators on L2(I~+) are unitarily equivalent to Toeplitz operators on 12. Indeed, the functions E,~(x) := e-~/2Ln(x), n = 0, 1, 2,..., L,~ referring to the nth Laguerre polynomial,

L,~(~)=~ ~ (~’~e-~), n_>O, form an orthonormal basis of ~2(I~+); thus, the operator

z: 12 -~ L~(~+),(~0, :~, ~,...)

(5.12)

232 is unitary,

CHAPTER 5.

REPRESENTATION

THEORY

and one can show that Z*W(a)Z = T(~) where

1 {t+i/2

° dt

~,~ = -~ ~ a ( t ) \ t _-~/ ] t 2 +1/ 4" For details see [153], Examples and Addendato Chapter 1, No. 6, and to Chapter 3, No. 1. Hence, many properties of Wiener-Hopf operators can be deduced from the corresponding properties of Toeplitz operators. In particular, if T(~) is invertible, then W(a) is invertible, and if ~ is moreovercontinuous, then the finite section methodfor T(~) with respect to the standard basis of is stable and, consequently, the fiaite section methodfor W(a) with respect to the basis (5.12) of L2(l~+) is stable, too. What we want to consider here is another approximation of WienerHopf operators. Wecompress W(a) to the subspace L2[0, t] of L2(ll~+), e. if Xt denotes the function Xt(x)

1 if = 0 if

O<xt

(the characteristic function of the interval [0, t]), W(a) by the operators

then we approximate

Wt(a):= xtW(a)lL~Io, Clearly, XtI -+ I strongly as t -+ c~, and so (Wt(a))t>o is an approximation method for W(a). This method is also called the finite section method for W(a) (but observe the ambiguity in this notion since the Wt(a) do not result from the finite section method with respect to a certain basis of L2(ll~+)). Notice further that the operators Wt(a) still act on infinitedimensional spaces, and so a suitable discretization of Wt(a) (e.g. by quadrature rule) is needed for aay numerical computation. If a is a piecewise continuous function, then the stability of the finite section method(Wt(a)) can be studied in full analogy to the Toeplitz case, and what results is the following theorem. Theorem5.24 Ira is piecewise continuous, then the finite section method (Wt(a)) is stable if and only if the operator W(a) is invertible L2(l~+). Whatwe are interested in is the case where a is an almost periodic function. The algebra AP of the almost periodic functions on I~ is the closure in L~(I~) of the set of all trigonometric polynomials a(x) = cle ialx + ... + Ckeic~kx where ci E (: and ai E ll~. In contrast to continuous functions, almost periodic functions can oscillate at infinity. Let ~/V(AP) stand for

5.2.

POSTLIMINAL

ALGEBRAS

233

the smallest closed subalgebra of L(L2(~+)) which contains all WienerHop] operators W(a) with a E AP (hence, all shift operators V~ with Theorem 5.25 VI;(AP) is a C*-algebra which is not postliminal. Sketch of the proof. The algebra W(AP) is C* since W(a)* = W(5) for every a ~ L~(l~) and since ~ belongs to AP whenever a is in AP. For the non-postliminality, one first verifies that the closure of )/Y(AP) in the weak operator topology is all of L(L2(~+)). Hence, the commutant of )/V(AP) coincides with the commutantof L(L2(lt~+)) which consists of the scalar multiples of the identity operator only. This implies the irreducibility of the identical representation of V~(AP) in L(L2(I~+)) by Theorem5.1. The second point one has to check is that the only compact operator in is the zero operator, which implies that ~V(AP) cannot be postliminal. ¯ The non-postliminality of W(AP)involves several serious difficulties; for instance, there is no ’proper’ Fredholmtheory for operators in VI;(AP), because wheneveran operator in )4;(AP) is Fredholm, then it is automatically invertible (but there is a generalized Fredholmtheory which associates with generalized Fredholmoperators a Breuer index taking real (not necessarily integer) values). Nevertheless one has the following result on the invertibility of Wiener-Hop]operators with almost periodic generating function. Theorem 5.26 Let a ~ AP. The Wiener-Hop] operator W (a) is invertible on L(L2(~+)) if and only i] (i) the ]unction a is invertible (in L~(~) or in AP), (ii) the mean motion w(a) := lim~_~+~ ~[arg a(x) ar g a( -x)] o]a is zero. These results where established in [63] and [36], and compare also [26], Sections 9.20 - 9.22, and [64], Chapter VII, for proofs of Theorem5.26 and [37] and [40] for further developments. Nowwe turn over to the finite section method for Wiener-Hop] operators. The experiences gathered in Chapter 4 suggest to introduce the C*-algebra of all bounded functions (A,),>o of operators A~ : Im X,I -~ Im X,I, to consider its smallest closed subalgebra, say .4, which contains all sequences (W,(a))t>0 with a almost periodic, and then to try to study .4 via localizing procedures (perhaps after factoring out a certain liftable ideal). It soon turns out that a direct transmission of this approach to the "algebra ‘4 is not very promising. Indeed, there is a natural homomorphism from A onto the algebra ~V(AP) via

234

CHAPTER 5.

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THEORY

Thus, any ideal of .4, and any element of ‘4 which commuteswith other elements of A modulo a certain ideal, has its counterpart in the algebra VI;(AP). As we know from Theorem5.25, this algebra has a quite involved structure. So we will pursue another strategy of localization which might be called outer localization in contrast to the inner localization employedin Sections 4.2 - 4.5 where the localizing elements could be found already amongthe approximation sequences we were interested in. Namely, we will enlarge the algebra ,4 by certain functions which will prove to commutewith the elements of .4 and can, thus, be used to localize the elements of .4 in a larger algebra. This approach is similar to that one used in Section 4.6. To make this precise, let 5v stand for the C*-algebra of all bounded functions (At)t>o of operators At E L(L2(~)) provided with pointwise operations and the supremum norm (it will be more convenient in what follows to work with operators acting on the whole real line rather than on the interval [0, t]). Further, let B refer to the smallest closed subalgebra of ~" which encloses the following functions: ¯ the functions (Gt) E ~- with IIG~I[ -~ 0 as t -~ c~; these functions form a closed ideal of 5v (and thus of/~) which we denote by ¯ the constant function (X+I) where X+ refers to the characteristic function of the positive reals; ¯ the function (XtI) where Re stands for the characteristic the interval [0, t];

function of

¯ all functions (ftI) where f is continuous on ll~, lim~-~_oo f(t) lim~_~+~of(t), and where (f~)(s) = f(s/t) for t > 0; ¯ all constant sequences (C(a)) where a ~ AP; ¯ all constant sequences (C(b)) where b is continuous on R and lim b(t)=

t-+--oo

lim b(t)=O.

The class of all of these functions will be denoted by Co. Thus, instead of (Wt(a)), we now consider the sequence

+(1

(5.13)

which evidently is stable if and only if so is the sequence (Wt(a)). Let, finally, .4 denote the smallest closed subalgebra of B which contains all

5.2.

235

POSTLIMINAL ALGEBRAS

sequences (5.13) with a E AP+Co. Wiener-Hopf operators with generating function in AP + Co are integro-difference operators: If

+

a(x)

with ~’~ ICkl < cx) and k e LI(~), then

= c f(z -

k(x - t)f(t)

r=O

The possibility of localizing is opened by the following proposition. Proposition 5.27 /f (At) ~ B and f ~ Co, then

IlftAt - At frill ~ o as

It is clearly sufficient to verify this result in case At = C(a) where a E AP or a E Co. In case a ~ AP it is enough to show that

as t -~ c~, which on its hand is a simple consequence of the uniform continuity of the function f. For a ~ Co, the proof is a little bit more technical; for details see [139]. ¯ Thus, the cosets (ftI)t>o + with ] q C+Co for m a central sub algebra of B/~ (observe that no ideal larger than ~ is needed!), and it is elementary to check that this algebra is *-isomorphic to the C*-algebra C + Co. Thus, we can localize BIG over the points of ]~ O {cx)} (with the points s ~ corresponding to the characters f ~ f(s) and with c~ corresponding to f ~ li~nt-~o f(t)). Given s ~ IR tA {o~}and (At) ~ 13, let ~s(At) refer to the local coset of (At) + ~ ~ BIG at s. It is immediate from the definitions that

O~(X+I) =

Os(O) ~(X+I) ~(I) Os(X+l)

{

if s 0 if s = oo

and ~,(XtI)

ff~(O) ~(X+I) ¢~(I) ~s(X(-oo, t]I)

{

if if if if

s¢[0,1] s =0 s e (0, 1) s = 1.

236

CHAPTER 5.

REPRESENTATION

THEORY

Thus, if we let A := ~ H W(aij) c~(1 -

x+)I and At := Z HWt(aij) +

c~(1- Xt)

with aij E AP + Co and a E C, then Os(At)

: ~s(I)

for

s ¢ [0,

1],

¢o(~) = ¢o(~ ~+c(~)x+~I +~(1 - ~)~1 i

= eo(~ H x+C(a,i)X+I

a( 1 -

X+)!)

= ~o(d),

¢1(A~) = a,(~ 1-[ x~x+c(.,~)x+x~I +-(1i j

= ~,(~ ~ Ut(1 x+)U-tC(ai~)Ut(1 i j = ~(U,(~ ~(1 x+)C(a~i)(1 -

X+)I +

x+)U-t +

~Utx+U-t)

aX+I)U-,),

where in the latter equality we used the translation invariance U_,C(a)Ut C(a), which holds for every function a ~ L~(~). A := ~ H(1 - x+)C(ai~)(1

- X+)I +

Proposition 5.28 Let A, ~, A~ be as above. I] A and ~ are invertible, then the sequence (A~) is stable. Proof. The stability of (Ae) is equivalent to the invertibility of the coset (Ae) + 6 in ~/6 and, thus, in B/6. By the local principle, it remains to check the invertibility of ~(A~) for s e ~ U {~}. In case s ¢ [0, 1] there is nothing to prove, and for s ~ {0, 1} one h~

¯ 0(A,) = ~0(A) ~(A, ) = ~, (U~AU_~)

5.2.

POSTLIMINAL ALGEBRAS

237

which implies invertibility in these cases, too. If, finally, s E (0, 1), then (I)s(At) = (I)s (.~) with ft := C(a~j)and it r emains to provethat the invertibility of A implies the invertibility of 4. One easily checks that the strong limit S(A) := s-limt-~o U-tAUt exists and that S(A) = ft. If now A is invertible (in L(L2(IR))), A is also invertible in the C*-subalgebra of L(L2(]R)) constituted by operators B for which S(B) and S(B*) exist. Since S is a *-homomorphism from this algebra into L(L2(IR)), we conclude that .~ = S(A) is invertible. ¯

Wefinally prove the converse of Proposition 5.28. Proposition 5.29 Let A, fl, At be as above. If (At) is stable then A and ft are invertible. Proof. If (At) is stable, then (At) + ~ is invertible in 9t-/G and, thus, A/G. One easily checks that for every coset (Bt) + in A/~the stro ng limits So((Bt) + := s-l im Bt andSI(( Bt) + 9) :=s-l im U-tBtUt exist and are independent of the concrete representative (Bt) of the coset, and that So((At) + ~) and St(( At) + ~) = (5.14) Since So, St are *-homomorphisms,we get the desired invertibility.

¯

Taking into account that the cosets of the form (At) + 9 with At ~i I-[i W~(aij) a( 1 - Xt)I li e dense in ‘4/ ~ andthat all o ccurring mappings (as (I)s, S, So, St) are continuous *-homomorphisms,one can easily derive the following theorem holding for arbitrary sequences in .4. Theorem 5.30 An approximation sequence (At) ~ .4 is stable if and only if the operators So((At) +9) and SI((At) +~) defined by (5.13) are ible. Let us consider the simplest case: A = x+C(a)x+I (1- X+)I anda in AP + Co. With (Jf)(t) := ](-t) one has JA*J = JX+C((*)X+J J( 1 - X+)J = (1 - x+)C(a)(1 - X +)I + X Hence, the invertibility of A implies that of 4. Thus we arrive at the classical theorem by Gohberg and Feldman [64]:

238

CHAPTER 5.

REPRESENTATION THEORY

Theorem 5.31 Let a E AP + Co. Then the finite section method (Wt(a)) is stable i] and only i] the operator W(a) is invertible. For an invertibility criterion for Wiener-Hopf operators with generating function in AP + Co we refer to Theorem9.22 in [26] and to Sections VII. 2- 3in [64].

5.3

Lifting ory

theorems and representation

the-

The goal of this section is to discuss lifting theorems from the view point of representation theory. It will be set forth that manyof the peculiarities of the lifting machinery, and many of the circumstances we observed when applying it to concrete algebras, appear as consequences of a few natural hypotheses. 5.3.1

Lifting

one ideal

Our starting point is a purely algebraic result. Proposition 5.32 (N ideals lemma) Let 92 be an algebra with identity, and let ~1, ..., 3N be ideals of 92 satisfying 31 ¯ ... ¯ 3N = {0}. Then an element a ~ 92 is invertible if and only if its cosets a + ~, . .., a + ~g are invertible in the corresponding quotient algebras. Proof. The invertibility of a implies the invertibility of every coset a + 3i. Let, conversely, all cosets a + 3i be invertible. Then, for every i, there are elements ci ~ P2 and ji ~ 3i such that cia = e + ji, where e refers to the identity element of 91. Hence,

Jl . . . iN=- ela)... with a certain element c e 92. But Jl...iN @31..-~N ---- {0}, whence ca = e. The invertibility of a from the right hand side follows analogously. ¯

Let now91 be a C*-algebra with identity e, 3 a closed ideal of 91, and W a unital *-homoraorphism from 92 into a certain unital C*-algebra ~. We say that W lifts the ideal ~ if the restriction of Wto 3 is a one-to-one homomorphismfrom 3 into ~. Clearly, then WI3 is a *-isomorphism from ~ onto a closed ideal of the C*-subalgebra W(91) Every homomorphismlifts the zero ideal, and the identical homomorphism lifts every ideal. For a less obvious example, consider Example 6

5.3.

LIFTING

THEOREMS

239

in 5.1.4 again, where the quotient homomorphismW/~ with W: (An) s-lim A,~P,~ lifts the ideal 31 := ((_PnKP,~)+~,co~npact}, an d th e qu otient homomorphismI~d/~ with W: (Am) ~ s-lim WnA~W~ lifts the ideal 32 := {(WnLW~) + ~, compact}. The lifting theorem for a single ideal is a simple consequence of the 2 ideals lemma. Theorem5.33 (Lifting one ideal) Let 3 be a closed ideal o] a C*-algebra 91 with identity, and let W : 91 -~ ~ be a unital *-homomorphismwhich lifts 3. Then an element a E 91 is invertible if and only if the coset a + ~ is invertible in the 91/3 and the element W(a) is invertible in Proof. The invertibility of a implies the invertibility of W(a) and a + 3. Let, conversely, W(a) and a ÷ 3 be invertible. Due to inverse closedness, the inverse of W(a) belongs to W(91), and the first isomorphy theorem, 91/Ker W~ W(91), gives invertibility of the coset a + Ker Win 91/Ker W. Since Ker W~ ~ -{0} by assumption, the 2 ideals lemmayields the assertion. ¯ Of particular interest for applications is the case where ~ is an elementary ideal, i.e. an ideal that is *-isomorphic to K(H) for some Hilbert space H. Proposition 5.34 Let 91 be a C*-algebra with identity and ~ an elementary ideal o] 91. Then there is an (up to unitary equivalence) unique irreducible representation of 91 which lifts 3. Proof. Let W be a *-isomorphism from 3 onto K(H). We know from Example 3 in 5.1.4 that Wis the only irreducible representation of 3 up to unitary equivalence. Theorem5.7 further implies that there is a unique extension of Wto an irreducible representation of 91. This extension lifts 3 by construction, and it is unital since every irreducible representation of unital algebras is unital. ¯ 5.3.2

The lifting

theorem

Let {W~}~eTbe a family of homomorphismsW~which lift certain closed ideals 3~. The lifting theorem states that the homomorphismsW~and the ideals 3~ can be glued to a homomorphism Wand to an ideal 3, respectively, such that Wlifts 3. Wefirst derive a general version of the lifting theorem, and then embark upon the case where the ideals 3~ are elementary. Theorem 5.35 (General lifting theorem) Let 91 be a C*-algebra with identity e and,/or every element t of an arbitrary index set T, let ~ be a

240

CHAPTER 5.

REPRESENTATION

THEORY

closed ideal of 92 which is lifted by a unital *-homomorphism Wt : 9.1 -~ Let further ~ stand for the smallest closed ideal of 92 which encloses all ideals ~t. Then an element a E 92 is invertible if and only if the coset a + is invertible in 92/~ and if all elements Wt(a) are invertible in 13t. Proof. If a is invertible, then a+~ and all Wt(a) are invertible. Conversely: If a + ~ is invertible, then there are elements b E 92 and k ~ 9 such that ba = e + k. Further, due to the definition of ~, one finds an element j and finitely many elements Jti E 9ti such that j = Jr1 + ... + jr,, and Ilk - Jll < 1/2. Multiplying the equation ba = e + k from the left hand side by (e+k-j) -1 and setting c := (e+k-j)-Ib and kt~ := (e+k-j)-tjt~, one arrives at ca = e + kt~ + ...

+ kt~ with kt~ ~

Repeating these arguments for ab one thus obtains the invertibility of a modulo an ideal ~ C_ 9 which is generated by a finite numberof the 9t, say The invertibility of all elements Wt (a) involves - as in Theorem5.33 the invertibility of the cosets a + Ker Wt~, . ¯ ¯, a + Ker Wt~. Since Ker Wt~ = {0} by assumption, one has ~.Ker Wt, . . . . . Ker Wt~ =(~t~ + . . . + gt~) " Ker Wt~. . . . . Ker Wt~ C_ 9t~ ¯ Ker Wt~ + ...+ 9~ ¯ Ker Wt, = {0}, and the N = n + 1 ideals lemma, applied to the ideals ~ and Ker Wt~ for 1 < i < n, yields the assertion. ¯ The family {Wt }teT induces a *-homomorphismWfrom 92 into the product of the C*-algebras ~t, t ~ T, via W: a ~-4 (t ~-+ Wt(a)). Corollary 5.36 Let the notations be as in the general lifting let W be the homomorphism(5.15). Then W lifts the ideal

(5.15) theorem, and

Proof. The homomorphismW is unital, so it remains to check whether its restriction to ~ is one-to-one, i. e. whether Ker Wfq 9 --- {0}. Let k ~ Ker Wf’l 9, and let a be an invertible element of 92. Then the following assertions are equivalent: (i) (ii) (iii) (iv) (v) (vi)

a is invertible. W(a) and a + ~ are invertible. a -t- Ker Wand a + ~ are invertible. a + k + Ker Wand a + k + 9 are invertible. W(a + k) and a + k + ~ are invertible. a + k is invertible.

5.3.

LIFTING

241

THEOREMS

The equivalences (i) ¢~ (ii) and (v) ~ (vi) are consequences of the lifting theorem, the equivalences (ii) ~ (iii) and (iv) ¢~ (v) follow the isomorphy theorem 91/Ker W ~ W(91), and (iii) ¢~ (iv) is obvious. The assertion is thus a consequence of the semi-simplicity of C*-algebras (Theorem 1.50). Nowwe turn over to the announced special setting. Theorem5.37 (Special lifting theorem) Let 91 be a C*-algebra with identity e. For every element t of an arbitrary index set T, let ~t be an elementary ideal of 91 such that 3s f~ ~t = {0} whenever s 7t t, and let Wt : 91 -+ L(Ht) denote the irreducible representation of 91 which lifts (which exists and is unique by Proposition 5.34). Let further ~ stand for the smallest closed ideal of 91 which encloses all ideals ~t. Then the assertion of the general lifting theoremcan be completed as follows: (a) The separation property holds, i.e. Ws(3t) {0} whenever s

¢ t.

(b) If the coset a + ~ is invertible, then all operators Wt(a) ¯ L(Ht) Fredholm, and there are at.most finitely manyof these operators which are not invertible. If the assumptions of this theorem are satisfied for an ideal 3, then we say that this ideal can be lifted by the special lifting theorem. Proof. (a) Let s,t ¯ T with s ¢ t. Then 3sfq~t = {0}, thus, every element j of the ideal 3s + 3t allows a unique representation as j = js + Jt with js ¯ 3~ and jt ¯ 3t. So the mapping l~Vs : 3~ + 3t -~ K(Hs), j = js + jt ~ Ws(j~) is correctly defined, and it is an irreducible representation of 3~ + 3t which coincides with Ws on 3s. Furthermore, l~Vs(~t) = {0}. From Theorem5.7 we infer that the irreducible extension of Wsfrom 3~ onto 91 is unique up to equivalence. Thus, 12ds is unitarily equivalent to the restriction to 3s + 3t of any irreducible extension of Ws. Since equivalent representations have the samekernels, and since ~t lies in the kernel of l~ds, we get the assertion. (b) Wefirst claim that Wt(j) is a compact operator on Ht for every j E 3 and t ¯ T. Let j E 3. By the definition of 3, given e > 0, there exists a finite subset {tl, ..., t,~} of T as well as elements jt~ ~ 3ti such that J = Jr1 +... + jr, + k with k ¯ 3 and Ilkll

< E.

(5.16)

242

CHAPTER 5.

REPRESENTATION

THEORY

Let t be arbitrary, and apply Wt to both sides of (5.16). The separation property (a) entails the existence of a compact operator Kt,e E K(Ht) such that W~(j) = Kt,~ + Wt(k) with [IWt(k)[I < Our claim follows from the closedness of K(Ht). Let now a+3 be invertible. Then there are elements b E 9/and j, k ~ ~ such that ab = e + j and ba = e + k. Application of the homomorphism W, to the equations ab = e + j and ba = e + k proves the ~edholmness of Further, applying Wtfor t ¢ {tx, ...,

t,~} to the equation (5.16) we find

Wt(a)W~(b) = ~ + W~()) and, similarly, Wt(b)Wt(a) = It + Wt(~) with IIWt())I[, I[Wt(k)ll < Choosing¢ < 1 one gets the desired invertibility of Wt (a) for all but finitely many t. ¯ Theorem 5.38 Let ~ be a closed ideal o] a unital C*-algebra ~. Then the following assertions are equivalent. (a) ~ can be lifted by the special lifting theorem. (b) ~ is dual. If these conditions are satisfied, then Spec ~ is homeomorphicto the set {Wt}teT, provided with the discrete topology. Proof. (a) =¢, (b): Let W be defined by (5.15). The homomorphism maps the ideal ~ into the product of the ideals K(Ht) as we have seen in the proof of the special lifting theorem, and it is injective due to Corollary 5.36. Hence, ~ is *-isomorphic to W(~). Wewill show that the image of under Wis just the restricted product of the ideals K(Ht), which is a dual algebra due to Theorem5.20, thus verifying the assertion. Indeed, let r ~ W(~), and let j denote the unique element of ~ with W(j) = r. Given z > 0, there is a decomposition of j as in (5.16). This decomposition immediately shows that k belongs to the restricted product of the K(Ht). On the other hand, the separation property entails that W(~t) = K(Ht); hence, W(~) cannot be smaller than this restricted product. (b) =~ (a): Let ~ be *-isomorphic to the restricted product of its elementary ideals ~, and let W~: ~ -~ K(Ht) be the (up to equivalence unique) irreducible representation of ~. Then ~s ~ ~ = (0} if s ~ t, and 3 is the smallest closed ideal of 9/which contains all ideals ~t. It is further clear as well that every mappingW~allows an (up to equivalence) unique extension to an irreducible representation of 9/into L(H~). This extension lifts the ideal ~t, which proves assertion (a).

5.3.

243

LIFTING THEOREMS

It remains to determine the irreducible representations of 3. Let ~r E Spec 3- Then, for every t E T, either

KerrC~3t

=3t

or Ker~rN3t={0}.

If the first case would happen for every t, then r would be the zero representation, which is not irreducible by definition. So there is at least one t E T with Ker ~r N 3t = {0}. In this case, the restriction of r to 3t is an irreducible representation of 3t by Theorem5.7 and, consequently, unitarily equivalent to Wt. But then, again by Theorem5.7, the representations ~r and Wt of the larger ideal 3 are unitarily equivalent, too. Further, if s, t ~ T and s ~ t, then the representations Wsand Wt of 3 cannot be unitarily equivalent. Indeed, otherwise the restrictions of W8 and Wt onto ~[s + 3t would be equivalent and, thus, have the same kernels. The separation property wouldthen yield 3s -- 3t, which is a contradiction. Consequently, any irreducible representation r of 3 is unitarily equivalent to exactly one of the representations Wt. Since, conversely, every Wt is an irreducible representation of 3 by definition, there is a bijection between Spec ~ and {Wt}t~T. Finally we infer from Theorem5.21 that the topology on Spec 3 is the discrete one. ¯ 5.3.3

Sufficient

families

of

homomorphisms

One intention attached with the application of lifting theorems is to render an algebra 91 accessible to further investigation by introducing an ideal 3 of 91 such that the quotient algebra 91/3 can be examined effectively and by introducing homomorphisms W~acting on 91 which describe the ’difference’ between 91 and 91/3. But there are many instances where the W~do much more: they do not only measure the difference between 91 and 91/3 but rather describe the algebra 91 itself. In this case we call {W~}a sufficient family of homomorphisms,and these families will be the subject of the present section. Sufficient families of homo~norphismsalso appear in the process of localizing an algebra via Allan/Douglas. Here is the exact definition. Let 91 be a unital C*-algebra and let {Wt}t~ T be a family of unital *-homomorphismsfrom 91 into unital C*algebras ~t such that the following implication holds for every element a~91: If W~(a)is invertible for every t ~ T, then a is invertible. Then we say that the Wt form a sufficient family of homomorphismsfor 91. If Wis a unital *-homomorphismfrom 91 into a unital C*-algebra g8 for which the singleton {W}is a sufficient family, then we call Wa

244

CHAPTER 5.

REPRESENTATION

THEORY

symbol mapping. Clearly, symbol mappings preserve spectra, and thus, a symbol mapping is nothing but a *-isomorphism between 91 and a C*subalgebra of ~. Observe that the same notions make sense for arbitrary Banach algebras, in which case there is a clear distinction between symbol mappings and isomorphisms. Every sufficient family of homomorphisms Wt : 92 -+ fSt gives rise to a symbol mapping Wwhich associates with a E A the function W: a ~-~ (t ~ Wt(a))

(5.17)

considered as an element of the product of the C*-algebras ~t. The converse is false: if a symbol mapping arises from a family of homomorphisms Wt as in (5.17), then the family {Wt}need not be sufficient. The point is that W(a) is invertible in the product of the ~Bt if and only if all elements Wt (a) are invertible and if their norms are uniformly bounded. This suggests to call the family {Wt}tET weakly sufficient if the following implication holds for every a E 91: IfWt(a) is invertible for every t e T, and ifsuptET [IWt(a)-l[[ < then a is invertible. For example, the family {~t}tE[0,1l of homomorphisms 5~ : f ~ f(t) is sufficient for C[0, 1], whereas{(it}tEl0, D is weaklysufficient but not sufficient. Theorem5.39 Let91 be a unital C*-algebra and let {Wt }tET be a family of unital *-homomorphismsfrom 91 into unital C*-algebras ~Bt. The following assertions are equivalent: (a) The family {W~}is sufficient. (b) For every a e 92, there is a t e T with [IWt(a)[[ = [[a[[. Proof. (a) ~ (b): Suppose there is an a E 92 such I[Wt(a)[[ < sup][Ws(a)[[ for all t sET

(5.18)

Since

= [[Wt(a)*Wt(a)[[ = = = , we can without loss of generality suppose that the element a in (5.18) is selfadjoint and non-negative. The norm of the self-adjoint element a coincides with its spectral radius p(a). Thus, (5.18) can be rewritten p(Wt(a)) < supp(W,(a)) for all t ~ T.

(5.19)

5.3.

245

LIFTING THEOREMS

Denote the supremumon the right hand side of (5.19) by Mand set c a - Me. The elements Wt(c) = Wt(a) - are inve rtible for all t 6 T since p(Wt(a)) < M, and hypothesis (a) yields the invertibility of c a - Me. Then, clearly, ba - me is invertible for all m belonging to some neighborhood U of M. On the other hand, since sups6T p(Ws(a)) for every neighborhood U of M there is an sv ~ T such that mv := p(Wsu (a)) 6 U. The element W~U (a) - mve~u is not invertible, because the spectral radius of a non-negative element belongs to the spectrum of this element. Hence, a - rove is not invertible. This contradiction proves the assertion. (b) ~ (a): Assumea 6 91 is not invertible. What we have to verify the existence of a Wt such that Wt(a) is not invertible. If a is not invertible, then at least one of the self-adjoint elements aa* or a*a is not invertible, for definiteness say a*a. Since a*a is non-negative, a simple application of Theorem1.48 (b) shows that II Ila*alle - ¯a’all = Ila*all

Setb :-- Ila*alle-a*a. Byhypothesis, there Ilbll, which together with (5.20) implies

is a t ~ T such that

(5.20)

IIWt(b)ll--

IIWt(lla alle * - a*a)l] = IIIla* alle- a’all =Ila*all. Since IIWt(a*a)ll< Ila*all, we can apply Theorem1.48 once more to obtain I] Ila * a]let - Wt(a*a)ll

the non-invertibility

of Wt (a’a) and, thus, of Wt (a).

Combiningthe previous result with Theorem5.2, one gets, for instance, the sufficiency of the family of the irreducible representations of a C*-algebra. Corollary 5.40 Let the notations be as in the previous theorem and suppose the ]amily { Wt } to be su~icient. Then, ]or every a ~ 91 and every e>0, t6T

Proof. Let A 6 tJteTa(e)(Wt(a)). Then A ~ a(e)(Wto(a)) for some to ~ T and, hence, either Wt0(a- Ae) is not invertible, or IIWto (a- Ae)-1 II >- 1/~. In the first case, a - Ae cannot be invertible (the family (Wt) is sufficient), whereasin the second case either a- Ae is not invertible, or II(a- Ae)-ll[ 1/s (since IIWtol] no, and moreover, condition (ii) reduces lys ~ ~-~t X[~s (vi) Ht (Et p~ ~ t:~H* k n KerWt(A~)*’~-n]k~-~n

KerW*(A.)’tZ~--n]

~-

for s # t.

Here are a few concrete examples. Example 1: Finite

sections

of Toeplitz

operators

-

6.1.

FREDHOLM SEQUENCES IN STANDARD ALGEBRAS

263

Proposition 6.8 Let a E PC and suppose (PnT(A)Pn) is a Fredholm sequence (equivalently, suppose T(a) and T(?~) to be Fredholmoperators). /fKerT(a) C ImPno and KerT(fi) C_ ImPno for a certain no, then ~ C P~erPnT(a)P~

(" : P~(erT(a)

+

l

nnP~erT(a)Rn

(6.10)

for all sufficiently large n. Indeed, condition (iii) is satisfied since p~t = p,~, and (v) holds because A,~ = PnT(a)Pn and W~AnWn= PnT(f)Pn. Condition (iv) is part the hypotheses, and (vi) is a consequence of the identity PnWnPno Wn = 0 holding for all n > 2n0. Observe that, due to Coburn’s theorem, one of the projections P~erT(a) and Pt~ rT in (6.10) is actually zero. Let us emph~ize in this connection that the results for the finite section methodfo~ Toeplitz operators T(a) derived in Section 4.2.2 ~ well as Proposition 6.8 above remain valid without changes for Toeplitz operators with matrix-valued piecewise continuous coe~cients a ~ (PC)NxN, in which c~e Coburn’s theorem is no longer valid. Hence, in this situation, the kernel of PnT(a)Puwill indeed consist of two subspaces, viz. the ’fixed’ part Ker T(a), and the ’wandering’ subspace R~Ker T(5). Let us further mention that in c~e of continuous (and scalar-valued) coe~cients a, the conditions KerT(a), KerT(h) ~ 0 are also necessa~ for the Moore-Penrosestability of the sequence of the finite sections PuT(a)P~. This remar~ble result belongs to Heinig and Hellinger [86]. Their proof is based on a very precise knowledgeon the kernel structure of Toeplitz matrices. Another proof is in [17]. Example 2: Polynomial collocation

for singul~

integral

operators

Let the notations be ~ in Section 4.4.2. Proposition 6.9 Let a, b ~ PC and suppose (Ln(aI holm sequence (equivalently, suppose al + b8 and 5I + bS to be Fredholm operators). I~ Ker (aI + bS) ImPnoand Ker (aI + ~S) ~ ImP~o ] or a certain no, then 2L

~ImP~

Ker(L~(~+~S)P~) P~e~(~+~s) + R~P~2~(aI+~S)R~

(6.11)

~or al~ su~ciently large n. The proof is as that of Proposition 6.8, with the identity for the ’reflected’ matrices An replaced by Ru(L~(aI + b~)Pn)Ru Example 3: Finite

sections

of singul~

integral

operators

264

CHAPTER

6.

FREDHOLM SEQUENCES

Let the notations be as in Example7 in Section 5.1.4. Recall that the finite section Rn(L(a)P + L(b)Q)Rn of the singular integral operator L(a)P L(b)Q can be identified with the block matrix [ PnT(~)Pn Png(~)Pn) A,~ -- \ P~H(b)P,~ PnT(a)P~ and that the corresponding limit operators So(A~)

kH(b)

S_~(A~) = T(b) L(/2), an

T(a)

e L(l:~12),

d SI (An)= T( 5) ~ L( l~).

Proposition 6.10 Let a, b ~ C(V) and let (Rn(L(a)P + L(b)Q)Rn) ~edholm sequence (equivalently, suppose L(a)P + L(b)Q~ T(b) and are Fredholm operators). Ker

H(b)

KerT(b)

T(a)

~

Im0 Pno ’

~ ImP~o and oKerT(a)

o)

~ ImP~

]ora certain no, then PKImP. $ImP. __ erA~ --

KerSo(A-)

+

0 ) ’~

KerS-~(An)

(

O)

+ for all su~ciently large n. The proof is ~ above. The details as well as the translation from operators on l 2 ~ l ~ to operators acting on/e(Z) resp. Le(~) are left as an exercise (compare also [148]).

6.2

~edholm sequences and the asymptotic behavior of singul~ values

Nowwe are going to establish a relation between quantities which ~e related to the ~edholmness of a sequence (An) and quantities characterizing the asymptotic behavior of the eigenvMuesof the A~.

6.2. FREDHOLM SEQUENCES: 6.2.1

The

main

SINGULAR

VALUES

265

result

Let A c ~-T be a standard algebra and (An) E A a Fredholm sequence. Then (An) is stably regularizable (Theorem6.4), hence, the singular values of (An) split in accordance with Theorem2.14(c), i.e. there are numbers d > 0 and en >_ 0 with lim e,~ = 0 such that a2(An) C_ [0, ~n] t~ [d,~).

(6.12)

The following theorem relates the nullity of (An) to the numberof singular values of Anin [0, en]. Theorem 6.11 Let .4 C_ ~T"T be a standard algebra and (An) ~ A a Fredholmsequence with the singular value splitting (6.12). Let ‘further (IIn) be a sequence in SH(An).Then, .for all sufficiently large n, n(A,~) = dimImIIn. In view of Corollary 2.23, one can restate this result as follows. Theorem 6.12 Let A C_ .~T be a standard algebra and (An) ~ J[ a Fredholm sequence with the singular value splitting (6.12). Then, ‘for n large enough, the numbero] singular values of An in [0, ~,~] (counted with respect to their multiplicity) is independent o‘f n, and this numberjust coincides with the nullity n(An) o‘f the sequence (An)~ These theorems will be proved in the subsequent three subsections. Before doing this, let us mention two consequences of Theorem6.11. The first concerns the fact that the abovedefinitions of the nullity, deficiency and index of a Fredholm sequence depends formally on the standard algebra A as an element of which (An) is considered. (Obviously, a quence (An) ~ ~" can belong to several standard algebras.) Theorem6.11 nowshows that these definitions are actually independent of the envelopping algebra ,4. In Section 6.3 we will pick up this observation in order to define Fredholmness of a stably regularizable sequence (An) without having recourse to its possible embeddinginto a standard algebra. The second consequence concerns the possibility of constructing stable regularizations An,~ of Fredholmsequences (An) without explicitly knowing a suitable cutting off parameter e > 0. Indeed, suppose (An) to be sequence of n × n matrices An with singular value decompositions An = UnEnV~ whereEn=diag(a~n), .. ¯ , a(n)~nj, andlet k =n(An).Thendefine ~n := diag(O, O, ¯O, " " ’ ,.,.(n) ~k+l’ " k zeros

" " ’a(nn))

266

CHAPTER 6.

FREDHOLM SEQUENCES

and set An := Un~nV~*for all n _> k. It is easy to check that An = An,e for every sufficiently large n and for every sufficiently small cutting off parameter e > O. 6.2.2

A distinguished

element

and its

range

dimension

Let (An) be a Fredholm sequence which belongs to the standard algebra .4 C_ :~T. Then the sequence (~,~) with Et-n ~tn := EEtnP~:rW’(A-)

(6.13)

tET

is correctly defined (only a finite number of summandsis non-zero) and belongs to the ideal fiT (the kernel of Wt (An) is finite-dimensional for every t E T) and, hence, to the standard algebra .4. Wewill prove Theorem6.11 by verifying the following relations: dim Im ~n = n(An) (this subsection), dimIm~n _> dimImIIn (Subsection 6.2.3), and dimIm~n _~ dimImIIn (Subsection 6.2.4), each holding for all sufficiently large Observe that the cosets (l~n) + 6 and ((An) n coincide by T heorem 6.6, but that the sequence (12n) will not belong to SII(An) in general, since the ~n need not be projections. Theorem6.13 Let A C_ .~T be a standard algebra, (An) ~T’T a Fredholm sequence, and let (~n) be given by (6.13). If n is sufficiently large, (a) Im gtn= EtET Im (Et~PKH2r (b) the restriction of ~n onto Im ~tn is invertible, (c) dimIman Weprepare the proof by three lemmas. For brevity, write Rt instead of t H t Et~RtEt_~. P~ierW’(A.) and fin for Lemma6.14 If s ~ T and n is sufficiently large, then Imgt~ ~ ( E Im~t~)

= {0}.

teT\{s}

Proof. Contrary to what we want, let us assume that there are an s ~ T and an infinite subset N~ of the natural numbers N such that Im~ ~ ( ~ Im~)

~ (0}

for

all

Then, necessarily, Rs ~ 0 and, for every t e T and n ~ N1, one can find t vectors v t ~ Im R such that n

~s

gnvn

+

~ t~Tk{s}

t t = 0 Env n

(6.14)

6.2. FREDHOLM SEQUENCES:

SINGULAR

VALUES

267

but v8n ~ 0. (6.15) Since Rt = 0 for all but finitely many t E T, one can further choose an infinite subsequence N2 of N1 as well as an index s’ E T, which is independent ofn e N2, such that ]lv~’l] >_ Ilvtn[] for all t ~ T and n ~ N2. Clearly, due to (6.15), n ¢0, andwe defi ne vect ors wn := vn/] [v n [[ for M1n ~ N2 and t ~ T. Rearranging the identity (6.14) we get ~

E~s ws’ n +

t t E~w~ =0

~

or, equivalently, s = _ p~8 wn (Recall

that

Pns’ is the initial

projection

ms ,~t t lZ, ¯ _ n ll, n W

(6.16)

n

of the partiM isometry

E~’ .)

The choice of v~ gu~antees that every sequence (w~)~en~ belongs to the unit ball of Im Rt, whichis compactsince Rt is a finite rank projection. Hence, given t ~ T, there finally exists an infinite subsequence N3 of as well ~ an element wt in the unit ball of Im Rt such that the sequence t (w~)nen~ converges to wt in the norm. Moreover, the set N3 can be chosen independently of t ~ Rt, since R~ is the zero projection for all but finitely manyt. So we conclude from (6.16) that P~’w ~’

=- ~ Z~nE~w t teTk{~’}

(6.17)

+ca

with a sequence (cn)~e~ tending to zero in the norm. The separation condition (6.1) implies that the right hand side of (6.17) converges wetly zero, where~its left hand side converges to w~’ in the normof the Hilbert space Hs’ (one has P~’ ~ I ~’ due to the requirements of the standard model). ButIIw~’ II = 1 (since IIw~II = ~ for a~n 6 ~ dueto thechoice of v~ ), which is a contradiction. Lemma6.15 If n is su~ciently large, then the restriction finite-dimensional space ~teT Im ~ has a trivial ke~el. ProoL Let w~ ~ ~t~T Im ~ be a vector with ~wn = and set v n~ := D~wn. Then, on the one hand,

~teT

v n ~ ImD~ whereas, since sv~ = - ~teTk{~} O~w~,on the other h~nd

Z teTk{~}

teTk{~}

of ~n to the t ~n wn = O, (6.18)

268

CHAPTER 6.

FREDHOLM SEQUENCES

The inclusions (6.18) and (6.19) together with Lemma6.14 reveal vnS = 0 for all s and all large n, whence wn E 71seTKer gt~. Since the operators 12~ are self-adjoint and have a finite-dimensional range, we have Ker 9t~ = (Im 9t~) ~-, i.e. wn is orthogonal to each of the spaces Im ~s~ and, consequently, also to their sum ~seT Im 12~. But wn belongs to the latter space; so it must be the zero element. Lemma6.16 Let H be a Hilbert space and let P, Qn ~ L(H) be orthogonal projections such that P has finite rank and Qn converges strongly to the identity operator on H. Then, for all sufficiently large n~ dim Im P = dim Im QnPQn. Proof. Evidently, dim Im P _> dim Im Q~PQ~for all n. Suppose there exists an infinite subsequence Na of N such that dim Im P > dim Im Q~PQ~ for all n ~ N1. Then, for n fi N1, the restriction of Q,~PQn to ImP has a non-trivial kernel (indeed, QnPQnlImPmaps ImP into the space Im QnPQn,whose dimension is lower than that of Im P; similar arguments will be used in several places in what follows). Hence, there are vectors v~ ~ ImP with I[v~ll = 1 such that QnPQ~vn = 0 for n ~ N1. The compactness of the unit ball of Im P ensures the existence of an infinite subsequence 512 of N1 and of a vector v E ImP with I]vll = 1 such that the sequence (v,~)neN2 converges to v in the norm of H. For n ~ N2, one has Q~PQnv = Q~PQ~vn + QnPQn(v - v~) = QnPQ,(v - v~) as n --~ oc on the one hand, whereas Q~PQnv = (QnPQ~ - P)v + Pv -~ Pv = due to the strong convergence Q~ -~ I on the other hand. What results is v = 0, which is a contradiction to [Ivl[ = 1. ¯ Proof of Theorem 6.13. The operators 9t~ map the space into Imf~n = Im Z fl~ C_ ZIm~ n.

Y~"t~T Im ~t~ (6.20)

t~T

This mapping has a trivial kernel by Lemma6.15. Thus, since all spaces under consideration are finite-dimensional, onto itself. So, assertions (a) and (b) of Theorem 6.13 are immediate consequences of (6.20). It is now moreover clear that dim Im l~ -- dim Z Im ~,

6.2.

FREDHOLM SEQUENCES: SINGULAR VALUES

269

and Lemma6.14 implies that the right hand side of this equality coincides with ¯ t t t ~ dim Im fl~ = Z d,m Im P~R P~. ~ET

It remains to apply Lemma6.16 with P~ and Rt in place of Q,~ and P, respectively, in order to get assertion (c) of the theorem¯ 6.2.3

Upper

estimate

of dim Im 1-In

The estimate dim Im 1-In _~ dim Im f~n is a consequence of the following lemma. Lemma6.17 Let (An) be a sequence in ~, and suppose there are a sequence (Q,~) E J: of finite-dimensional projections Qn such that I[AnAnQn][ --~ 0 as n-~ c~, as well as a sequence (Rn) ~ ~: o] finite rank operators Rn such that (A*nA,~+ R,~) is a stable sequence. Then, ]or all sufficiently large dim Im Qn dim Im ~,~ in an analogous manner as the lemmain the previous subsection. Difficulties arise in the step from (6.21) to (6.22) since the operators ~,~ are no longer projections. On the other hand, the operators ~n, considered as mappings from Im~n into Im~,~, are invertible by Theorem 6.13(b). Thus, if, analogy to (6.21), we would have BnA*nAnv~ = vn with

Vn ~ Im~n,

then BnA~A~lvn = v~, and now we could proceed as in the proof of the preceding lemma if we only would know that the sequence (~lVn) remains bounded. This boundedness is an immediate consequence of the following theorem, which essentially sharpens Theorem6.13 (b). Theorem 6.18 Let A be a standard algebra and (An) E A a ~-~redholm sequence, and let (~) be given by (6.13). Then sequ ence (~n[ Imfl.) is stable. Weprepare the proof of Theorem6.18 by several lemmas. Lemma6.19 Let M and N be finite-dimensional and non-zero subspaces of a Hilbert space H with M N N = {0}, let P be the (in general, nonorthogonal) projection operator from M + N onto M parallel to N, and let P~ and Pff denote the orthogonal projections from H onto M and N, respectively. (a) Then [IP[[ = dist(Sl(M),g) -1 where St(M) := {me M : IIm[[ = 1} is the unit sphere of M. (b) If I]P~Pff[] < 1, then []Pll 0 as n~oo.

¯

In the following lemmawe check that the spaces M,~ := Im ~n and N,~ :-Im 12~ with s, t ~ T and s ¢ t satisfy the hypotheses of Corollary 6.20.

6.2.

FREDHOLM SEQUENCES: SINGULAR VALUES

273

Lemma6.22 Let s, t e T and s 7k t. Then Im ~t n C~ Im ~ = {0}/or all sufficiently large n, and IIPiHm~P~Ime~.l I --+ 0 as n ~ ~. Proof. The first ~sertion h~ been already verified in Lemma6.14. For the proof of the second assertion recall the following simple observation: If H1 and H2 are closed subspaces of a Hilbert space H3 with ~H~H3 and ~H~ H H~ H~ C _ H2 C _ H3, then P~ =. H, " H~ " H, ~ = P~, ’

(6.28)

Since Im D~ ~ H~ ~ H, this observation yields

~rther, dueto theunit~rity of theoperators E~ on Im P~ andof E~n on Im Pn,onehas H. = pfl. Phn

f~

ImE~P,~RsP,~E~_.

= Es’n* pH: IrnP,~R~P,~-n

and consequently, I]P~H~P~Hm~ II < []pIH~p~I~spgES-nE~npIHm"p~R’p~ Dueto (6.28), therighthandsideof thisinequality canbe estimated from aboveby

Once using Lemma 6.21 with H ~ H~, Q~ ~ P~, P ~ R~ and once employing the same lemma with H ~ Ht, Q~ ~ P~, P ~ Rt yields

with a sequence (g~) tending to zero. Nowthe separation condition gives the weak convergence of E£~E~to zero, which implies the strong convergence of RSES~E~to zero because of the compactness of Rs and, finally, t to zero because of the compactness the norm convergence of RSES~E~R of Rt. Thus, the ~sertion is a consequence of (6.29). Corollary 6.23 If s, t ~ T with s ~ t, and if n is sufficiently large, then the no~ of the (in general, non-orthogonal) projection operator ~om Im ~ + Im ~ onto Im ~ parallel to Im ~ is less than 2. In what follows, we will need an analogous result for the projection from Im ~ + ~teTk{s} Im ~ onto Im ~ parallel to ~teTk{s Im ~, which we will derive from Coroll~y 6.23 by induction. The basis for this is provided by the next lemma.

274

CHAPTER 6.

FREDHOLM SEQUENCES

Lemma6.24 Let (Mn), (N,) and (T,) be sequences of finite-dimensional subspaces of a Hilbert space H which satisfy

(i) M.n (g. + Tn) = g. n (Mn+ T~)= T. n (M.+ N.) = {0} sufficiently large n, and (ii) HP~.P~.]] ~ O, ~[P~.P~]] ~ O, ]~P~.P~] ~ 0 as n ~ ~. Then ~]P~ P~.+N.~] ~ 0 as n ~ ~. Proof. H Mnand Nn are finite-dimensional subsp~ces of a Hilbert space H with M, ~ N, = {0}, then the orthogonal projection P~+N. can be expressed in terms of P~. ~nd P~. by Aronshain’s formula as P~.+N. = ~=~ P,.~ with

(see [2]), where the series converges in the strong operator topology. the present setting, ~sumption (ii) guarantees that the convergence of the series is even uniform and absolute if only n is large enough. Thus, given e > 0, there are numbers k0 and no such that

II ~ P~,~ll~ e k:ko

for n >_ no. Then, clearly,

k:ko

for all n >_ no, and it remains to check whether ko-1

IIP~ ~ P,~,~ll-<e k=l

for all large n. But this is also a consequenceof hypothesis (ii), each of the (finitely many) summandsin this sum.

applied

Corollary 6.25 If s E T and n is sufficiently large, then the norm of the (in general, non-orthogonal) projection from ~teT Im fttn onto Im [2sn parallel to the space ~teT\{s} Im f~t n is less than 2. As already remarked, this corollary follows by induction on the (finite) number of elements t E T at which Im flt~ ~t {0}: Corollary 6.23 serves as the starting point and Lemma6.24 is needed for the step from r to r + 1. Lemma6.14 ensures that the hypothesis (i) of Lemrna6.24 is satisfied every step.

6.2. FREDHOLM SEQUENCES:

SINGULAR

275

VALUES

Weare now prepared to prove Theorem 6.18, which together with the results of the previous subsections also finishes the proof of Theorem6.11. Proof of Theorem 6.18. Assume (12nlI~nn.) is a non-stable sequence. Wewill derive a contradiction from this assumption. All operators (f~nlIrn n,) are invertible by Theorem6.13(b). Hencethere exist an infinite subset N1 of N and vectors vn E Imftn with IIv~ll = 1 for every n e ~1 such that H~nvnll ~ O. Write vn ~ ~t~r v~ with v~ e Im ~. This representation is possible by Theorem6.13(a), and it is unique due to Lemma 6.14. Moreover, IIv&ll IIv ll = 2 for t T and for all sufficiently l~rge n ~ N~ by Corollary 6.25. The inclusion v t Im ~ is equiv~ent to t ~ V n = PIm.~V tn = ~n(~n)

(compare Theorem 2.4), whence = nnnn(nn) tET

s~T

(6.30)

n.

s,t~T

Further, the separation condition and the compactness of the operators tR yield, ~ at the end of the proof of Lemma6.22, that

Ilat

s

for

ll 0 as

and t

Thus, and since only finitely manyof the operators ~ are non-zero, there are operators G~ tending to zero in the operator norm such that t t~T =

t EVn+ t~T

E

t Gn(~n) t + Un=Vn+ t ~ Gn(~n)t t~T t~T

t

+Vnt

or, equivalently, Vn

=

~’~nVn

-- Z tET

t t +t Gn(f~r, ) vr,.

(6.31)

The sequence (f~,~Vn),~Er~l tends to zero by assumption, and for the MoorePenrose inverses (f~tn)+ of ~ we have (~)+

=(E~RtE~,)+

= t t t P~)E_ n. E~(P~R

Thus, the Moore-Penrose inverses of ~ are uniformly bounded with respect to n if and only if the Moore-Penrose inverses of the operators

276

CHAPTER 6.

FREDHOLM SEQUENCES

PtnR~Ptn are uniformly bounded, and since the Moore-Penrose inverse of a self-adjoint operator A is given by A+ = (A + (I-

P~A)) -1.

P~A

(compare Example 2.17), the sequence of the operators (ptnR*Pt~)+ is uni~ + I - Pimt~ formly bounded if and only if the sequence ( P~ ~ R~ P~ p~ R’ p~ ) is ~ p~g stable. From Lemma6.21 we infer that [[ Imp*~’p’, - R*I] 0, and since also [[Pt~R*Pt~- R~[] ~ 0, this stability is evident. Consequently, the right hand side of (6.31) goes to zero as n tends to infinity, whereas the left hand side has norm 1 for all n ~ N1. This contradiction proves Theorem 6.18. : ¯ 6.2.5

Some

examples

Weare going to illustrate examples: Example 6.26

the results of the previous subsection by a few

: Norm convergent

sequences

Let H be a Hilbert space, 5v the C*-algebra of all bounded sequences (An) of operators An ~ L(H), th e C*-subalgebra of ~- consisting of thenormconvergent sequences in ~’, and ~ the ideal of the sequences which converge to zero in the norm. Given (An) ~ A denote the limit lira An by W(An). The quotient algebra A/~ is *-isomorphic to L(H), the isomorphism being given by (An) +~ ~ W(A~). Thus, A is a standard algebra (which can be viewed as a C*-subalgebra ~-T where T is a singleton and E,~ = E-n = I). Specifying Theorem 2.19, Theorem 6.4, Theorems 6.6 and 6.7 and Theorem 6.11 to this context yields: Theorem 6.27 Let H be a Hilbert space, and let An, A ~ L(H) be operators such that IIA,~ - All ~ 0 as n (a) The sequence (A,~) is stably regularizabl~ i] and only if A zs normally solvable, and (An) is Fredholmff and only if A is Fredholm. (b) IrA is a Fredholm operator, then the sequence (B,~) with Bn = (A~An H P~erA)-I A~ ]or all sufficiently ~arge n is a regularizer o] (An) in the sense that [Id~Bndn - dn[[ ~ 0, [IBnAnBn - Bnl[ -~ O, II(A~B~)* - A~B~]I ~ O, II(B~A~)* - B~A~I[ -~

6.2. FREDHOLM SEQUENCES:

SINGULAR VALUES

277

stable if IIP~erA this holds if (c) If if (As)and isonly a Fredholm sequence then -~ this0. Particularly, sequence is Moore-Penrose ° -- P~orAII AnPI~A= 0 for all sufficiently large n. (d) If(An) is a Fredholm sequence and :=dimKer A, the n the k smal lest singular values of An tend to zero as n --~ oo, whereasthe remaining part of the singular values remains bounded awayfrom zero by a positive constant d independent of n. In particular, this theorem applies to projecion methods for Fredholm integral equations of second kind as considered in Section 1.1.2. ¯ Example 6.28 : Toeplitz function

matrices

with polynomial

generating

Here we consider the singular values of the finite sections of Toeplitz operators with polynomial generating function. If a is a trigonometric polynomial, then the sequence (PnT(a)P~) belongs to the algebra S(C), which a standard algebra. Hence, if a has no zeros on ~1", then T(a) and T(5) are Fredholm operators, and the kernel dimension identity implies that n(Pr~T(a)P~) = dim Ker T(a) + dim Ker T(a). In Figures 6.2 and 6.4, there are plotted the singular values of the Toeplitz matricesP, T(a)Pn and P~T(b)Pn with n between 1 and 150 for a(t)

= 5t -3 + t -2

+ 3t -1 + 1 + 4t + 7t2 3+ t

and b(t) 5= 0.7t + t respectively. The generating functions a and b have winding numbers 1 and 4 (Figures 6.1 and 6.3), and Figures 6.2 and 6.4 showexactly the predicted splitting of the singular values. Thus, the singular value splitting is an effect which can be observed numerically. It is a recent result by BSttcher and Grudsky[21] that, if a is a rational function without zeros on "1i" but with non-zero winding number, then the smallest singular value of P,T(a)Pn converges exponentially to zero. This excellent convergencebehaviour is nicely illustrated in Figures 6.2 and 6.4.

Example 6.29 : Cauchy-Toeplitz

matrices

A Cauchy-Toeplitz matrix is a matrix which is both a Cauchy matrix (i.e.

278

CHAPTER 6.

.I01 -10

I -$

I 0

I 5

FREDHOLM SEQUENCES

I 10

I 15

I 20

25

Figure 6.1: Image of the unit circle under the generating function a. of the form( ~-y~Ji,j=l z: . n ~ ~,~ )andaToeplitzmatrix(i.e. oftheform(~_j)ij=l). Every n × n Cauchy-Toeplitz matrix is necessarily of the form

g+ "-- j)h i,j=l with complex numbers g and h such that g+kh ~ 0 for k 6 {l-n,..., n-l}. The case h = 0 is not interesting here, so we assume h ~ 0. Moreover, since we wish to consider the matrices Tn for every n, we suppose that 0 ¢ g+Zh. Under these restrictions, T~ is just a complex multiple of a matrix of the form

T~,. := (i - i) + g ~,~=1

with

gEC~Z,

(6.32)

and these are the matrices we will be concerned with here. Encouraged by S. Parter, and motivated by a lot of applications, E. Tyrtyshnikov studied the asymptotic behaviour of the smallest singular value of Ta,, as n ~ ~. To restate his results published in [173] ~d [174], let 0 ~ a~n ~ a2~ ~ ... ~ an, (6.33)

6.2. FREDHOLM SEQUENCES:

SINGULAR

279

VALUES

25

20

0

50

100

150

Figure 6.2: Singular values of P,~T(a)Pn for n between 1 and 150. denote the singular values of Tg,n. Tyrtyshnikov proved that, if g is real and Igl -> 1/2, then al~ -+ 0 as n -+ c~ and, if moreoverIgl = 1/2, then c_~_~_< al,~ _< c2 log n log n

(6.34)

with certain positive constants Cl, c2. Observe that, due to the estimate (6.34), it is practically impossible to detect the asymptotic behaviour the al,~ by numerical tests. Wecan complete Tyrtyshnikov’s results as follows. Theorem6.30 Let g E C \ Z, and let Tg,,~ denote the matrix (6.32) with singular values (6.33). (a) If IRegl < 1/2, then there is a constant d > 0 such that a,,~ >_d for all sufficiently large n. (b) /] ]RegI E (k - 1/2, k + 1/2) for some integer k > 1, then akn "-~ 0 as n --~ cx), and there is a constant d > 0 such that ffk+l,n >_ d ]or all sufficiently large n. (c) ff IRegl -- k + 1/2 with a certain integer k, then ajn --} 0 ]or every fixed j.

280

CHAPTER 6.

FREDHOLM SEQUENCES

2 1.5

0.5

-0 -1 -1.5

Figure 6.3: Image of the unit circle under the generating function b. Proof. First we remark (as Tyrtyshnikov did) that the Cauchy-Toeplitz matrices Tg,,~ are related to the finite sections of a Toeplitz operator with a piecewise continuous generating function. Indeed, if h refers to the function ¯ ~r h(e~=) _ sin ~rge-,~=,:~ ~ [-~-, ~-) (which is continuous on ~ \ {-1} and possibly has a jump at -1) and if denotes the operator

then JnP,~T(h)PnJn = Tg,~. Thus, the singular values of Tg,n coincide with those of the finite section matrix P,~T(h)P,~, and so we are left with studying these singular values. The sequence (P~T(h)Pn) belongs to the standard algebra $(PC), and a little thought shows that, if IReg] < 1/_2, the operators W(PnT(h)P~) T(h) and I~V(PnT(h)Pn) = T(h) with h(t) = h(t -1) are invertible, and that in case IReg[ e (k - 1/2, k + 1/2) the operators T(h) and T(~) are

6.2. FRE,

281

DHOLM SEQUENCES: SINGULAR VALUES

1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0

100

150

Figure 6.4: Singular values of P,~T(b)P,~ for n between 1 and 150. Fredholm with dimKerT(h)

+ dimKerT(~)

So, Theorem6.11 yields assertions (a) and (b). The proof of assertion (c) is based on the following observation: Toeplitz operator T(a) with piecewise continuous generating function is normally solvable if and only if 0 does not belong to the curve a ~ introduced in Section 4.1.3. In particular, this result shows that T(h) and T(~) cannot be normally solvable if Re g E Z + 1/2. To get this observation recall from Theorem2.4 that T(a) is normally solvable if and only if it is Moore-Penrose invertible and from Corollary 2.18 that T(a) + ~ T(PC) in this case. Since T(PC)/K(I2) is a commutative algebra, this further yields that the symbol of T(a) does not vanish. The reverse assertion is also clear: if 0 ~ a ~ then T(a) is Fredholm, hence normally solvable. Now let Reg ~ Z + 1/2 and, contrary to what we want, assume that ajn 7l+ 0 for a certain j. Let j0 be the smallest of all j having this property, and let (nk) be an infinite subsequence of N such that infk ajo,n ~ > O. Then one has aj,n~ -~ 0 for all j < jo, hence, the singular values of the matrices in the sequence (Pn~T(h)Pn~)split in sense of assertion (c) of Theorem2.14. Thus, (Pn~T(h)Pn~) is a stably regularizable sequence,

282

CHAPTER 6.

FREDHOLM SEQUENCES

which, together with the fractality of the standard algebra S(PC) and the analogue of Theorem2.24, yields that T(h) and T(~) are normally solvable. This fact contradicts the afore-mentioned observation. ¯ For another proof of assertion (c) we refer to Theorem6.67.

6.3

A general

Fredholm theory

The theory of Fredholmsequences as sketched above is still unsatisfactory. The main point is that, so far, Fredholmnessis defined only for sequences in a standard algebra. Thus, at least formally, the Fredholmness of a sequence (A,~) depends on the algebra as an element of which (A~) is regarded. course, the characterization of the alpha-number of a sequence via singular values reveals that, actually, the quantities a(An) and f~(An) do not depend on the embedding of (A,~) into a standard algebra. But, for example, the identity ind (AnB~) in d (A,~) + in d (Bn) ca n only be guaranteed if (An) and (B~) are elements of one and the same standard algebra. The goal of this section is to propose a general Fredholm theory which principally applies to every approximation sequence (A,~) E ~’, and which reduces to the above sketched theory in case of sequences in a standard algebra. In particular, the identities (6.3) (which are no longer definitions but consequences of the theory) will be generalized to a muchlarger class of algebras which includes standard algebras. Moreover, it will be pointed out how the Fredholm theory of approximation sequences is related to the theory of Fredholm elements in Banach and C*-algebras as described, e.g., in [10]. Andfinally, a few new insights into the structure of algebras of approximation sequences (i.e. of subalgebras of ~) will b e d erived. Throughout this section, we consider only sequences of matrices, i.e. we let ~" refer to the C*-algebra of all bounded sequences (A,~) with An ~

6.3.1

Centrally

compact

and

Fredholm

sequences

Compact elements in C*-algebras. Let B be a C*-algebra. An element k ~ B is of rank one if, for every b ~ B, there is a complex number #(b) such that kbk = tt(b)k. An element of B is of finite rank if it is the sum of a finite numberof elements of rank one, and it is compactif it lies in the closure of the set of all finite rank elements. Wedenote the set of all compact elements in B by G(/~). It is easy to check that both the elements of finite rank and the compact elements form two-sided ideals in B. In case B = L(H), an element b ~ B is of rank one, of finite rank, or compact

6.3. A GENERAL

283

FREDHOLM THEORY

and only if the operator b has range dimension less than or equal to one, finite range dimension, or is compact, respectively. Proposition 6.31 Let A be a C*-subalgebra of J: which contains the ideal ~. Then G(A) = Proof. Let (An) ~ 0 be a rank one element of c. ThenAk ~0 fo r a certain k. Let Ak+. denote the Moore-Penrose inverse of Ak, and consider the sequence B := (0,..., 0, A~+, 0,...) e with the Ak+ standing at the kth position. By assumption, there is a #(B) C such that (An)B(An) = #(B)(An), whence #(B) = 1 and A1 ....

= Ak-1 = Ak+~ = Ak+2 = ....

O.

Thus, every rank one sequence in ~" is necessarily of the form (0, ..., 0, Ak, 0, ...)

(6.35)

with some Ak ~ Ck×k. Further, Ak k×k, must be a rank one element in C that is, it is zero or has one-dimensionalrange. It is clear that, conversely, all sequences (6.35) with dim Im Ak _< 1 are elements of rank one in. ~’. Nowthe assertion follows immediately from the definitions. ¯ Centrally compact elements. One might call a sequence (An) ~ Fredholmif it is invertible modulothe ideal G(~’) = G. This indeed yields a reasonable Fredholmtheory (see the following subsection), but it doesn’t give the desired notion of Fredholmness, since Fredholmness of a sequence in this sense simply means stability of that sequence. Here is a modified notion of compactness which fits exactly to our purposes. Recall that the center of an algebra is the set of all elements which commutewith every element of the algebra. Definition 6.32 Let I3 be a unital C*-algebra. An element k ~ B is of central rank one i], for every b ~ 13, there is an element #(b) belonging the center o]13, such that kbk = #(b)k. An element orb is oj*finite central rank i] it is the sum of a finite numbero] elements of central rank one, and it is centrally compacti] it lies in the closure o] the set o] all elements o] finite central rank.

284

CHAPTER 6.

FREDHOLM SEQUENCES

Wedenote the set of all centrally compact elements in B by J(13). It is easy to check that both the elements of finite central rank and the centrally compact elements form two-sided ideals in B. In case 13 = L(H), the rank one, finite rank, and compact elements coincide with their central analogues, since the center of L(H) consists of the scalar multiples of the identity operator only. On the other hand, the center of the algebra ~" coincides with l ~ (where the number sequence (an) ¯ ~° i s i dentified w ith t he matrix s equence (anIn)). Hence, t he i deal J(~-) should be muchlarger than the ideal G(~-) = G of the zero sequences. Proposition 6.33 A sequence (An) ¯ J~ is centrally compact i] and only i], ]or every ~ > O, there is a sequence (Kn) ¯ ~: such that supllAn-Knll_l is stable. Then (An + UnPkV~)n>Iis a stable sequence, too. Thus, there are sequences (Cn) E ~" and (Gn) ~ such th at (Cn)(An + UnPkV~) = (In) or, equivalently, (Cn)(An) = (In) + (Gn) - (CnUnPkV~). Since (CnUnPkV~)has finite central rank, (Gn) - (CnUnPkV~)is a centrally compact sequence. Thus, (An) is invertible modulo J(~’) from left hand side, and its invertibility from the right hand side follows analogously. ¯ The preceding theorem suggests to define the a-number of a Fredholm sequence (An) (corresponding to the kernel dimension of a Fredholm operator) as the smallest numberk for which (6.38) is true. Equivalently, a(An) is the smallest number for which there exists a sequence (Bn) ~ ~ as well as a sequence (Jn) of finite central rank such that BnA~An= In + Jn and lira supn_~o~ dim Im Jn = a(An). The index of a Fredholm sequence is the quantity ind (An) := a(A~) - a(A~). Observe that, in the case at hand, this index is always zero. This is a consequence of the following peculiarity of finite matrices, which has no counterpart for arbitrary linear operators acting on an infinite dimensional Hilbert space. Lemma6.36 Let A ~ Cn×n. Then AA* and A*A are unitarily equivalent and, hence, a( AA*) = a( A * A ) with corresponding eigenvalues having same multiplicity. Proof. Let A = UEV* be the singular value decomposition of A as in Section 2.1.1, and set W := UV* and G := VEV*. Then W is unitary, G is non-negative, and A = WGis some kind of a polar decomposition of A (yet not the canonical and unique one considered in Theorem2.10). Further we have A*A = V~U*U~V* = V~3V* 2, = G giving AA* = WGGW* = WA*AW, which implies that A*A and AA* are unitarily

equivalent.

288

CHAPTER 6.

FREDHOLMSEQUENCES

Hence, the matrices A*A n n and AnAn *have the same eigenvalues even with respect to their multiplicity which implies that the alpha-numbers of the sequences (An) and (A~,) coincide. So, at the first glance, the most interesting quantity associated with a Fredholm sequence of matrices seems to be its Mpha-number.A closer look shows that also the vanishing of the index of (An) has some remarkable consequences and applications as it will be pointed out in Section 6.5. Let (An) be a Fredholm sequence and k := a(An). Is there an analogue of the splitting property which holds for Fredholm sequences in standard algebras (Theorem 6.11)? The following simple example says that the answer is no in general. Example. Let (a,~) be an enumeration of the rational and set

numbers in [0, 1],

An:= P,~(a,~P1 + (I - P1))P,~ = diag (an, 1 ....

,1).

Since (Pn)(An) = (a,~)(P1) + (Pn(I = (Pn) - ( 1 an)(P~) and since (1 - an)(Pa) is a sequence of central rank one, the sequence (An) is Fredholm, but the smallest singular values ~r~n) of the matrices Amlie dense in [0, 1]. ¯ Thus, one cannot expect that limn~o a(k n) = 0 if (An) is Fredholm sequence with k = a(An), but one obviously has liminfo-k (’~) = 0. Hence, every Fredholm sequence in ~" possesses an infinite subsequence which owns the splitting property. Finally, let us agree upon the following. The phrase ’Fredholm sequence’ is reserved for sequences in ~" which are invertible modulothe ideal J(.T). Occasionally, we also will have to deal with sequences or elements which are invertible moduloother ideals J of compact or centrally compact sequences or elements. To these kind of Fredholmnesswe will refer as J-Fredholmness. 6.3.2

Fredholmness

modulo

compact

elements

Weproceed with a brief sketch of the Fredholm theory in a C*-algebCa A modulo the ideal G(A) is given. Someof these results are well known (see [10]); they are recalled here for the reader’s convenience with their (as a rule, short) proofs. As mentionedbefore, a direct application of this Fredholmtheory to the algebra 9~ does not yield anything of interest.

6.3. A GENERAL

FREDHOLM THEORY

289

But, as will be pointed out in the forthcoming subsection, applying this Fredholmtheory in case of a standard algebra ‘4 C_ ~" twice (namely in the algebra .A/G(,4) modulothe ideal G(.4/G(A))), one will exactly obtain the Fredholm theory described in Section 6.2. Ideals generated by elements of rank one. In what follows, H is again a separable Hilbert space, L(H) the C*-algebra of the bounded linear operators on H, and K(H) the ideal of the compact linear operators on H. Westart with a result on irreducible representations of the ideal J(A). Theorem 6.37 Let A be a unital C*-algebra and ~r : A -~ L(H) an irreducible representation of A. Then ~r(J(A)) c_ K(H). Proof. Wewill prove that, if k E J(A) is of central rank one, then ~r(k) is an operator with range dimension at most one. This clearly implies the assertion of the theorem. If ~r(k) = 0, then nothing is to prove. So let r(k) ~ 0. For every a E there exists an element # in the center of .4 such that kak = #k. Then ~r(#) is in the center of ~r(A), and the identity

=

(6.41)

shows that ~r(k) is a central rank one element of ~r(A). Since ~r(#) the center of r(.4), the operator ~r(#) is a scalar multiple of the identity operator due to the irreducibility of zr (Schur’s lemma;see Theorem5.1). Hence, the ~r(#) in (6.41) can be chosen as a complexnumber, and ~r(k) a (common) rank one element Let nowx, ~ be vectors in Im ~r(k) with H such that x = ~r(k)y and ~ = zr(k)~. Again due to the irreducibility, ~r(A)x = H. In particular, there is an a E A such that r(a)~r(k)y zr(a)x = ~). Multiplyingthis identity by ~r(k) we get ~r(k)~r(a)~r(k)y = ~r(k)~) which, together with (6.41), yields ~r(#)r(k)y = rr(k)~ or ~r(#)x = &. Since ~r(#) is a number,this showsthat Im ~r(k) -- span {x}. In particular, ~r(k) has range dimension one. ¯ Since K(H) has no proper closed ideals besides the zero ideal, this result implies that ~r(J(.4)) is either {0} K(H). Now we turn over to the ideal G(A) of the compact elements. For every non-zero rank one element k of A, we denote by I(k) the smallest closed ideal of .4 which contains this element. From Theorem 6.37 one immediately gets

290

CHAPTER 6.

FREDHOLM SEQUENCES

Corollary 6.38 Let ‘4 be a unital C*-algebra. Then, ]or every irreducible representation ~r : A --~ L(H) and every rank one element k of ~r(I(k)) C_ K(H). Actually,

much more can be shown.

Theorem 6.39 Let .4 be a unital C*-algebra and k a non-zero rank one element o] A. Then there exists an irreducible representation ~r : A -~ L(H) such that ~r(I(k)) = If(H) KerQrl1(k)) = {0 In particular, every ideal I(k) is *-isomorphic to the ideal of the compact operators on a Hilbert space. Wesplit the proof into several steps. The first partial result says that every ideal I(k) is generated by a rank one projection. Proposition 6.40 Let ‘4 be a unital C*-algebra and let k E ‘4 \ {0} be a non-zero rank one element. Then there exists a rank one projection p ~ ‘4 such that I(k) = I(p). Proof. Let k be a non-zero rank one element of .4, i.e. given A ~ .4 there is a complex number p(a) such that kak = #(a)k. Then the elements k*, kk* and k*k are rank one and non-zero, too. Indeed, for every a ~ .4, k*ak* = #(a*)k*,

k*kak*k = ~(ak*)k*k

and kk*akk* = #(k*a)kk*.

So, these elements are rank one, and moreover 0 ¢ Hk[I2 = IIk*ll 2 = []kk*ll = In the next step we verify that I(k) = I(k*k). The inclusion I(k*k) C I(k) is obvious. For the reverse inclusion, consider kk*k = I~(k*)k. Since k ¢ 0, the number #(k*) is uniquely #(k*) = 0. Then kk*k = 0 and, consequently,

(6.42) determined.

Assume that

~ = Jlk*kk*kJJ Ilkl?- IIk*~ll =o~ which contradicts

k ~ O. Thus, #(k*) ~ O, which implies k = #(k*)-lkk*k

~ I(k*k)

and hence, I(k) C_ I(k*k). From (6.42) we further conclude k*kk*k = p(k*)k*k.

(6.43)

6.3. A GENERAL

FREDHOLM THEORY

291

Both sides of (6.43) are non-negative elements of A and #(k*) ¢ 0. #(k*) > 0, and taking norms in (6.43) gives I[k*kk*k[[= [[k*k[[~ = #(k*)[[k*k[[. Since [[k*k[[ = [[k[[ 2 ¢ 0, this implies #(k*) = [[k*k[[. Nowit is evident from (6.43) that p := [[k*k[[-lk*k is a projection in A which is rank one and that I(p) = I(k*k) = I(k). Proposition 6.41 Let A be a C*-algebra with unit element e and let p A \ {0} be a non-zero rank one projection. Then the identity

pap= defines uniquely a pure state Proof. The uniqueness follows from p # 0. For a = e one gets ~-(e)p pep = p2 = p, hence T(e) = 1. Since Iip[[ = 1, one moreover has IT(a)[ for every a E A, whence[[r[[ = 1. It is also clear that the functional r is linear, hence r is a state of A. It remains to showthat this state is pure. Let L~ := {a e A:~-(a*a) 0} denote the left kernel of T. The state T is a pure if and only if Kerr = Lr + L*~

(6.44)

([91], Theorem10.2.8). Since the inclusion L~ + L~ C_ Ker T holds for every state, it remains to check the reverse inclusion. Let a ~ Ker T, i.e. pap = T(a)p = O. Since pap = O, a=pa+qa=paq+qa

with

q--e-p.

For b := paq one gets r(b*b)p = pb*bp = pqa* paqp = whence T(b*b) : and b ~ L~. Analogously, fo r c :=qa onefind s T(Cc*)p

~-

pcc*p = pqaa*qp= 0,

hence, T(CC*) = and c ~ L~. Co nsequently, a = b+c ~ Lr+ L~,and T is a pure state.

¯

292

CHAPTER 6.

FREDHOLM SEQUENCES

Since T(a* a)p = pa*ap= (ap)*(ap), it is T(a*a) ----- if andonlyif ap= O. Thus, L~- = Aq = {aq : a E A}, and the Hilbert space associated via the GNSconstruction with the pure state T is H := .A/L~ = .A/.Aq with inner product (a + Aq, b + Aq>:= T(b* (it is not necessary to take the completion since T is pure, cp. [91], Theorem 10.2.3). The pureness of Z also guarantees that the representation ~r : .4 --~ L(H), a ~ (b + Aq ~-+ ab + Aq)

(6.45)

is irreducible. The following proposition finishes the proof of Theorem6.39. Proposition 6.42 Let ~ as in (6.45). Then ger (~rl1(k))

=

Proof. Let r ~ I(k) = I(p) and ~r(r) = 0. Then, by (6.45), rb+Aq=O resp.

rb~Aq

resp.

rbp=O

for

allb~A.

This implies rbpc = 0 for all b, c ~ A and, consequently, r ~i bipci -- 0 for all bi, ci ~ A. The elements ~ bipc{ lie densely in I(p). Hence, rj = 0 for every j ~ I(p). In particular, rr* = 0, i.e. r = 0. ¯ Since K(H) has no proper closed subideals besides {0}, one has the following consequence of Theorem6.39. Corollary 6.43 Let kl, k2 be non-zero rank one elements of the unital C*-algebra A. Then either I(kl) = I(k2) or I(kl) I( k2) = {0 Lifting theorems. The preceding results suggest to introduce an equivalence relation in the set of all non-zero rank one elements of a unital C*-algebra A by calling kl and k2 equivalent if I(kl) I( k2). Le t T abbreviate the set of all equivalence classes and, given t E T, choose a representative Pt of the coset t, abbreviate the ideal I(pt) by It, and let ~rt : A -~ L(Ht) stand for the associated irreducible representation (6.45). Thus, G(~4) is generated by its minimal subideals It where t ~ T. With these notations, the following version of the lifting theorem holds (for proof see Section 5.3, Theorem5.35). Theorem 6.44 (Lifting theorem, part 1.) Let A be a unital C*-algebra and T the set of the equivalence classes of the non-zero rank one elements of.4. Then an element a ~ .4 is invertible in ~4 if and only if the operators 7~t(a) are invertible in L(Ht) for every t ~ T and if the coset a + G(A) invertible in the quotient algebra A/G(A).

6.3.

A GENERAL FREDHOLM THEORY

293

In other words: If a E .4 is a G(A)-Fredholmelement, then all operators ~t (a) are Fredholm, and the Fredholmelement a is invertible if and only all Fredholmoperators ~rt(a) are invertible. Together with the following separation property, the lifting theorem can be essentially completed (cp. the special lifting theorem 5.37 in Section 5.3). Proposition 6.45 Let the notations be as be]ore. Then (a) Let tl, ..., tm~ T and t~ ~ tj ]or i ~ j. Then (I~ 1 +...+h.~-l)V~Itm

{0}.

(b) Let s, t e T with s ~ t. Then ~rs(It) = {0}. Proof. (a) Clearly, (I,1 + ... + It.._~) n hmis an ideal in m. Si nce is isomorphic to K(H,~.), its only closed subideals are {0} and/t.~ itself. Thus, if the assertion wouldbe wrong then, necessarily, I,m C_ I,, + ... It~_~. In this case, let I, m = I(p) with a rank one projection p, and choose elements kt~ ~ It~ such that p = k,~ +... + kt~_~. Multiplying this identity by p from both sides yields p = pktlp + ... + pkt~_,p. If pkt, p = 0 for every i, then p = 0 which is impossible. So pkt~p ~ 0 for somei. Since p is rank one, there is a complex number # such that #p = pkt~p ~ Its. Hence, p ~ It~ which contradicts It~ ~ I,~ = {0}. (b) Let r E ~rs(/~), i.e. r = ~r~(k~) for a kt ~ /~. By Theorem6.37, belongs to K(Hs), and by Theorem6.39, there exists a k~ ~ I~ such that ~r(k~) -- r*. Then ~rs(k~k,) = r*r. On the other hand, since I~ V~/, = {0}, one has k~k~ = 0 which implies ~r~(k~kt) = O. Thus, r = 0. ¯ Theorem 6.46 (Lifting theorem, part 2.) Let the situation be as in Theorem 6.37, and let a ~ A be a G(A)-Fredholm element. Then all operators ~,(a) are Fredholm, and there are only finitely manyt ~ T ]or which rt (a) is not invertible. The rank of an element. Let k ~ G(A) be an non-zero element of finite rank. Wesay that k has rank r, if k is the sum of r elements of rank one, but not a sum of r - 1 rank one elements. The rank of k will be denoted by rank k. Further define rank 0 = 0. Proposition 6.47 Let k ~ G(A) be o] finite rank. Then, ]or every t ~ there exist finite rank elements kt ~ It with kt = 0 ]or all but a finite number o] t such that k = ~t~T kt" The kt are uniquely determined, and rank k = ~ rank t~T

294

CHAPTER 6.

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Proof. Let k E G(.A) be the sum of the rank one elements kl, ..., k~. Every kj belongs to exactly one of the ideals It (namely to I(kj)). So one gets a decomposition of k as a sum ~-~teT kt with only finitely many non-vanishing elements kt ~ It of finite rank. The uniqueness of this decomposition can be checked as follows: Let hi +... ÷ hm = 0 for certain elements hi ~ Itl with ti ~ tj for i ~ j. Then htl = -ht2 - ... - ht.~, i.e. ht, ~ I~1 N (It2 + ... + I~.,). BYProposition 6.45 (a), ht, = It remains to show the rank identity. Let k = ~ kt with kt ~ It and let rank kt = rt. Then every kt is the sum of rt rank one elements, hence, k is the sum of ~ rt rank one elements. Consequently, rank k _< Z rt = Z rank kt. Conversely, let k be the sum of r = rank k elements kl, ..., kr of rank one. For every t ~ T and every rank one element h, define nt(h) to be 1 if h ~ It and set nt(h) = 0 if h ¢~/t. Further let kt := ~"~ir=~nt(ki)ki. Then every kt is the sum of rt rank one elements where rt is the number of the ki which lie in It. Since every kl belongs to exactly one of the ideals It, one has ~ rt = r. Thus, Zrankkt _ 2, this is a contradiction. Assumethat dim Im P~/,t -- 0 for infinitely manyn. Then the fractality of .4 implies (Theorem1.66) that (P~’~) belongs to G which contradicts definition of pi,t ~ 0. Hence, dim ImPn/’t < 1 for all n, and this range dimension is zero for at most finitely many n. Modifying the sequence (p~,t) by adding a sequence which tends to zero one can obviously reach that all projections Pr~,~ have range dimension one. ¯ Proposition 6.56 Let A be as in Theorem 6.54 and (p~,t) the lifting of rank one projection Pi,t ¯ A/6 such that dim Im Pin’t = 1 for all n. Then the sequences (Pin’t) can be modified by adding sequences in ~ in such a way that the modified sequences still consist of rank one projections and that the orthogonality condition (6.50) is satisfied for all sufficiently large Proof. Weproceed by induction. For one lifted sequence there is nothing to prove. Assumethat already k of the sequences (Pin’t) are modified such that (6.50) holds. Let (Pn) abbreviate the sum of these k sequences, and let (Qn) be a further sequence of rank one projections Qn. The orthogonality condition (6.50) implies that the operators Pn are projections for n large enough. Consider the operators (In - Pn)Qn(I,~ - Pn) =: (~n with In referring to the n x n identity matrix again, and let p and q denote the cosets (Pn) + and (Qn) + 6, respectively. Since dim Im Qn = 1, the 10n are operators with range dimension at most one, and (~2~ = (In-Pn)Qn(In-Pn)Q,~(In-Pn)

= #n(In-P,~)Q,~(I,~-Pn)

(6.52) with an /°°-sequence (#n). Since the operators ~n are self-adjoint and non-negative, the numbers #n can be assumed to be real and non-negative. Further, since pq = qp = 0, one has (e - p)q(e - p) = and, co nsequently, Jl(In -- Pn)Qn(In - P,~) - Qn]] ~ re sp. H( In -- Pn)Qn(I,~ - P n)][ ~ 1

302

CHAPTER 6.

FRED.HOLM SEQUENCES

as n -~ ~. Together with (6.52), this shows that #n # 0 for sufficiently large n, and the operators l{~,n

= l

([u

--

lim#n = 1. Hence,

P=)Q~(In - P~)

are projections with rank one which are orthogonal to Pn. Modifying a finite number of entries of the sequence (~=) one gets a sequence all entries of which are projections. Nowwe can finish the proof of Theorem 6.54. By the preceding propositions, we can ~sume that all sequences (p~,t) consist of projections with range dimension one and that (6.50) holds for all large n. Let (P~) stand for the sum of all these sequences. The orthogonality (6.50) ensures that the operators Pn are projections and that dim ImPn = ~ dim KerWt(An) t~T~

for all sufficiently large n (the term on the right hand side is just the number of the different sequences (p~,t)). ~rthermore, (A~)*(A~)+(P~) is a stable

sequence, and (A~)*(A~)(P~) (6 .53

Indeed, the sequence (A~) is ~edholm by assumption and the sequence (P~) belongs to the ideal J(A) by construction. Thus, (A=)*(An) + is a ~edholm sequence. ~rther, all operators Wd(A~)*(A=) + (P~)) = Wt(A,~)*Wt(~=) are invertible, and Theorem 6.53 implies the stability of the sequence (An)*(A=) + (P~). Similarly, the sequence (An)*(A=)(Pn) belongs both to the ideal J(A) and to the kernels of all representations Wt with t e T~ss. Again by Theorem6.53 and due to the semi-simplicity of C*-algebras, the intersection of J(A) with all these kernels is the ideal 6 whencethe second assertion of (6.53). Recall from Section 2.2 and (6.53) that the coset (A~) + 6 is MoorePenrose invertible and that (P~) + 6 is the associated Moore-Penrose projection. ~omTheorem 2.22 we further conclude that there is a sequence (H=) of projections such that every projection H~ belongs to the C*-algebra generated by A~A~and by the identity matrix I~ and that (6.53) holds with H= in place of P,. Moreover, since Moore-Penrose projections are uniquely determined modulo the ideal ~, IIP, - H=II~ 0. The latter convergence implies that dim Im P~ = dim Im H= for sufficiently large n, and it remains to verify that the range dimension of

6.3. A GENERAL

303

FREDHOLM THEORY

the Hn for large n (which is independent on n and equal to the sum of the kernel dimensions of the operators Wt(A,~) as we have already checked) coincides with the a-number of the sequence (An). The property Ha E alg (A~An, I,~) ensures that the matrices A*A n n and 1-In can be diagonalized simultaneously:

with 0 _< a~n) < a~n) O. The stability of the sequence (An)*(An) (1-In) requires that liminf(a(~ ~) +p(rn)) > 0 whence lim p(,.n)

1 forr < k,

and the condition I]A~A,~H,dl -~ 0 implies lim a(n)~ (n) = 0 whence lim p(r n) = 0 for r > k since the numbers p(~n) can take the values 0 and I only. Here we also used the fact that the range dimension of the projections II n stabilizes as n -~ c~. This observation finishes the proof of the kernel dimension formula (6.48) and of Theorem 6.54. Remarks. 1. The following example shows that, without the hypothesis of fractality, one cannot expect that the ideals J(Kn)/~ are isomorphic to K(H) for essential rank one sequences (Kn). Let k E ~ be the sequence (P1, P2, P3, ...) where every Pn is a projection from Cn onto a one-dimensional subspace of C’~. This sequence has essential rank one. The ideal J(k) contains the sequence

k’ := (P1,0, P3,0, Ps, ...) = (P1,P2,P3,P~,Ps, ...) (~1,0, ~, 0, z~, which, on its hand, is essential rank one, too, and generates a proper ideal J(k’) of J(k) which is strictly larger that G. Since K(H) has no proper non-zero ideals, the quotient J(k)/(~ cannot be isomorphic to K(H) for some Hilbert space H. 2. It turns out that the Fredholminverse closedness of the algebra .4 is also rnecessary for the kernel dimension formula. To be more precise, let .4 _C 5 be a unital algebra with center c which contains the ideal G. Suppose that, for every Fredholm sequence (An) E A (i.e. for every sequence in A which

304

CHAPTER 6.

FREDHOLM SEQUENCES

is invertible modulo J(~’)), the operators Wt(A,~) are Fredholm, only a finite numberof these operators is not invertible, and the identity a(A,~)

= di m KerWt(A,~) ~ET~

holds. Then, necessarily, A is Fredholminverse closed. Indeed, let (A,~) A be a Fredholm sequence, let PKerW,(An)denote the orthogonal projection onto the kernel of Wt(A,~), and choose sequences (p~t) in J(.A) such that Wt( Ptn) = PKerWt(~4=).Then the sequence (B=) := (A~A,~ + ~_, P~) is a Fredholm sequence, too (since (A,~)*(An) is Fredholm and J(A) C J(~’)). Furthermore, dim KerWt(B~) = 0 for all t E Tess. From the dim Ker formula we conclude that a(B,~) 0, hence (B=) is stable sequence. This implies the invertibility of the sequence (A,~)*(A,~) modulo J(A). Similarly, one gets the invertibility of (A,~)(An)* modulo this ideal. But then, the sequence (As) itself is invertible moduloJ(.A). Thus, every sequence in A which is invertible modulo J(~’) in ~ ( and, hence, modulo J(~) t3 A in A), is also invertible modulo J(A). So, A is Fredholm inverse closed. 3. In case of a standard algebra A, its Fredholm theory reduces to the theory developped in Section 6.2. Indeed, it is clear from the definition that A is unital and contains the ideal 6. Further, all homomorphisms Wt(An) = s-lim tt~Et~*A n~ n =Eare fractal, hence, A is a fractal algebra. Finally, condition (B) in the definition of a standard algebra ensures the Fredholminverse closedness of A (actually, this condition guarantees much more: It requires that a sequence (A~) is stable if all operators Wt(An) are invertible, whereas Fredholminverse closedness essentially meansthat (An) is stable if all operators Wt(A,~) are invertible and if (An) is a Fredholm sequence). It is also easy to identify the ideals It and J(.A) in case of a standard algebra, namely t t ¯) + ~: K e K(Ht)} It = {(EnK(En) and J(A) = .~. The irreducible representations Wt are just the strong limit homomorphismsWt, and the kernel dimension formula (6.48) reduces exactly to the first identity in (6.3).

6.4.

6.4

305

WEAKLY FREDHOLMSEQUENCES

Weakly Fredholm sequences

In this section, we are going to examineanother generalization of the notion of a Fredholm sequence in a standard algebra. Our starting point here is the characterization of Fredholm sequences in standard algebras via the splitting property of the singular values (established in Theorem6.11). What we will get is a further notion of a ’Fredholm sequence’ which is weaker than our former ones. In this section, we let H, Hn, Pn, ~ and G be as in Section 6.1.1 6.4.1

Sequences

with

finite

splitting

property

Let (An) E ~" be a stably regularizable sequence. From Theorem2.14 infer the existence of real numbersd > 0 and en _> 0 with ~n ~ 0 as n -~ oo such that aL(,.)(A~An) [O,cn]U[d, oo) for all sufficiently large n. Let k be a positive integer. Wesay that the sequence (An) has the k-splitting property if the number of eigenvalues of A~An which lie in [0,~n] (= the dimension of the (-oO,~n]-spectral subspace of A~An)is independent of n and equal to k for sufficiently large n. If a(A~An) C_ [d, oo) with a constant d > 0 and for all sufficiently large n, then (An) is said to have the O-splitting property. Finally, the sequence (An) has the finite splitting property if there is a non-negative integer k such that (An) has the k-splitting property. In this case we call the number n(An) := k the nullity of the sequence (An). Definition 6.57 A sequence (An) E J~ is weakly Fredholm if it is stably regularizable, and if both (An) and (A~) have the finite splitting property. If (An) is a weakly Fredholmsequence then the index of (An) is the number ind (A,~) :-- n(An) - n(A~). Observethat, for a stably regularizable sequence (An), the adjoint sequence (A~) is stably regularizable, too, which is an immediate consequence the characterization of stably regularizable sequences given in Theorem 2.19. If (A~) has the finite splitting property, then we call the number d(An) := n(A~) the deficiency of Clearly, if the sequence (An) belongs to a standard subalgebra of ~’, then it is weaklyFredholmif and only if it is Fredholmin the former sense. 6.4.2

Properties

of

weakly

Fredholm

sequences

The properties of weakly Fredholmsequences are less specific than those of ’proper’ Fredholmsequences. So it is certainly not true that the product

306

CHAPTER 6.

FREDHOLM SEQUENCES

of two weakly Fredholm sequences is weakly Predholm again. To have trivial example, consider the n x n diagonal matrices diag(1,O,l,l,1...,1) An := diag (O, l, l, l,1 .,1)

ifniseven ifnisodd

and B,~ := diag(0,1,1,1,...,1). Both sequences (An) and (Ba) are Predholm and have the 1-splitting property, but their product (AnBn) fails to have the finite splitting property since the multiplicity of the eigenvalue 0 of AnBnis 1 if n is odd and 2 if n is even. But we will see that if the product of the weakly Predholm sequences (Aa) and (Bn) is weakly Predholm again, then the identity ind (AnBn) = ind (An) in d holds as for proper Fredholm sequences. Westart with the following observation. Theorem 6.58 Let (An) be a weakly Fredholm sequence. Then the operators An E L(Hn) are Fredholm, and ind (An) = ind An for all suJficiently large n. Proof. Let (An) be weakly Fredholm. Then both sequences (An) and (A~) are stably regularizable, and we let (Hn) and (l~In) denote the (essentially unique) elements of SH(An) and Sll(A~), respectively. That is, and I~In are projections which commutewith A~Anand AnAl, respectively, such that (A~An + IIa),

(AnA*~ + fin)

are stable sequences

(6.54)

and

Purther, the assumption of weak Predholmness ensures that both Ha and I’In are compact projections with dim Im Hn = n(Aa), dim Im l’In = d(Aa) for all sufficiently large n. Set Xn := (I - fIn)An(I - IIn). Then, due to

(6.55), A~An+IIn = (I-IIn)A~(I-fI,~)An(I-II~)+IIn+Gn AnA~ + fIa = (I-fIn)Aa(I-Hn)A~

(I-fIn)+fIn

= Z~Zn+Hn+Gn,

+ Hn = XnX~ +fIn

with certain sequences (Gn), (H,~) ten~ing to zero in the operator norm. Hence, due to (6.54), the sequences (X~Xn + Hn) and (XnX~ + fI~) are

6.4.

WEAKLY FREDHOLMSEQUENCES

307

stable, and this implies the invertibility of X~X, + Hn as well as that of X,‘X~ + ~,‘ for all sufficiently large n. Further, since Ha and l:In are finite rank operators, we conclude that X,‘ is a Fredholm operator on Hn for every large n. (If ~r,‘ denotes the canonical projection from L(Hn) onto the Calkin algebra L(Hn)/K(Hn) then ~’,‘(H,~) = ~rn(l:In) = 0, hence, ~r,‘(X,‘) is invertible L(Hn)/~K(H,‘) from both sides.) Moreover, we clearly have XnIIn = 0 and X,~[In = 0. Together with the invertibility of X~X,‘ + l-I,, and of XnX~+ l-I,‘, this implies H. HKer X~ : Hn

and

YIKer

H,~ ~ lZln X~*

and, consequently, dimgerXn =

dimIml-ln = n(dn), dimgerX~ = dimIml~I,‘

= d(An)

for all sufficiently large n. Thus, the index of the Fredholmoperator Xn is equal to dimKerXn - dimKerX~ = n(An)

- d(An) = ind(A,‘)

for all sufficiently large n. Nowobserve that An and Xn differ by a compact operator only, and recall that the index of Fredholmoperators is invariant under compact perturbations to get the assertion. ¯ As an immediate consequence, we obtain: Corollary 6.59 (a) Let (An) and (Bn) be weakly Fredholm sequences assume that the product (AnBn) is weakly Fredholm, too. Then ind (AnBn) = ind (An) + ind (Bn). (b) Let (An) be a weakly Fredholm sequence. Then there is an ~ > 0 that, for every weakly I~redholmsequence (B,‘) with II(A=)- (B=)II < ind (A,‘) = ind (Bn). Thus, the index is still a continuous function on the set of all weaklyFredholm sequences. But we cannot claim that this set is open (whereas the set of all Fredholmsequences in the sense of Section 6.3 is open).

6.4.3

Strong limits

of weakly Fredholm sequences

Here we are going to discuss the Fredholm property of the limit operator of a strongly convergent weakly Fredholm sequence. Let ~-c refer to the set of all sequences (An) 6 ~" such that both (AnP,~) and (A~Pn) are

3O8

CHAPTER 6.

FREDHOLM SEQUENCES

strongly convergent sequences. Werecall from Theorem 1.18 that ~-c is a C*-subalgebra of ~, and that the mapping W : .T "C -~ L(H), (An) s-lim AnPn is a unital *-homomorphism. Theorem 6.60 Let (An) E ~c. (a) I] (An) is stably regularizable, then W(An)is normally solvable.

(b) II (An)is weakly Fredholm, thenW(An) is Fredholm. Proof. (a) By Theorem2.19, the sequence (An) is stably regularizable and only if the coset (An) + ~ is Moore-Penrose invertible in ~/~. Let (Bn) + ~ be the Moore-Penrose inverse of (An) + ~. Since C*-algebras inverse closed with respect to Moore-Penroseinvertibility (Corollary 2.18), we conclude that (Bn) + G .T ’c/6. Hence (B n) ~ ~- c, an d th e st rong limit s-lim BnPnexists. Thus, letting n go to infinity in

(AnBn)"

= AnBn + ),(BnAn)*=

BnAn+ G~ )

with certain sequences (G~)) e ~, 1 < i < 4, we obtain the Moore-Penrose invertibility and, thus, the normal solvability of W(An). (b) From assertion (a) we know that W(A~) has a closed range. Let us prove that the kernel dimension of W(An) is finite. Suppose that (An) has the k-splitting property, i.e. if e > 0 is sufficiently small, then the (-~,~]-spectral projections Rn of A~An have range dimension k. If n is large enough then, by Corollary 2.23, we have R~ = H~ where (1-In) is the (essentially) unique element of SII(An). In particular, dim Im l-In = k for n large enough.

(6.56)

Further, by definition, (Ha) + G is the Moore-Penroseprojection associated with (A~An) + ~, and this projection belongs to ~-c/~ since the coset (A~An) + belongs to thi s alg ebra (Th eorem 2.1 5). Hen ce, (Ha ) ~ 9 re , and the strong limit s-lim HnPnexists. Invoking the definition of (Ha) and the inverse closedness of C*-algebras with respect to Moore-Penrose invertibility once more, we get sequences (G(~I)), (G~)), ~)) in ~ as well as se quences (Bn),(C,~) cc such that A~A~H,~ = H,~A~An for all

large n,

B~(A~A~ + H.) = Ils ~ + GO) (A~An + Ha)Ca = )IIH~ + G~ A~,A.Hn 3) = G(~

6.4.

WEAKLY FREDHOLMSEQUENCES

309

Letting n go to infinity in these equalities we obtain W(A~A~) W(H~) = W(H~) W(A*~A~) = as well as the invertibility of W(A~An)+W(Hn). Hence, W(H~)is nothing but the orthogonal projection from H onto the kernel of W(A~A~) W(A~)*W(An), which coincides with the kernel of W(An). It remains to verify the estimate dim Im W(Hn) ~_ lim sup dim Im H~,

(6.57)

which, in combination with (6.56), yields that the kernel of W(An) has a finite dimension: dimImW(IIn) = dimKerW(An) _< k < ~. We prove the estimate (6.57) in Lemma6.63 for a more general situation. Analogously one checks the finite dimensionality of the kernel of W(A~)*. Thus (apart from the proof of (6.57), which is still open for a moment), see that the strong limit of a weakly Predholmsequence (An) is a Fredholm operator and that dimgerW(An) _< n(An),

dimKerW(A~) _< d(A~).

(6.58)

In what follows we will generalize these estimates to the case where a finite numberof different strong limits of (A~) is considered simultaneously. To this end, we pick up the situation of Section 6.1.1, but nowwith a finite ~ index set T = {1,2,... ,r}, say. That is, we have Hilbert spaces H with partial isometries E~n : Ht -4 H such that both the initial and the range projections of the Etn converge strongly to the identity operator, and such that the separation condition holds. Further, we let ~-T again stand for the collection of all sequences (An) E c for which t he s trong l imits W~(A~) := s-lim

Et ~A~Et~ and WI(An) * := s-lim

exist for every l E T. Theorem 6.61 Let (An) be a weakly Fredholm sequence in yrT. Then the limit operators WI(An),... ,Wr(An) are Fredholm, ~dimKerW’(An)

k. Choose vectors el,. ¯., ek+l such that the Qe~,..., Qek+lare linearly independent. ~+~ is invertible, and since the norms Then the Gram matrix ((Qei, Qej))i,j=~ {l((Qe~,Qej))~,~=:ta+l - ((P,~e~,

P,~e~))~,j=~lla+l -- II((Pe~, e~.))- ((P,~e~,e~))ll

6.4. WEAKLY FREDHOLM SEQUENCES

311

becomeas small as desired if only n is large enough, we conclude that the Gram matrices ((Pne~, Pnej))~,j=l are invertible, too, for all large n. This implies that lim sup dim Im Pn _> k ÷ 1 which is a contradiction. Hence, dim Im P _< dim Im Q _< k. ¯ The estimate (6.57) in the proof of Theorem 6.60 is an immediate consequence of this lemma. Nowwe return to the setting of Theorem 6.61. Weglue the Hilbert spaces H1,... ,Hr to a new Hilbert space/~ := H1 @... @Hr, i.e. /~ is the set of all r-tupels (hi, ..., hr) T of vectors h~ E H~, provided with the inner product ((hi,...,

h~)T, (gl, ..., I=l

(it is convenient to think of the elements of/?/as column vectors), and glue the partial isometries E~,..., E~n to the operator 1 J~n: I?t -+ H,( h l , h 2 , . . . , h, ) T ~ -~ ,= Clearly, one can think of the operators ~:n and ~ as the matrices 1 E1 /~,~ =: ~ ( n,’",

1 E,~) and T. ~ =: -~ (E~,...,

Er__n)

Because 1 1

~n~ =_1r (E~E_,

r r

+...

+ E,E_,)

1(

=rrPn) =

we have

=

...,

=

Hence ~,~ is a partial isometry from /~ into H with range projection P,~, whereas the initial projection of ~n is the matrix operator

~

=

¯ _1 ( E~nE~ r~

E~E~

:

Er E~ Er E2

...

E~E~

¯

~

:

~.

E~ ~ E

Lemma6.64 (a) I] (An) e ~T, then the weak limit w-lim ~An~n exists. (b) The mapping ~: ~T ~ L(~), (A~) ~ r-w-lim~g~ is a unital *-homomorphism.

312

CHAPTER 6.

FREDHOLM SEQUENCES

Proof. Let (An) E T with /~ An/~,= g~l tEk_nA,Elnlk,l=l.~r If k = l, then the entries Ek_nAnE~ n = Ek_nAnE~of that matrix converge strongly to Wk(An), whereas in case k ~ l the entries Ek_nAnE~ = Ek_nPnAnE~n= (Ek_nEt~) (Et__nAnEt~) -~Wt(A,~)

converge weakly to zero by Lemma6.62. Hence, w-lim/~An/~n = 1_ diag (WI(An),...,

W~(An)),

r

which implies assertion (a) and, since the t are u nital * -homomorphisms, also assertion (b). Lemma6.65 Let (Rn) ~ T’T bea s equence of ort hogonal pro jections. Then IfV (Rn) is an orthogonal projection, and dimIml/iz(Rn)

_< limsup dimImRn.

Proof. The first assertion is an immediate consequence of Lemma6.64 (b). For the second assertion, set k := lira sup dim Im Rn. Since dim Im (ABC) < min (dim Im A, dim Im B, dim Im C) for arbitrary linear operators A, B, C, we have lim sup dim Im/~Rn/~n < k. The operators

~Rn~n are orthogonal projections,

~Rn~,n"

J~Rn~n = J~RnPnRn~n

= ~,~Rn~n,

which converge weakly to ~I?V(Rn). Lem~na6.63, applied to the orthogonal projection 12V(R~), yields dim Im I]d(Rn) _< k. Proof of Theorem 6.61. The ~edholmness of the operators Wt(An) follows as in the proof of Theorem6.60. For the first estimate in (6.59), repeat the arguments of the same proof with ~ in place of Wto conclude that ~(Hn) is the orthogonal projection from ~ onto Ker ~(A~). Hence, r

Z dim Ker Wt (An) = dim Ker lYd(An) = dim Im l]d(IIn), l:l

313

6.4. WEAKLY FREDHOLM SEQUENCES and now apply Lemma6.65 with Ha in place of Rn in order to find dimIml2d(Hn) _< limsup dimImIIn n( An).

¯

The second estimate in (6.59) follows analogously.

Observe that Theorem6.61 also provides another proof for one part (actually, the simpler one) of Theorem6.11. 6.4.4

Weakly

Fredholm

sequences

of

matrices

Weconclude this section by mentioning special effects which are related with weakly Fredholm sequences of matrices, i.e. we suppose Hn is an n dimensional subspace of H and we identify L(Hn) nxn. with C As a consequence of Lemma6.36 we find that, if (An) is a bounded sequence of matrices An E Cnxn, then (An) has the finite splitting property if and only if (A~) has this property, and in this case (6.60)

n(An) = n(A~). Fromthis identity we easily derive the following theorem.

Theorem 6.66 Let (An) be a bounded sequence of matrices An ~ nxn. (a) The sequence (An) is weakly Fredholm i] and only i] it has the finite splitting property. In this case, n(An) = d(An) and, hence, ind (An) = (b) If (An) has the k-splitting property and if (An) belongs to some algebra ~T’T a8 in Section 6.4.3, then all operators Wt(An) are Fredholm, and r

~’~dimgerWt(An)

< k.

l=-I

(c) I[ (An) is as in assertion (b) and, moreover, belongs to some standard subalgebra ,4 of ~T, then ~ dim Ker Wt(A,~) = di mgerWt(An) * = k, /=1

l:l

and ~indWt(An)

=0.

/=1

This is an immediate consequence of (6.60) and of the definition of weak Fredholmness for assertion (a), of Theorem6.61 for assertion (b), and Theorem6.11 for assertion (c). The following theoremhighlights a further effect of the finite-dimensionality of Ha.

314

CHAPTER 6.

FREDHOLM SEQUENCES

Theorem 6.67 Let (A,~) be a bounded sequence of matrices Aa E n×n which belongs to some algebra j:w with iFT as in Section ’6.4.3. Let ]urther aln ~_ ... ~_ ann rej~er to the singular values o] An. I] at least one o] the operators WI(An),..., Wr(An) Jails to be Fredholm, then ajn --~ 0 as n -~ oc ]or every ]ixedj.

(6.61)

Proof. Assertion (b) of Theorem 6.66 yields that (An) cannot have finite splitting property. Moreover, since all homomorphisms W~ are strong limits and, hence, fractal, we even conclude that no infinite subsequence of (An) can have the finite splitting property. Clearly, this implies (6.61). Observe that Theorem 6.67 offers another way of proving Theorem 6.30 (c). Wealso recommendthe readers to pursue the effects of these index theorems in the concrete examples of Chapter 4.

6.5 Some applications Wefinish with a brief discussion of several examples and applications of Fredholm approximation sequences. 6.5.1

Numerical sion.

determination

of

the

kernel

dimen-

In case of a Fredholm Toeplitz operator T(a) with continuous generating function, the kernel dimension of T(a) is simply the maximumof 0 and of the negative winding number of the curve a(’l~) around the origin (where this curve is provided with the orientation which is naturally inherited from the counterclockwise orientation of the unit circle). A similar simple geometric argument applies to the determination of dim Ker T(a) if a is piecewise continuous (cp. Theorem 4.11). In contrast to this, the determination of the kernel dimension of a compactly perturbed Toeplitz operator T(a) + ca n pr ove to be a s er ious problem even in case of a nice generating function a. An application of kernel dimension identity to the sequence (An) where An --’- Pn(T(a) + K)Pn with a piecewise continuous and K compact yields c~(An) = dim Ker (T(a) + K) di m Ker T( 5) where again 5(t) := a(1/t). The kernel dimension of T(5) can be determined via the winding number. Thus, if one is able to observe the c~-number of (AN) numerically, then this identity yields the desired kernel dimension of T(a) +

6.5. 6.5.2

SOME APPLICATIONS About the operators

finite

315 section

method

for

Toeplitz

Sufficient stability conditions. As above, let T(PC) stand for the smallest closed subalgebra of L(l 2) which contains all Toeplitz operators T(a) with piecewise continuous generating function a. Again we consider the finite section method(P,~AP,~), but nowfor operators A in T. Accordingly, let B refer to the smallest closed subalgebra of 5v which contains all sequences (PnAPn) with A ¯ T(PC). It is not too hard to prove that the strong limits W(A,,) and I~(An) (defined as in Theorem1.54) exist every sequence (An) ¯ B. Thus, the invertibility of the operators W(An) and I~(A,~) is a necessary condition for the stability of the sequence (An). Assumethe invertibility of these operators is also sufficient for the stability of (An). Then (and under the preliminary assumption that B is standard algebra) the index identity 0 = ind (An) = ind W(A,,) ind l? d(An) should hold for every Fredholm sequence (An) ¯ B, i.e. for every sequence (An) for which W(An) and ITV(An) are ~redholm operators. There are simple examples showing that this identity cannot be true for arbitrary Fredholm sequences in B. Indeed, let a(ei~) := Then a(e~X)2 :=

1 if z¯ (0,2~r/3) e 5i~/6 if x ¯ (2~r/3, 4~r/3) e7i~/~ if x ¯ (4~r/3, 2~r). i~/3 e -i~/3 e 1

{

if if if

x ¯ (0, 2~r/3) x ¯ (2~r/3, 4~r/3) x ¯ (4~r/3, 2~r),

hence, for An = PnT(_a)2pn, the operator W(An) = T(a) 2 has index -2 whereas the index of W(An) T(2) is 0. Thus, the index identity predicts that the invertibility of the operators W(An) and 17V(An) cannot be sufficient for the stability of a sequence (A,~) ¯ B in general. (In a similar way, the kernel dimension identity implies that the invertibility of the operator W(An)is not sufficient for the stability of a sequence (An) ¯ A in general, although it is sufficient for sequences of the form (PnT(a)Pn).) A detailed analysis (essentially performed by Werbitzky, Rathsfeld, BSttcher and the authors) yields the following stability result for sequences in B where, besides the invertibility of W(An) and I~(A=), certain local

316

CHAPTER 6.

FREDHOLM SEQUENCES

stability conditions occur. Let the spline space Sn[0, 1] as well as the partial isometrics E,~ : ImPn-+ S,~[0, 1] and E_,~ = E~ : S~[0, 1] -~ ImPn be defined as in Section 4.2.3. Further, for T E ~’, let Y~stand for the operator o~ Yr: 12 -+ 12, (xk)~=l ~ (~-k x~)~=l.

One can show that, for every sequence (An) E B and for every T ~ ~’, the strong limit W~" (A,~) := s-limn-~E,~Y~-IAnYrE-,~ exists and that it defines a bounded linear operator Wr(An) on L2([0, 1]). Theorem6.68 A sequence (A,~) ~ B is stable if and only if the operators W(An), IYV(An) and Wr(An) are invertible for every For a proof see, e.g., [77], Theorem4.1. This proof also shows that B is a standard algebra. Global vs. local stability conditions. For a more refined version of Theorem6.68, let X stand for a closed subset of the unit circle ~ and denote by PCx the C*-algebra of all piecewise continuous functions on "1~ which are continuous at the points of ~ \ X. Accordingly, let "Ix stand for the smallest closed subalgebra of L(l 2) which contains all Toeplitz operators T(a) with generating function a ~ PCx, and let Bx refer to the smallest closed subalgebra of ~- which contains all sequences (P~AP,~) with A ~ "Ix. A closer look at the proof of Theorem6.68 reveals the following. Theorem6.69 A sequence (An) ~ BX is stable if and only if the operators W(A,~), I~V(An) and W~(An) are invertible for every T e Z Of particular interest is the case when X is a singleton, say X = {1}. For A ~ 7~1}, Theorem 6.69 says that the sequence (P~APn) is stable if and only if the three operators W(PnAP~)= A, I~V(PnAP~)’and I(P,~AP,~) are invertible. Actually, the invertibility of W~(P,~APn)proves to be redundant in this special setting which is again a consequence of the index identity. Theorem 6.70 Let A ~ "~1}. Then the sequence (PnAP,~) is stable if and only if the operators W( P,~ AP,~) = A and ITV ( P~ AP~) are invertible. Proof. The index identity, X = {1}, yields

specified to the setting of Theorem6.69 with

ind W(A,~) in d ~( A~) + in d W~(A~) = 0

6.5.

SOME APPLICATIONS

317

for every Fredholmsequence (AN) E 7~1}. Let, in particular, An P,~AP,~ with A E 7~1}, and suppose W(P,~AP,~)and I~V(P,~AP,~) are invertible. It is not hard to check (using the Gohberg/Krupnik symbol calculus) that then W1 (PnAPn) is a Fredholm operator. Hence, (P,~AP,~) is a Fredholm sequence, and the index identity yields ind W~ (PnAPn) = O. Further, the special form of the sequence (A,~) = (P,~AP~) implies that W~ (P~AP,~) is a Mellin operator (see [77], Sections 2.5.1 and 2.5.2), which is subject Coburn’s theorem. Hence, W~(PnAPn)is invertible. ¯

6.5.3 Discretization

of shift

operators

Several important classes of concrete operators including Toeplitz, Mellin and Wiener-Hopf operators as well as singular integral operators can be interpreted as functions of the shift operator. This special point of view goes essentially back to the pioneering monographby Gohberg and Feldman, [64], and has been further developped in [123], Chapters 4 and 5. In the present section, we will illustrate howthese techniques, in combination with the index formula in standard algebras, can be applied to study a whole variety of approximation methods for the mentioned operators. Westart with recalling some facts about C*-algebras generated by an isometry. Thus, we let ,4 be a C*-algebra with identity e, and we let v ~ A be an isometry which is not unitary, i.e. v*v = e, but vv* ~ e. By B(v) we denote the smallest closed subalgebra of ,4 which contains v and v*. Given m a trigonometric polynomial p on T, p(t) = ~j=-m ajt~, we abbreviate the element m

--1

~ ajv j + ~ aj(v*)

-~

(6.62)

to p(v), and we let L(v) stand for the closure in B(v) of the set of all elements of the form (6.62). Clearly, L(v) is a closed subspace of B(v). Further, we write QC(v) for the quasicommutator ideal of B(v), i.e. for the smallest closed ideal of 13(v) which contains all elements of the form (pip2) (v) - pl (v)p2 withp~ and p2 tr igonometric polynomials. Proposition 6.71 (a) L(v) n QC(v) = (0} and L(v) + QC(v) = B(v). (b) The quotient algebra B(v)/QC(v) is *-isomorphic to C(T) isomorphism sending the coset p(v) + QC(v) to the ]unction p for every trigonometric polynomial p. Thus, there is associated with every element a ~ B(v) a continuous function on ~ which we call the symbol of a and denote by smb a. (c) An element a ~ L(v) is invertible, invertible only from the right side or invertible only from the le]t hand side if and only if (smb a)(t)

318

CHAPTER 6.

FREDHOLM SEQUENCES

]or every t E T and i] the winding numberof the ]unction smb a with respect to the origin is zero, negative or positive, respectively. A proof can be found in [123] where also generalizations to the Banach space setting are considered. For an alternative proof recall Coburn’s result in [35] which says that the algebra B(v) is *-isomorphic to the algebra 7"(C) generated by the Toeplitz operators on 12 with continuous generating function (see Section 1.4.2). Thus, the results from that section together with Coburn’s theorem immediately imply the preceding proposition. In what follows, we specify A to be the algebra L(H) of all bounded linear operators on a Hilbert space H, and we let V be a non-unitary isometry on jH. Further we assume that dim KerV* = 1 and that the operators (V*) converge strongly to 0 as j -~ oo which guarantees that the quasicommutator ideal QC(V) C l~(V) coincides with the ideal K(H) of the compact operators. What we want to study is approximation methods for operators in ~(Y). For this goal we choose a sequence (Hn) of subspaces of H having the property that the orthogonal projections Pn from H onto Hn converge strongly to the identity operator as n --~ oc. By ~c we denote the C*algebra of all bounded sequences (A,~) of operators A,, : Im P,~ -~ Im P,~ such that both sequences (A,~Pn) and (A~Pn) converge strongly. The strong limit of a sequence (A,~) E ~c will be denoted by W(An). Further, we recall from Theorem 1.19 in Section 1.2.4 that the set ~c := {(PnKPn + Ca) : K ~ K(H), lira IICnll ~ 0} forms a closed two-sided ideal of Nowsuppose we are given a certain discretization of the operators in B(V), i.e. a symmetric and unitat bounded linear mappingD 13(V) -~ such that W(D(A)) fo r ever y oper ator A ~ B(V), whic h owns the following properties: ¯ D(K) E Gc for every compact operator

K,

¯ D(I) - D(Y)D(Y*) and ¯ D(Vj+k) - D(VJ)D(V~:) e 6C for every j, k _> 1. Let C refer to the smallest closed subalgebra of ~-c which contains all sequences D(A) with A ~ 13(V) and which contains the ideal 6 of all sequences tending to zero in the norm. The symmetry of D implies that is a symmetricsubalgebra of ~-c and, since D is unital, this algebra contains the identity element (P~). It is moreover easy to see that C contains the complete ideal ~c and that the quotient Gc/G is *-isomorphic to K(H). Indeed, if K ~ K(H) then, by assumption, there exist a compact operator

6.5.

SOMEAPPLICATIONS

319

L e L(H) and a sequence (Ca) ¯ ~ such that D(K) = (PnLPn + Cn). The assertion follows from the identity K = W(D(K)) = W(PnLPa+ Ca) For (Aa) ¯ C, let (A,~)" abbreviate the coset (Aa) v. By Theorem 1.20, a sequence (An) ¯ C is stable if and only if the operator W(An) is invertible in L(H) and if the coset (An)" is invertible in A/~C. Our assumptions further imply that the algebra C/Ge is generated by the coset D(V)" and its adjoint (D(Y)~r)* = D(V*)~ and that D(V)"D(V*)" Thus, two situations can occur: Either D(V*)" is a unitary isometry in C/GC, or it is a non-unitary one. In the first case, we claim that the Fredholmnessof W(An) already implies the invertibility of the coset (An)’. Indeed, from Proposition 6.71 we know that the algebra B(V)/K(H) is generated by its unitary element V + K(H) and that the spectrum of that element is the complete unit circle ~’. On the other hand, the algebra C/~c is generated by its element D(V)" which is unitary by assumption and, hence, has a subset of ~" as its spectrum. So, in the first case, the claim follows from the Gelfand-Naimark theorem for commutative C*-algebras. If D(V*)~ is a non-unitary isometry, then Proposition 6.71 applies to the algebra C/6c = I~(D(V*)~) to establish a *-isometry, I~d, from cC/~ onto the algebra B(V) which maps the coset D(V*)" onto the shift operator V. Summarizing we get the following. Theorem 6.72 Let the notations and assumptions be as above. (a) I~ D(V*)" is a unitary element of c, the n a s equence (An) ¯ ~ stable if and only if the operator W(An)is invertible in L(H). (b) I] D(V*)~ is a non-unitary element of C/6C, then a sequence (An) ¯ is stable i~ and only i] the operators W(An)and I~V(An) are invertible L(H). Corollary 6.73 In any case, if A ¯ B(V), then the sequence D(A) stable if and only if the operator A is invertible. Proof. There is nothing to prove if D(V*)~ is unitary. So assume ~ D(V*) is a non-unitary element of C/~c, and let A be invertible. In this case, taking into account Proposition 6.71 once more, the assertion can be proved in the same manner as Corollary 1.36 in Section 1.3.3. ¯ It is an interesting consequence of the index equality that D(V*)" cannot be unitary in case of matrix sequences. Corollary 6.74 Let dim Ha < ~ for every n and let the further assumpC. tions be as above. Then D(V*)" is not a unitary element of C/6

32O

CHAPTER 6.

FREDHOLM SEQUENCES

Proof. Suppose that, contrary to what we want, D(V*)~ is unitary. Then, by Theorem 6.72, C is a standard algebra for which C/~ is *-isomorphic to t3(V) via the mapping W. Hence, a sequence (An) E is a F re dholm sequence if and only if the operator W(An) is a Fredholm operator on H, and ind (A,~) = ind W(An) for every Fredholrn sequence (An) in C. Moreover, we know that ind (A,~) is necessarily 0 since (A,~) is a matrix sequence. Onthe other.side, clearly, there are Fredholm operators in 13(V) which do not have index 0 (every power of V provides an example). Contradiction. Example 1: Finite sections of Toeplitz operators on weighted 12 spaces. Given # E IR, let l 2’~ refer to the space of all sequences x = (xn),~>o of complex numbers which satisfy [[x[12u := ~(1 + n)2"[x,,[ 2 < oo. The inner product (x, y)~ :-- ~(1 n)2~Xn~nn n_>0 makes 12’~ to a Hilbert space, and the operator Au: l: ~/2,,,

(x~) ~ ((1 n) -~x~)

:’ 12,° ~. and l is an isometry between the tIilbert spaces l ~ = Whatwe are interested in is discretizations of Toeplitz operators on 12 ,~ which can be thought of as functions of the shift operators

V : 12’"-~ 12,~t, (Xn)~-4(0, XO,Xl,...), V(-1): 12,,u _.}/2,#, (Xn) (Xl, x2, x3, .. .). The point is that these operators are not subject to Proposition 6.71: the backward shift V(-1) is not the adjoint of V with respect to the inner product (., .)~ if # ~ 0, and V is not an isometry. But the weighted shift operator 17 := A~VA-~ : 1~,~ _~ 12,~ is a non-unitary isometry on 12’~ with adjoint 17" = A"V(-1)A-u and, hence, Proposition 6.71 applies to the algebra B(17). Moreover, it turns out that the operators V - 17 and V(-1) - 17" are compact and that, consequently, V and V(-1) belong to/3(17). If we define for every trigonometric

6.5.

SOME APPLICATIONS

321

polynomial p(t) = ~ ajt j the Toeplitz operators corresponding to V and ~ by T(p) := ~-~aj(V(-1)) j0

respectively, then it is also clear that T(p) E B(V) and that T(p) - ~(p) is a compactoperator. Finally, if a(t) = ~-~j ajtJ is a sufficiently smooth function such that the series ~’~ la~l II(W(-1))-~ll + ~ la~l IIW~ll j0 converges, then we can define T(a) and 7~(a) as uniform limits of operators of the form T(p) and 7~(p) and get that T(a) ~ B(V) and T(a) - ~(a) is compactfor all sufficiently smooth functions a. The discretization of operators in /~(V) by the finite section method is defined as follows. Set Pn := I- (~’*)n~, introduce accordingly the algebra ~-c and its ideals, and associate with every operator A E B(V) the sequence D(A) := (PnAPn)n>_o. A little thought shows that all assumptions concerning the discretization made above are satisfied, and the finite-dimensionality of the discretization implies that assertion (b) of Theorem 6.72 is relevant in the actual situation. Hence, specifying Theorem 6.72 and Corollary 6.73 to this context yields: Theorem 6.75 A sequence (A,~) ~ C = C(~) is stable if and only if operators W(A,~)and ITV(A,~) are invertible on 12’’. Corollary 6.76 (a) The finite section method applies to an operator A ~ B(V) if and only if this operator is invertible. (b) If a is sufficiently smooth, then the finite section methodapplies to the Toeplitz operator T(a) L(/2,") if andonlyif th is operator is in vertible. Example 2: Approximation via Fejer-Cesaro means. Given a function a e C(qr) with Fourier coefficients a~, j E Z, its nth Fejer-Cesaro mean an(a) is the function a,~(a)(t)

:= ~ (1n+llJl

~. )ajt

It is well knownthat the functions a,~ (a) converge uniformly to a as n -~ and that lim [lan(ab) - an(a)an(b)l I = 0

322

CHAPTER 6.

FREDHOLM SEQUENCES

for arbitrary continuous functions a and b (see, e.g., [26], Section 3.15). Wewill use these means in order to define a discretization for Toeplitz operators with continuous coefficients on 1~, that is we let V : 12 ~ 12 be the forward shift operator, which is a non-unital isometry, and consider operators in B(V). This algebra coincides with T(C) and, hence, it consists of all operators T(a) + where a is continuous on "l~ and K isa compact operator. Choose Hn := l ~ for every n, define 5cc in accordance with this choice, and introduce a discretization D : T(C) -~ joe as follows: D(T(a) + := (T( a,~(a)) + K), >o. Again it is easy to check that all assumptions for D madeabove are satisfied and that now, in contrast to the previous example, D(V*)~ is a unitary coset. So we get Theorem 6.77 A sequence (A,) E C = C(V) is stable operator W( An ) is invertible.

if and only if

Corollary 6.78 Let a E C(’~) and BY ~ K(12). Then (T(an(a)) stable sequence if and only if T(a) + K is an invertible operator.

Notes and references The main results of the present chapter are due to the authors. The notion of a Fredholm sequence in a standard algebra was introduced in [149], and the same paper also contains the characterization of the nullity of a Fredholm sequence in term~ of the singular values of the entries of the sequence (Theorems 6.11 and 6.12). The Example 6.3 is taken from [151]. The general definition of a Fredholm sequence and the results of Section 6.3 were derived in [142]. The material of Section 6.4 is published here for the first time. It is partially based on an observation by T. Ehrhardt concerning the special setting of the finite section methodfor Toeplitz operators. Finally, the first parts of Section 6.5 are cited from [142], whereas the results of 6.5.3 are new. For a deeper analysis of the asymptotic behaviour of the singular values (and of the condition numbers as well) of Toeplitz matrices we refer to the brand-new textbook by BSttcher and Grudsky [21].

Chapter 7

Self-adjoint approximation

sequences Thebeginningof all this is a theoremof G. Szeg5on the determinantsof Toeplitz matrices. H. Widom

In this concluding chapter, we will focus our attention on self-adjoint approximation sequences for self-adjoint operators. Since self-adjoint operators as well as self-adjoint approximation sequences are completely determined by their spectra, we shall be mainly concerned with questions of the spectral asymptotics.

7.1

The spectrum of a self-adjoint mation sequence

approxi-

The main points of this section are a classification of the limit points of the eigenvalues of Amdue to W. Arveson and the proof of the existence of a fractal subsequence for every approximation sequence.

7.1.1 Essential and transient points Let H be an infinite-dimensional Hilbert space and (H,~) be a sequence of finite-dimensional subspaces of H such that the orthogonal projections Pn from H onto Ha converge strongly to the identity operator I on H. 323

324

CHAPTER 7.

SELF-ADJOINT

APPROXIMATION

SEQUENCES

Further, denote by 5r the C*-algebra of all bounded sequences (An) of operators An E L(Hn), by ~ the associated ideal of the zero sequences, and by fc the C*-subalgebra of 9r consisting of all sequences (An) such that both strong limits s-lim AnPn and s-lim A~Pn exist. The mapping ~c _~ L(H), (A,~) s-li m AnPnwill be de not ed by W. Given a self-adjoint sequence (An) E r and a n open i nterval U C_l~, let Nn(U) refer to the number of the eigenvalues of An in U, counted with respect to their multiplicity. Definition 7.1 (a) A point ~ ~ ~ is called essential ]or (An) i], .for every open interval U containing A, lim Nn(U) (b) A point A ~ I~ is transient if there is an open interval U containing such that sup Nn(U) n

Every essential point of (An) lies in the uniform limiting set lim inf a(An), whereas all points outside the partial limiting set limsupa(An) are transient. The set of all essential points will be denoted by lim infess The set lim infess a(An) is a closed subset of lira inf a(An). Indeed, if A is non-essential, then there is an open neighbourhood U of A as well as an infinite subsequence ~ of N such that sup Nn(U) < c~. Evidently, every point in U is non-essential, too, which implies that the non-essential points form an open subset of ll( Observe that there might be points in lim sup a(An) which are neither essential nor transient: If diag(0,1,1,...,1,1) An :- diag (0, 0, 0, . ,0,1)

ifneven ifnodd

then limsupa(An) li minfa(A~) = {0, 1} , bu t no ne of the limi t poin ts is essential or transient. For situations where every point in lim sup a(An) is either essential or transient see Sections 7.1.3 and 7.1.4. For A ~ L(H), let aess(A) refer to the essential spectrum of A, i.e. to the set of all A ~ C for which A - AI is not a Fredholm operator or, equivalently, to the spectrum of the coset A + K(H) in the Calkin algebra.

7.1. SPECTRUM OF SELF-ADJOINT

SEQUENCE

Theorem 7.2 Let (An) E ~c be a sequence of self-adjoint Then a(W(An)) C_ liminfa(An)

325 operators.

O’es s (W(An)) C_ l imi nfessa(An).

In particular, the limiting sets lim inf a(An) and lim infess a(An) are never empty under the conditions of the theorem. Proof. For the first inclusion, let A be a real number which is not in lim inf a(An). Weclaim that then W(An) - AI is invertible. Since A E ~ \ liminfa(A,0, there is an e > 0 as well as an infinite subsequence ~/of N such that a(An) M(A-~,A+e) = 0 forall

nE~/.

Thus, the distance of a(An) to A is at least e, which shows that the operators An - AI[H,, are invertible and that their inverses are uniformly bounded:

sup II(An - AII.n)-Xll < 1/e.

(7.1)

Let ~c, and :-re denote the C*-algebras of all sequences (B,(n)) where (Bn) ~ :" and (Bn) ~ :-c, respectively. Clearly, the sequence (An ), IlHn) belongs to 5c~c and is invertible in ~-, due to (7.1). Inverse closedness -1) ~ 9c, c, i. e. th e st rong li mit B := C*-algebras gives ((An MIH~) s-limne~(An - M[Hn)-IP~exists. Letting n go to infinity (An - AIIH.) -~ (A,, - AIIH.) (I IH,) yields B (W(An) AI) = I. Thus, W(An) - A is invert ible. For the second inclusion, suppose A ¢ lim infess a(An). Wewill show that then W(An) - AI is a Fredholm operator. By assumption, there are an infinite subsequence ~/ of N as well as positive numbers e and k such that supNn(A-e,A+~)

= k < o~.

For n E ~/, let Qn denote the orthogonal projection from Hn onto the (A - e, A + e)-spectral subspace of An. These projections are compact, and dim Im Qn oo due to (7.4); so we conclude that 0 an eigenvalue of An for all sufficiently large n, and that the multiplicity of this eigenvalue tends to infinity. Hence, 0 E lim infess a(PnAPn).

7.1.2 Fractality

of self-adjoint

sequences

The results of this section hold without restriction to matrix sequences. So, here we let again (C~) be a sequence of C*-algebras with identity elements en and consider their product 2- and their restricted product 6. If (An) is a self-adjoint fractal approximation sequence, then lim sup a(An) = lim inf a(An),

(7.5)

as we knowfrom Theorem3.20(b). Wewill now see that, conversely, (7.5) is the only obstruction for a self-adjoint boundedsequence to be fractal. Theorem 7.3 Let (An) ~ :T be a sequence o] sel]-adjo~nt (A~) is fractal if and only if the equality (7.5) holds.

matrices. Then

Proof. The ’only if’-part is Theorem3.20 (b). For the reverse conclusion suppose that (7.5) holds. Let .4 denote the smallest closed subalgebra 5r which contains the sequence (an) and the identity sequence (en) r. Further, given a monotonically increasing sequence ~/, write An for the algebra RnA. We claim that A/(A ~ 6) is isomorphic

to An/(A n ~ 6~)

(7.6)

328 CHAPTER 7.

SELF-ADJOINT

APPROXIMATION

SEQUENCES

with the isomorphism given by (bn) + (A N 6)

(bT(~)) + (- An N

(7.7)

To get the claim recall that A/(A~ 6) and A,/(A, N 67) are isomorphic to (A + 6)/6 and (A,~ + 67)/6v, respectively. The latter algebras are singly generated by their elements (an)+ and (aT(n)) + and th e sp ectr a of these cosets are lim sup a(an) and lim sup a(au(n)) due to Corollary 3.18, respectively. The assumption (7.5) guarantees that these spectra coincide; hence, by the Gelfand-Naimark theorem for singly generated C*-algebras (Theorem 4.16), the isomorphy (7.6) follows. Let now ~r stand for the canonical homomorphismfrom .4 onto 6) and let r/ be a monotonically increasing sequence. Then, evidently, ~r = ~rTR~ = ~¢,R, where ¢7 is the canonical homomorphism from A onto Av/(A, N ~) and where ~7 if the inverse of the isomorphism (7.7). Hence,~r is fractal. As a first application of the previous theorem we derive a fractality result for the sequence of the finite sections of a self-adjoint operator. Observein this connection that the spectrum of a self-adjoint operator A for which the finite section method(PnAPn)is fractal can be as complicated as possible: Given an arbitrary compact subset K of the real line, choose a dense subsequence ( ki)i=l oo in K, and consider the operator A = diag (kl, k2, k3, ¯ . Then lim sup a(P~AP,~) = lim inf a(P~AP,~) = Theorem7.4 If A E L(H) is a self-adjoint operator with connected spectrum, then the sequence (PnAPn)is fractal. Proof. If A - AI is invertible, then A - AI is either positively or negatively definite and, hence, (Pn(A- AI)P,~) is a stable sequence by Theorem 1.10(b). Conversely, if this sequence is stable, then the operator A - AI invertible due to Polski’s theorem. Thus, a(A) = a.rlg( (P,~AP~) Further we know from Corollary 3.18 that a~=/O ((PnAP~) + 6) = lim sup a(P~AP~), and from Theorem 7.2 that a(A) C_ lim inf a(P,~AP,~).

7.1. SPECTRUM OF SELF-ADJOINT

SEQUENCE

329

These inclusions give lim sup a(PnAPn)li m inf a(PnAPn), and th is id entity is equivalent to the fractality of (PnAPn)as we have seen in the previous theorem. ¯ There are simple examples such as A=diag

((0 1)(0 1 0 ’ 1 0 ,...

e n(/2)

(7.8)

which showthat the finite section methodfor selfoadjoint operators is not necessarily fractal: For A as in (7.8) one has a(P2nAP2n) = {-1,1}

¢ (r(P2n+lAP2n+l)

= {-1,1,0}

for all n. In the case at hand, it turns out that the (non-fractal) sequence (PnAPn) possesses a fractal subsequence (formed by the matrices of even order). So one might ask whether every self-adjoint approximation sequence has a fractal subsequence. For a long time we conjectured that the answer is no (and tried to find examples amongthe finite section sequences for the almost Mathieu operators; see Section 7.2.4). Moreover, motivated by Theorem7.4, we conjectured that if (An) is completely no n-fractal sequence (i.e. if no infinite subsequence of (A,~) is fractal), then a(A) is a set of Cantor type. Then our collegue T. Ehrhardt drew our attention to the Hausdorff compactness criterion Theorem3.7 which, in combination with Theorem7.3, gave a surprisingly simple proof of the (for us) surprising fact that the converse of our conjecture is true. Theorem 7.5 Let J: be as in Section 7.1.1. Every selJ-adjoint (An) E ~ possesses a ~actal subsequence.

sequence

Proof. Consider the sets M,~ := a(An). By Theorem 3.7, there exists subsequence (Mn(n))n>_l of (M,) such that

a

lim sup(M,(.)) = li.m~f(M,(.)). Then the sequence (An(n))n>1 is fractal due to Theorem7.3. Furthermore, based on that result, we axe now in a position to derive the existence of a fractal subsequence for every (not necessarily self-adjoint) sequence of matrices. Actually, we will showa little bit more: Theorem7.6 Let jc be as in Section 7.1.1, and let ,4 be a separable C*subalgebra o] ~F. Then there exists a sequence ~ C N such that the algebra An = R~AC_ J:n is fractal.

330 CHAPTER 7. Since every finitely ately implies:

SELF-ADJOINT

APPROXIMATION

SEQUENCES

generated C*-algebra is separable, this result immedi-

Theorem7.7 Let J: be as in Section 7.1.1. Then every s’equence (An) possesses a fractal subsequence. One cannot expect that Theorem7.6 holds for arbitrary C*-subalgebras of 9r; for example it is certainly not true for l ~. On the other hand, there clearly exist non-separable but fractal subalgebras; the algebra S(PC) of the finite section method for Toeplitz operators with piecewise continuous generating function can serve as an example. In the proof of Theorem 7.6 we will several times make use of the following equivalent characterization of fractal algebras which is in turn a simple consequence of the third isomorphy theorem for C*-algebras. Lemma7.8 The C*-algebra .4 C_ J: is ~ractal if and only if the restriction of the canonical homomorphism ~ : J~ --~ J:/~ onto .4 is ]ractal. Proof of Theorem 7.6. Let .4 be a separable C*-subalgebra of ~" with a countable dense subset ((A,~)~_>1)k_>1. (k) B(~TMand ~,~ we denote the real and the imaginary part of.-ha(k) , respectively, and we write B (C_ for the set of all sequences (B(~k))~>l with k _> 1 and 7)(C_ A) for the set all difference sequences (B(~a))n>_I- (B(~O)n>_~ with k, 1 >_ 1. The set BtJ7) (k) is countable, and each of its elements is self-adjoint. Let ((Dn)n>~)k>l be any numeration of the elements of B U 7). By Theorem7.5, every sequence (D(n~))n>_~ possesses a fractal subsequence. Wewill employ a standard diagonalization process in order to construct a sequence y E N such that (k) > 1. the sequence ~D ~ ~?(n))n>_l is fractal for every k -(a) Let r h C m(n))n_>l is a fractal sequence _ N be a sequence such that t ~D and, for every k _> 2, choose a subsequencer/~ of r/~-i such that ~D(~) ( r~(n))n>l is a fractal sequence. Then define r/by

:= The sequence ~/ coincides (with the possible exception of at most finitely manyentries) with a subsequence of ~/k for every k. Hence, every sequence in I, (k) ~(n)/n_>l, k~---1, 2, .. .} 7)~ := Rn7) : "te~D is fractal. Weclaim that the algebra ~1, := RvA is fractal. What we have to verify is that, given a subsequence # of r/, there is a homomorphism such that

7.1.

SPECTRUMOF SELF-ADJOINT SEQUENCE

331

where ~ is the canonical homomorphism from ~’~ onto ~’~/6~ (Lemma7.8). Observe that the set of all sequences i

(A,(n))n>_l

with k = 1, 2, ... is dense in A,. Without loss of generality, we can assume ~7 = N in what follows. So we will have to deal with the following situation: .4 is a C*-subalgebraof Y: with a countable dense subset ((A(nk))n>_l)k>_l such that each of the self-adjoint sequences~B(k)~ ~ n )n>~ andtB(k)~ ~ n )n>~ (B(nl))n>_~with k, l >_1 is fractal, where2k) andB(n2~+1) are the realand imaginarypart of A(, k), respectively. Wehave to showthat .4 is fractal, i.e. given a subsequence # of N, we have to define a homomorphism~r, such that ~r]¢t = onto Write B and ~D in place of/3~ and :Dn. Let # be a subsequence of N. We start with defining the mapping~r, on the set of the self-adjoint elements of A,. So let (A,~) E A, emdassume (A,(,)),~>I to be a self-adjoint sequence. Claim 1. There is a sequence ((c(~k))n>~)k>~ C_ B such that ~C(k) ~

~’

(7.9)

Indeed, write A,~ as Re An + i Im An. Since ((A~)),>l)k>l is a subset of A, we can approximate the sequence (Re A,),>~ as closely desired by sequences of the form (Re A(nk))n>i = (S(n2k))n>~. Then, clearly, the sequence (Re Av(,0)a>l can be approximated as closely as desired (~) sequences of the form (B,(n))n>~ E B,, and since Re A~(n) Av(n) by hypothesis, this gives the claim. Now,given a self-adjoint sequence (Av(n)) ~ choose and fix a se quence ~~C(k) ~(n)~>~k>l C_ B~ with property (7.9) and, for every k, choose a sequence (~(~k)) e /3 with R~(~(~~)) = t~(~) Let ‘4(k) refer to the smallest C*-subalgebra of ‘4 which contains the sequence (~(nk))n>~. The algebras A(k) are fractal by construction. Hence (Lemma7.8), there are homomorphisms~r(~ ~) such that for every k. In particular,

=(5(2))g.

332 CHAPTER 7.

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APPROXIMATION

SEQUENCES

Moreover, the coset (~(nk)) + 6 turns out to be independent of the choice of the ’representative’ (~(~k)) of the sequence~~(k) ~(n)~" Claim 2. Let (Cn), (Dn) E B be sequences with C~(n) ) for ev ery n.

Then

(Cn) + 6 = (On) Indeed, the sequence (Cn - Dn) belongs to 73 and is, thus, fractal by construction. By Theorem1.71, the limit lim [[Cn - Dn[I exists and is equal to [[(Cn - D,,) + 6[[, but this limit is zero since infinitely manyof the differences Cn - Dn are zero by assumption. Hence, (Cn -Dn) ~ 6, proving our claim. (k) Thus, knowing only the subsequences ~C ~ ~(n))n_>l, one can rediscover the cosets (C(, k)) + g uniquely. Claim 3. The cosets (C(nk)) + 6, k = 1, 2, ... converge in This follows from ffT(~) ii(c(~’~) +¢~- (cC~’~) +~11~/~ fC(k) ~ ~c.C~ ~ II(C.(n~)--’ (~) (°)11~,

(7.1o)

where the equality is a consequenceof the fractality of the sequence (C~~) C~0) ~ ~, and from the convergence of the sequences ~ .(~)~ to the sequence (A.(~))~2~. Hence, ~)) + g) ~ is a Cauchy seque nce and thus convergent, which verifies Claim a. Let (C~) + g denote the limit of the sequence ((C~~)) + g)~. Claim 4. The eoset (C~)+g does not depend on the choice o] the sequence (~) ((C~(~))n2~)~ which approximates (A~,(n)). ~ Indeed, choose besides ~C(~) it .(~))~kW~2~ another sequence ((D(~) .(~))n~)~2~ in B. which also converges to (A.(n)) and which generates (in the same (~) way ~ the sequence ((C~(~))) does) a coset (D~) + g. Then we

(~.10) < _ limsup k~

tC ~- D( , (~) .(.), (~

.l

= (An) +

(7.12)

C ,(n))n>l)k>l which Indeed, there are possibly several sequences (((k) verge to (A~(n)). But among these sequences there is by assumption at least one such that (C(~k)) -~ (An) v. in 9 For this special sequence, one evidently has (C(~ ~))+6~(An)+6

in ~’/6.

The limit limk-~o ((C(~k) ) + ~) is, as we have seen in Claim 4, independent of the choice of (c(nk)). Hence, ~rt,(A.(n)) = (An) which settles the construction of r. on the set of the self-adjoint sequences of A~. If now(A,(n))n>l is an arbitrary sequence in A~, then we define 7r~ (A~(n)) :-- ~r~ (Re A~(n)) + i~r~ (Im Due to (7.12), r~ (A.(n)) = An) + ~ + i (( ImAn) +~) = (An) ¯

whence~r~R~l.4 -- ~rlA as desired. 7.1.3

Arveson

dichotomy:

band

operators

Wesay that the self-adjoint sequence (An) E f has the Arveson dichotomy if every point in lim sup a(An) is either essential or transient. Observethat the Arveson dichotomy of a sequence is not immediately related to the fractality of this sequence. For example, the sequence (A,~) with 1, An = diag diag(0, (0, 0, 1, ...’,"’0,

1) 1) ifif nn is is even odd

334 CHAPTER 7.

SELF-ADJOINT

APPROXIMATION,

SEQUENCES

is fractal by Theorem7.3 since limsupa(An) = liminf a(An) = {0, but neither 0 nor 1 is an essential or a transient point. Conversely, 0 is a transient and 1 is an essential point of the sequence (An) given diag(O, 1,1,..., 1) if niseven An = diag(1, 1,..., 1, 1) if nisodd. This sequence fails to be fractal (again due to Theorem7.3). The goal of this section is to establish a result by Arvesonwhich states that the finite section methodfor self-adjoint band operators (as well as for certain self-adjoint band-dominated operators) has the Arveson dichotomy. Whencompared with [8], we shall consider a slightly more special setting, namely, we let H be a Hilbert space with orthogonal basis (ek)k>_O and we suppose that Pn is the orthogonal projection from H onto the linear span of e0, el, ..., e,~-i (whereas Arveson allows dim Im P,(- dim Im Pn-1 be greater than 1). Wewill identify H with the Hilbert space 12 =/2(Z+), which will be thought of as being embeddedinto/2(Z) in the natural manner. Given an operator A = (aij)i,jez L(/2(Z)) an d an int eger k, dkA := sup lai+k,il, and let B stand for the class of all operators A E L(/2(Z))

IIAII,

:=

+ +2


_ 1}, and we write T(L°°) for the smallest closed subalgebra of L(H) which contains all Toeplitz operators T(a) with a E L~°(’/I’). FromExample7.17 we infer that 7-(L~°) is a unital C*-subalgebra of the F¢lner algebra 5(7/). Wewill show in this subsection that 7-(L ~) is a Szeg5 algebra, and we will identify the unique state in Roo(T(L~)). The following lemma provides the basis for this. Lemma7.20 Let al, ..., ak ~ L~(q~), and let Pn refer to the functional pn = "~1 tr (PnAPn). Then lim p~(r(al)...T(ak))

= ~ (al...ak)(eit)dt.

Proof. Since T(ai) ~ ~(74), we conclude from Lemma7.18 that ( pn(T(al)’"

.T(ak))....

l tr(PnT(al)Pn

PnT(ak)Pn))

as n --~ ~. Nowwe think of/2(Z+) as being embedded in/2(Z) and of Toeplitz operator T(ai) as being the compression of the Laurent operator L(ai). Then, clearly, P,T(ai)Pn = P,L(a~)P,, and thus tr (PnT(al)PnT(a2)Pn " " PnT(ak)Pn) = tr (PnL(al)P~L(a~)P,...P,L(ak)Pn)). Let 74~ stand for the sequence of the spaces Im Pn, which we now consider as subspaces of/2(~). Repeating the arguments of Example 7.17 one can show that L(a) ~ ~(74’) for every a ~ L°°(’F); hence applying Lemma7.18 once more implies ( ~ tr(PnL(al)Pn"" PnL(a~)Pn) _ l_n tr (PnL(al)¯ .. L(ak)Pn))

"7.2.

SZEG(~-TYPE

THEOREMS

347

as n --~ c~. Since L(al)L(a2)...L(ak) = L(ala2...ak), and since this 1 r2~r~ ~*) Laurent operator has the constant value ~ J0 (al ... ak)(e dt on its main diagonal, we obtain

~

(al . . . ak)(ei*)

tr (P,~L(ala2 . . .ak)Pn)

which proves the ~sertion. For the concluding step in deriving SzegS’s theorem we need the fact that the mapping T(a) ~ ca n be ext ended to a * -homomorphism from T(L~) onto L~(~) which we denote by smb. It is not hard to establish the existence of this homomorphism;nevertheless we defer this discussion to the following subsection where a more general situation will be considered. So assume we are given a *-homomorphism stub : T(L~) ~ L~(~) mapping T(a) to a. Since both 1

A are linear 7.20 that

and continuous mappings, we immediately obtain from Lemma lim

for all operagors A ~ T(L~). ~heorem ~.~1 (S~ega - SeLegue) ~et A ~ T(L ~) be selJ-~djoint, let {~, ~), ..., ~} be the eigenwlues 4 PnAP~, ~nd let I: N ~ a be ~ comp~ctl~ s~pported continuous Junction. Then

n~lim

n

= 2~

(] o smb A) (e it) dr.

Proo£ The mapping smb is a *-homomorphism. So we conclude smb/(A) = ] o smbA, and (7.35) yields lim pn(f(A))

= lim tr (P~f(A)P~) 1 n = ~£ (f°smbA)(e~)dt"

Further, invoking Lemma7.18, we find for every polynomiMp that lim ~(tr (P~p(A)P~) tr (p( PnAPn))) = O

that

348 CHAPTER 7.

SELF-ADJOINT

APPROXIMATION

SEQUENCES

and since the polynomials are dense in C[a,/3] for every compact interval [a,/3], we obtain lim -l(tr

(Pnf(A)Pn) tr (f( PnAPn))) = 0

for every compactly supported continuous function f. Finally, tr (](P, AP,)) = ~(A~n)) + ...

we have

+ f(£~n)),

which finishes the proof.

~

This generalization of SzegS’s theorem belongs to Dylan SeLegue. For a further generalization (b~ed on similar arguments) we refer to [27], The~ rem 5.23, and also to the next section. 7.2.3

A further

generalization

of SzegS’s

theorem

There is yet another version of SzegS’s theorem which reads as follows: If a E L~(~’), T(a) is invertible, and if the finite section method (PnT(a)Pn) is stable, then the limit det (Pn-lT(a)Pn-x) lim n~ det (PnT(a)Pn)

(7.36)

exists and is equal to P1T(a)-XP1(which is actually the entry in the left upper corner of T(a)-~). If, moreover, a is a locally sectorial function, i.e. if a is of the form eb ¯ eiv where b is a continuous and v a real-valued function in L~(~") with Ilvlloo _< ~r/2, then (PnT(a)Pn) is automatically stable (Gohberg/Feldman, see [27], Theorem2.18), and the constant G(a) := 1/(P1T(a)-~P1)

(7.37)

can be identified with G(a) = exp (loga)0

(7.38)

with (log a)0 refering to the zero-th Fourier coefficient of log a := b ÷ iv. Although not obvious, the two versions of SzegS’s classical theorem are essentially equivalent (compare Theorems5.9 and 5.10 in [27]). In this section we are going to verify the existence of the limit (7.36) for operators A in place of T(a) which belong to a much larger algebra of operators rather than 7"(L°~). The algebra we will examine here is the algebra 7-~ of the Toeplitz-like operators introduced in Section 4.3.3. All notations such as R~, V+~, T(A), H(A), ~(A) and/-~(A) are as in section.

7.2.

SZEG~)-TYPE

THEOREMS

349

Proposition 7.22 Let A E T£ be invertible and (PnAPn) be stable. the limit det (P,~-~APn-1) lim ~-~oo get (P~AP,~)

Then

exists, and it is equal to PI~(A)-I P~. Proof.

We have det (Pn-~APn-~) det (P,~AP,,)

det (Rn-~AR,~-~) det (R~ARn)

where, by Cramer’s rule, the right hand side of this equality is equal to the first componentof the solution xn of the equation RnAR,,x,~ = T. (1, 0, 0, ...,

0)

Note that the stability of the sequence (PnAPn)involves that of (RnARn). Hence, as n --~ c~, the first componentof xn tends to the first component of the solution x of the equation ~(A) x = (1, 0, 0,...)T. Since ~(A) = s-lim RnARnis invertible by Polski’s theorem, we see that the first component of x actually coincides with P~(A)-~P1. In order to identify the constant PI~(A)-~P1 (at least in some instances) we need some further information about the algebra Proposition 7.23 The linear and continuous operators T, ~, H and [-I map the algebra TE into TE again, and ]or the compositions these operators one has o

T

TT~

¢~TO HH[-IO [-I [-I

~

H O

0 0 0 H O O.

Proof. The linearity is obvious, and the continuity is an immediate consequence of the uniform boundedness of the sequences (R~) and (V+~). let us verify the first row of the table for example. Given an operator A ~ T£, write T(A) =: V-nAVn + C~ with operators C,~ converging (together with their adjoints) strongly to zero. Then, for +, every m ~ Z V-mT(A)Vm = V-,~V-nAVnVm + V_,,~C,~Vm = V_,~-,~AVm+n + V-mCnVm

350 CHAPTER 7.

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SEQUENCES

and letting first n and then rn go to infinity we obtain (7.39)

V-mT(A)Vra = T(A), andT ( T ( A ) ) =T ( A respectively. Similarly, RmT(A)Rm

= PmRm+nARm+nPm.+

RmCnRm,

and passage to the strong limit as n -+ cx) yields RmT(A)R,~ = P,~(A)P,~, whence ~(T(A)) = ~(A). Furthermore, V-mT(A)Rm = V-m-nARm+,~Pm + V-mC,~Rm, which gives finally,

V-mT(A)Rm = g(d)Pm as well as H(T(A)) = H(A) and, RmT(A)V,~

= PmR,~+,AVm+n + R,~C,,Vm

involves RmT(A)V,~ = P,~ft(A) and [-I(T(A)) of the table can be checked analogously.

=/~(A). The other entries ¯

As a first consequence of the relations between T, ~, H and/~ established in the table we mention that ImHtJIm/~

C_ kerT = ker~ C_ kerHnker/~.

Proposition 7.24 The algebra T£ splits T£ = T(L°°(T))

(7.40)

into the direct sum KerT.

(7.41)

Proof. The operator T 6 L(Tf) is idempotent (i.e. 2 =T)by Proposition 7.23, which implies that TE = Im T ~ Ker T. From (7.39) we conclude that every operator in Im T is a bounded Toeplitz operator and hence of the form T(a) with a function a 6 L°°(~). Conversely, if a e Lc°(T), then T(a) belongs to 7"£ and T(T(a)) = T(a) by Theorem 4.36. Thus, ImT = T(L~(V)). Proposition 7.25 The kernel of T is an ideal of 7-£., and the quotient algebra Tf /Ker T is *-isomorphic to L~°(~). Proof. The ideal property of Ker T follows from the identity T(AB) = T(A)T(B)

+ H(A)f-I(B)

(7.42)

7.2.

SZF~GO-TYPF~ THEOREMS

351

derived in Theorem 4.36 (b) and from the inclusion ImH U Im/~ C_ kerT observed in (7.40). To identify the quotient algebra T£/Ker T recall that, by the preceding proposition, for every operator A E 7-£ there exists exactly one function a E L°° (~I’) such that A - T(a) belongs to Ker T. Clearly, the mapping A ~t a is linear, and the function a depends on the coset of A modulo Ker T only. Hence, T/2/KerT

-~ L°°(~),

A KerT ~ a (7

.43)

is a correctly defined linear mapping, and since T(A*) T(A)*, th is ma ping is even symmetric. To get the multiplicativity of (7.43), let A, B ~ T/2 and let a, b ~ L°°(’!~) be the (uniquely determined) functions such that A - T(a), B - T(b) KerT. Then, by (7.42), Proposition 7.23, Theorem 4.36(c), and Identity (1.16), T(AB) = T(A) T(B) + H(A)[-I(B) = T(T(A))T(T(B)) + H(T(A))[-I(T(B)) = T(T(a))T(T(b)) + g(T(a)) = T(a) T(b) + H(a) H(~) which yields the multiplicativity.

¯

Given A ~ T/2, we call the associated function a E L~°(~) with A-T(a) Ker T the symbol of A and denote it by smb A. The mapping smb (when restricted to T(L°~)) is just the *-homomorphism the existence of which we claimed in the preceding section. Observe also that the mappingsmb (more precisely: the mapping A + Ker T ~+ smb A) is a symbol mapping in the sense of Section 1.4.4: It is indeed evident from Proposition 7.25 that the invertibility of smb A implies the invertibility of A + Ker T in 7-/2/Ker T. Let us return to the context of Proposition 7.22. Let A ~ 7-/2 and a := smbA. Then we find for the constant G(A) := 1/(PI~(A)-Ip1) that G(A) = 1/(PIT(5)-lP1). If we further assumea to be locally sectorial, then a comparisonwith (7.37) and (7.38) yields G(A) exp (l og a) o. Su mmarizing these fa cts wegetthe following. Theorem7.26 Let the operator A ~ T/2 be invertible, the finite section method (PnAP,,) be stable, and the symbol a of A be locally sectorial. Then lim det (Pn-~ AP,~-I) = 1/exp (log n-~ det (PnAP,~)

352 CHAPTER 7.

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APPROXIMATION

SEQUENCES

Observe that the 0 th Fourier coefficients of the functions exp(log a) and exp(log 5) coincide under the conditions of the theorem. Remark1. The algebra 7-Z: of the Toeplitz-like operators is essentially larger than the Toeplitz algebra T(L~). For example, every Hankel operator H(a) with a E L°°(~) belongs to T£, but there exist Hankel operators which do not belong to T(L~) (Barria ([11]). Remark 2. It is not hard to extend the results of Proposition 7.22 and Theorem 7.26 to an appropriate class of operators on L(/2(Z)) which cludes, for example, the singular integral operators A = L(a)P + L(b)Q considered in Section 4.4.1. Under suitable assumptions for a and b, one can identify the associated constant G(A) with the product exp (log 5)0 exp (log b)0. 7.2.4

Algebras

with

unique

tracial

state

Let H be the Hilbert space/2(Z) and, given O E IR, let U and V refer the operators U: (x,~) ~-~ (xn-1), V: (xn) ~-~ (e-2~i° xn). These operators satisfy the commutator relation UV = e 2~ie) VU.

(7.44)

(7.45) Let ~l~ denote the smallest C*-subalgebra of L(/2(Z)) which contains unitary operators U, V (and the identity operator). In case O is irrational, ,4~ is called an irrational rotation algebra. The algebra Ao turns out to be simple in case O is irrational (see [44], TheoremVI.1.4). Hence, every C*algebra which is generated by two unitary elements U, V satisfying (7.45) is *-isomorphic to .Ao. Let Hn denote the subspace of/2(Z) consisting of all sequences (xk) with Xk = 0 for k < -n and for k > n, and set ~/= (H,~)~>I. Then both operators U and V belong to the Folner algebra ~(7-/) and, hence, Ao is C*-subalgebra of ~(7-/). The identification of Roo(A~) in the case where ~ is irrational is essentially simplified by the fact that fl, o has a uniquetracial state, T say, which clearly implies that R~o(¢4~) = (r} ([44], Proposition VI.l.a). Let now the situation be as in Section 7.9~.1, and let A be a unital C*-subalgebra of the F¢lner algebra ~(~/), but assume in addition that possesses a unique tracial state, r. Then every self-adjoint operator et ~ A determines a natural probability measure b~a on I~ by ~° f(x) d#A(X) = T(f(A)). (7.46)

/_

7.2.

353

SZEGO-TYPE THEOREMS

ttere is the specification of the results of Section 7.2.1 to this context. Theorem 7.27 Let A be a unital C*-subalgebra of the F¢Iner algebra ~(~l) which possesses a unique tracial state 7". Let further A E ‘4 be self-adjoint operator, associate with A a measure #A by (7.~6), and let ~(’~) refer to the eigenvalues of PnAPn.Then, for every corn)~n) , "’’’ "’dimH~ pactly supported continuous function f : ~ ~ ~, (n) oo lim f(~n))q_ ... _l_f(Adimn~) f_ f(x) dim Hn = oo

d#m(X).

Proof. Set p,~(A) := tr (PnAPn)/trP,~. Since T is the only tracial state of .4 we observe *-weak convergence of Pn to 7" by Theorem7.19, i.e. lim p,~(f(A)) = r(f(A)). As in the proof of Theorem7.21 one can check that lim p,~(f(A))

1

- tr (f( PnAPn dim H,~

1 = lim -- (tr (P~I(A)Pn) ,~-~ dim H,~

tr (I( P, AP, ))) = 0

and since tr(f(PnAPn)) = f(A~n))

)))

(n)

-t - f( /~di mHn)’

this completes the proof. Weconclude with an application of this result which has been intensively studied by W.Arvesonin a series of papers ([5], [6], [7]). Many one-dimensional quantum mechanical systems can be descibed by a Hamiltonian which is an unboundedself-adjoint operator on L2(I~) the form 1 (H f)(x) = --~ f" (x) + f(x), V : I~ --~ I~ being a continuousfunction representing the potential. In [5], [6] Arveson argues that the appropriate discretization of the Hamiltonian H, which preserves the uncertainty principle as far as possible, is the bounded self-adjoint operator 1 2 H~ := --~ Pd + V(Q,~) where

1 (Paf)(x) -- ~ ((f(x

q- 5)

354 CHAPTER 7.

SELF-ADJOINT (Qaf)(x)

APPROXIMATION

SEQUENCES

1 (sinSz) f(x),

and where 5 is the (small, positive, rational) numerical step size. If further U and Wstand for the unitary operators (Uf)(x)

= eiaXf(x),

(Wf)(x)

25),

then H~ can be written as H~ = aA + flI where a and fl are real numbers and A is the operator A = W + W.* + v(~(U - U*)), v being an appropriately rescaled version of V. Obviously, A belongs to the C*-algebra generated by the unitary operators U and W, and these operators satisfy the commutator relation WU = e 2i~ UW. Thus, the C*-algebra generated by U and Wis *-isomorphic to the irrational rotation algebra A,2/~ as mentionedat the beginning of this section, and we can identify U and Wwith the operators (7.44) and, hence, A with the tridiagonal operator on 12(Z), given by U1 + D + U~’, where U1 is the shift operator (xk) ~-~ (Xk-1), and D is the diagonal operator diag (d,~) with dk = v(-- sin (2d2k)). Even in the simplest case where v(x) = 2x (which physically corresponds to the case of the one-dimensional harmonic oscillator, and in which case the operator A is called the almost Mathieu operator), basic properties of the spectrum of A are unknown.So it is still an open problem to characterize the parameters ~ for which this spectrum is totally disconnected or not (consult [34] for somevalues of ~ (related with Liouville numbers) where a(A) is totally disconnected, and [131] where not). Consequently, it is of particular interest to understand how one can use numerical computations (based on efficient algorithms for calculating eigenvalues of self-adjoint tridiagonal matrices) in order to determine the spectrum of operators of the type discussed above. Theorem 7.27 gives a clear explanation in which sense the eigenvalues of PnAP,~approximate the spectrum of A.

Notes and references Most of the comments and references were already given in the text. Let us only mention once more Arveson’s papers [7, 8, 9] from which we took

7.2.

SZEG~)-TYPE THEOREMS

355

large parts of Section 7.1. The results of Section 7.1.2 can be found in [141], whereas those of Section 7.1.4 are perhaps new. Section 7.2 owesa lot both of its contents and of its presentation to the papers by Arveson[8, 9], B~dos[12, 13] and SeLegue[158]. Only the results of Section 7.2.3 are due to the authors, but they are certainly well-known to specialists. For similar discussions see, e.g., [27].

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Index absolute value of an operator 340 algebra 15, 35 antiliminal 223 Banach algebra 36 Calkin algebra 36, 152 C*-algebra 53 Douglas algebra 170 dual 227 elementary 214 F¢lner algebra 341 l~redholm inverse closed 298 irrational rotation alg. 352 liminal 223 normed 36 postliminal 223 primitive 216 semi-simple 55 simple 210 singly generated 154 standard 258 Szeg5 algebra 345 Toeplitz algebra 56 unital 35 Wiener algebra 334 algebras product of 226 restricted product of 226 ultraproduct of 67 annihilator 227 approximation method 26 applicable 26 collocation method 31,183 e-stable 127

finite section method31 Galerkin method 187 Moore-Penrose stable 87 projection method 28 spectrally stable 114 stable 26 stably regularizable 87 Arverson dichotomy 333 Bergman space 30 center 149 character 146 circulant matrix 190 paired circulant 190 collocation method 31, 183 singular integral op. 183 Bergman Toeplitz op. 52 composition series 225 length of a 225 condition number 631 79 generalized 100 convex hull 66 convolution operator 230 deficiency of an operator (cokernel dimension) 36 of a sequence 305 Douglas algebra 170 eigenvalue 12 eigenvector 12 element of an algebra 373

374 centrally compact 283 invertible 35 Moore-Penrose invertible 89 normal 113 of central rank one 283 of finite central rank 283 unit element 35 e-invertibility 119 e-kernel 123 of a non-negative op. 122 e-pseudospectrum 119 e-range 123 e-regularization 84 of a matrix 79 of a non-negative op. 123 e-stable sequence 127 families of homomorphisms243 sufficient 243 weakly sufficient 244 finite section method 31 band-dominated op. 200 singular int. op. 187, 220 Toeplitz op. on l ~ 49, 159 on Bergman space 52 F¢lner algebra 341 Fourier transform 230 fractal algebra 67 fractal approximation method 66 fractal *-homomorphism 67 fractal sequence 67 function almost periodic 232 characteristic 167 locally sectorial 348 piecewise continuous 151 quasicontinuous 170 piecewise 175 set function 106 Galerkin method 187

INDEX Gelfand transform 148 generator of an algebra 154 homcomorphism 149 homomorphism 35 *-homomorphism 54 unital 35 ideal 35 dual 227 largest 227 liminal 223 maximal 146 maximal ideal space 147 primitive 210 projection lifting 94 *-ideal 54 trivial 35 idempotent 89 index of an operator 36 of a sequence 287 involution 53 isomorphism 35 Laguerre polynomial 30, 231 least square solution 76 lifting of an ideal 238 lifting theorem 158 limes inferior of ~ set seq. 106 limes superior of a set seq. 106 limiting set 106 partial 106 uniform set 106 mean motion 233 Moore-Penroseinverse 12, 76 invertibility 89 invertible operator 80 projection 90 stable sequence 87

INDEX N ideals lemma 238 nullity of an operator (kernel dimension) 36 of a sequence 305 numerical range 134 algebraic 134 spatial 134 operator almost Mathieu operator 354 band 198,342 band dominated 198 bounded below 42 Fredholm operator 36 of second kind 29 Hankel operator 45 Hilbert-Schmidt operator 30 Hille-Tamarkin operator 29 Laurent operator 45 limpotent 112 Moore-Penrose invertible 80 non-negative 84 normally solvable 42 nuclear 340 of convolution 230 of regular type 42 partial isometry 84, 257 reflection 57 shift 33, 174,231 singular integral op. 178 Toeplitz operator 33, 45 on Bergman space 52 on the quarter plane 229 trace class 340 Wiener-Hopf operator 230 operator polynomial 128 orthogonal sum of Hilbert spaces 228 partial isometry 84, 257 partition of the identity 83

375 point essential 324 transient 324 polar decomposition 84 projection 89 interpolation projection 30 of Lagrange type 183 Moore-Penrose projection 90 projection lifting ideal 94 projection method 28 quadrature

methods 189

Raileigh quotient 134 regularization e-regularization 84 Tychonov regularization 79 representation 208 faithful 208 irreducible 209 unitary equivalence 210 sequence Mpha-number 287 deficiency 305 ¢-st~ble 127 Fredholm sequence 285 index of a 287 weakly 305 Moore-Penrose stable 87 nullity 305 spectrMly stable 114 stable 26 stably regularizable 87 separation property 241 set function 106 shift operator 33, 174, 231 singular integral 178 singular value 77, 119 splitting property of 88, 305 singular value decomposition 77 space

376 invariant subspace 209 maximalideal space lzl7 of splines 167 state space 134 totally disconnected 113 spectral theorem 83 spectrum 35 essential 324 of an algebra 210 spline space 167 splitting property 88, 305 finite 305 state 134 tracial 344 state space 134 subalgeb~a 35 inverse closed 55 symbol 61,244 symbol mapping 61 Szeg5 algebra 345 Toeplitz algebra 56 topology discrete 227 hull-kernel 211 Jacobson 211 *-weak 148 strong 13 trace of an operator 336 two projections theorem 181 Wiener algebra 334

INDEX