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[
_
(5)
This is the prototype of what is called a Galerkin method. On introducing the orthogonal projections Pn : L2(~’) --> span {z-n,...,zn},
f ~’~ E (f’ zk) zk’
16
CHAPTER O.
INTRODUCTION
we can rewrite the linear system (5) Anun := PnAPnun = Pn(aI + SbI)Pnun
= PaY.
Hence, we can think of the approximation operators An as being composed again, e.g. by A and Pn. Since we already agreed upon considering the whole sequence (An) as the basic object (and not the single operator An), we are thus led to the idea of considering the sequence (An) as being composed, e.g. by the ’atomic’ sequences (Pn) and (A), or by (Pn), (aI), (bI) and (S): ( n)n=l
"-~
(Pn)n=l(A)~n-l(Pn)n_=l
:
(Pn)[(aI)
T (S)(bI)]
(Pn).
(6)
It is thus natural to define algebraic operations for approximation sequences, as well as to introduce algebras of approximation sequences. The following definitions are suggested by (6): For arbitrary sequences (An), (Bn) and complex numbers a we set (An)+(Bn) := ~An+Bn), (An)(Bn)
:= (AnBn),
a(An) := (aAn).
(7)
Let 9~ denote the set of all boundedsequences of operators on a fixed Banach space X. Provided with the operations (7), ~ will becomean algebra and, if a normon ~" is defined by II(An)ll := sup IIAnll, even a Banachalgebra. If (An) is an approximation method for an operator A, then the BanachSteinhaus theorem implies the boundedness of the sequence (An). Thus the algebra ~" contains every approximation method. Nevertheless, it is not yet the algebra which is adequate for the purposes of numerical analysis. The point is that the first few elements of an approximation sequence do not influence the asymptotic behavior of the sequence. Thus, it would be desirable to identify, for instance, the sequences (A1, A~, ...,
A~:~, 0, 0, ...)
and (0, 0, ...).
(8)
Algebraically seen, this requires putting all of these sequences into one ideal of $" which then has to be factored out. More precisely: The set Go of all sequences (An) with only finitely manynon-zero entries An is an ideal the algebra ~’, i.e. it is a subalgebra and every product of a sequence in Go with a sequence in ~ belongs to Go again. Modulo 60, the sequences in (8) indeed coincide. Since the ideal Go is a linear subspace of ~-, one can form the quotient space ~/6o which is a linear space again. Moreover, in ~’/60, there is a natural definition of the multiplication of cosets by ((An) + 60)((Bn) + (AnBn) + 60, whi ch makes 9 ~’/6o to an a lge bra, the quotient algebra of ~" by 60. (The independence of this definition from
0.3.
ALGEBRAIC LANGUAGE OF NUMERICAL ANALYSIS
17
the concrete choice of representives of the coset is a consequence of the ideal property.) In order to introduce a norm on a quotient algebra, and to verify the completeness of the quotient algebra with respect to this norm, one needs the closedness of the ideal factored out. In our setting, the ideal Co is not ¯ closed in ~, but it is an easy exercise to determine its closure G, which consists of all sequences (An) with IIA~II -~ 0 as n --~ oo. Clearly, ~ is again an ideal of ~, one can form the quotient algebra ~/~, and II(An)
Gll :=~.)einfg II (An) - (Gn)lly = inf su IlAn- G~II (
(~,,)e~
is a norm on ~-/G which makes the latter to a Banach algebra. It is one purpose of the present book to convince the reader on the fact that the algebra 5r/G indeed provides the adequate frame in which many of the problems of numerical analysis find their natural place. (Of course, not all problems: for instance the speed of convergence of the approximate solutions to the solution x of Ax = y does not only depend on the coset (An) + ~ (and the right hand side y) and can therefore not be reflected within ~-/6.) So, in a sense the role of ~-/G in numerical analysis can be compared with that of the algebra L(X) in operator theory. Wewant to mention here at least one point which makes this adequacy evident. For, we introduce the notion of a stable sequence. Definition 0.3 A sequence (An) E a~ is stable if the operators An are invertible, beginning with a certain no, and if the norms of their inverses are uniformly bounded. Stability of an approximation methodis closely related to the applicability of this methodas the following b~sic result shows. Theorem 0.4 (Polski’s theorem) Let A ~ L(X) be an invertible operator and (An) C_ L(X) be an approximation method for A. This method applicable if and only if the sequence (An) is stable. The next result relates invertibility problem:
the stability
problem for a sequence (An) to an
Theorem 0.5 (Kozak’s theorem) A bounded sequence (An) is stable if and only if the coset (An) + ~ is invertible in the quotient algebra jz/~. Thus, the applicability of an approximation method (An) is essentially equivalent to the invertibility of the coset (An) + G in the Banach algebra
y/a.
18
0.4
CHAPTER O.
INTRODUCTION
Microscoping
The translation of stability problems into invertibility problems in Banach algebras offers the possibility of employinggeneral invertibility theories (such as Gelfand’s theory for commutative Banach algebras) to study the applicability of numerical methods and, actually, this has been the main reason for introducing algebraic concepts into the world of numerical analysis. To explain what is meant, consider once more the Galerkin method (P,~(aI+SbI)Pn) for the singular integral operator on the unit circle ~, and suppose first that a and b are continuous functions on ~?. Wewill think of this approximation sequence as an element of the smallest closed subalgebra of the algebra :~ (consisting nowof all boundedsequences (A,~) of operators on L2(~I’)) which contains the sequences (Pn), (S) and all sequences (cI) with c E C(~). Clearly, this algebra also encloses all sequences of the form (Pn(aI
+ ~S)P,~)
with a,~ E C
(9)
which correspond to the Galerkin method for singular integral operators with constant coefficients. Naturally, it proves to be mucheasier to study the stability of the sequences (9) with constant coefficients rather than the stability of sequences with arbitrary continuous coefficients. This leads to the obvious idea to try to reduce the study of approximation methods for operators with continuous coefficients to that of approximation methods for operators with constant coefficients, possibly by freezing in the coefficients, i.e. by replacing the functions a and b by their values a(A) and b(A), which has to be done for every point A ~ "~. Is it possible in this way to replace a complicated stability problem (that for the sequence (P,~(aI + SbI)Pn) for instance) by a whole family of much simpler stability problems (those for the sequences (Pn(a(A)I + b(A)S)P,~) with A running through ~ in our setting)? This idea (which has its origins in operator theory) proves to be x;ery fruitful. It is indeed possible (although not obvious) that this programme can be carried out for large classes of approximation methods for large classes of operators (including the Galerkin method for singular integral operators with continuous coefficients). In simple situations, an elementary partition of unity will suffice to decomposea complicated ’global’ invertibility (or stability) probleminto a family of ’local’ problems, and glue the results for the local problems together to obtain the solution for the global problem. This quite naive decomposing and glueing ’by hand’ denies for more complicated sequences. If, for example, the coefficients a, b of the singular integral are discontinuous at some points A ~ ~ (if they have jumps or
0.4.
MICROSCOPING
19
oscillations there), then it is already not clear what the ’right local representations at A’ should look like (but it is clear that simple evaluation at A will not provide local representatives). Roughlyspeaking, one has to consider two coefficient functions as locally equivalent at the point ~ E ~7, if they appear as almost equal whenlooking through a microscope directed to the point )~, and if the coincidence gets better whenever we increase the magnification. For example, the ’very different’ functions f and g in Figure 1, which both have a jump at ~ E ~, coincide locally at A.
10x
1000x
Figure 1: Microscoping Also the functions ] and g in Figure 2 coincide locally at ~ (which becomesevident by taking into account that the difference of coefficient functions is measured in the supremum norm), whereas the functions and g in Figure 3 are not locally equivalent at A.
Figure 2: Locally equivalent functions Moreover,it turns out that certain kinds of oscillating functions (quasicontinuous functions e.g.; see Section 4.3) behave locally as constant func-
2O
CHAPTER O.
INTRODUCTION
Figure 3: Not locally equivalent functions
tions again. But to see this, one needs a much stronger instrument than a commonlight microscope, an electron microscope say, which shows that a quasicontinuous function (which can oscillate at the points of ~) can identified with a continuous function on a certain ’balk’ (Figure 4) and, thus, localizing over this balk (and no longer over the unit circle) provides constant functions as local representatives.
Figure 4: Localization of quasicontinuous functions It is obvious that one needs a machinery which formalizes the intuitive notion of local equivalence and which can take the place of the ’decomposing and gluing by hand’. Such machineries are well known, and the most convenient and elegant of them are formulated in the language of Banach algebras. Here is one of these machineries: Theorem 0.6 (Allan’s local principle) Let ~4 be a Banach algebra with identity e, and let B be a closed subalgebra in the center o].4 which contains e. For every maximal ideal x o] l~, let Is denote the smallest closed ideal o] A which encloses x. Then an element a E ,4 is invertible in A if and
0.5. A FEW REMARKS
ON ECONOMY
21
only i] the cosets a + Ix are invertible in A/I~ ]or all maximalideals x with I~A. The whole Chapter 4 is devoted to this theorem and its several applications. Here we only reveal that the maximalideals x just correspond to the points of the unit circle (for coefficients with jump discontinuity) or to the ’balk’ (for quasicontinuous coefficients), that two approximation sequences coincide locally at x if and only if their difference belongs to the ideal Is, and that the cosets moduloIs are the ’right’ local representatives.
0.5
A few remarks
on economy
As a rule, every Banach algebra of approximation sequences will contain manymore sequences than those which we were previously interested in and which gave rise to the construction of the algebra. So, in manyapplications the concrete sequences we want to examine are just the generators of the algebra, but the algebra obviously also contains sums of products of its generators. Moreover, new elements will arise in the process of completing the algebra (or of closing it within a larger algebra), and these newelements are often of a quite different nature than their generators. To have an example: the smallest closed subalgebra of 3v which contains all Galerkin sequences (Pn(aI + SbI)Pn) with a,b E C(T) contains not only all sums of products ~i I’Ij(Pn(aiJ I + SbijI)P,~), but also all sequences (P,~KP,~) where K is compact on L2 (T). Since we need complete algebras in order apply Allan’s local principle, we cannot avoid these new and (at the first glance) unwelcomeelements. Hence, in order to obtain stability results for a few special sequences, we will require whole Banachalgebras of sequences, in which most elements lie outside our previous interests. So one might ask whether there are other, more economic, approaches which only deal with the sequences we are really interested in. The first answer is that there are manystability problems for approximation methods which can actually be investigated only by the algebraic approach via local principles; so there is indeed no other choice of approaches at the moment. The second answer is that the very same approach which allows us to study the stability of the generators of the algebra, in manysituations also allows us to examinethe stability of an arbitrary sequence in the algebra. In other words: we often comeinto a situation which allows us to be able to compute the spectrum with respect to invertibility in 9r/~ of an arbitrary coset (An) ÷ of thealge bra of a pproximation methodswe are considering. So one often gets stability criteria for other interesting approximation sequences almost automatically. For some typical instances see [77] where, based on approximation sequences for singular
22
CHAPTER O. INTRODUCTION
integral operators with piecewise continuous coefficients, a Banachalgebra is defined and studied, which besides its generators also contains approximation methods such as quadrature or collocation methods for Mellin operators, and where the derived stability criteria are of a form which applies both to the parents of the algebra and to the new membersof the family. But the real importance of the latter facts only becomes clear if one considers C*-algebras of approximation sequences. A Banach algebra is C* if there is an involution a ~ a* on the algebra such that [[aa*l] = I[a[I 2 for every element a. The algebra ~- of all bounded sequences (An) of operators on the Hilbert space L2(~") is a C*-algebra with respect to the involution (An)* := (A~), and also its smallest closed subalgebra A containing all sequences (Pn(aI + SbI)Pn) with a, b E C(~’) is C*. The point of the C*-setting is that the knowledgeof a suitable invertibility (-- stability) criterion will enable us to describe the algebra completely in the following sense: Wecan construct a mapping from the algebra into another, in some sense simpler, C*-algebra which preserves the algebraic and metric structure, i.e. the mappingis a so-called *-isomorphism. In the case of the algebra .4 defined above, the algebra .A/G proves to be *-isomorphicto a C*-algebra, the elements of which are triples (B1, B2, B3) of operators on L2(~I’). So, instead of working with cosets of (infinite) sequences, one can simply deal with triples of operators! This isomorphy also opens a way to tackle a whole series of other problems (including the problems B’ and C’ mentioned above). Thus, the description of concrete algebras of approximation sequences will occupy a main place in this book.
0.6 Brief description
of the contents
Weconclude this introduction by a short description of the contents of the several chapters of the book. The first three chapters are devoted to the problems A’-C’ for approximation sequences, respectively. Their main goal is to point out the close relationship between these problems and certain properties of the coset (An) + in the quotient alge bra 5v/G, thus highl ighting the o utst anding role of this algebra in numerical analysis. For problem A’ this connection is roughly explained above. Parallel to this we single out a special class of subalgebras of 9v which we call fractal. The reason for this notion is that whenever (An) belongs to a fractal algebra A then every infinite subsequence of (An) contains all essential information about the sequence (An) itself or - more precisely the knowledgeof an infinite subsequence of (An) allows us to reconstruct (An) within .4 up to a sequence in
0.6.
BRIEF DESCRIPTION OF THE CONTENTS
23
The property of fractality proves to be of a similar fundamental importance as that of stability. Particularly, it guarantees that several approximations can be done uniformly. This will be illustrated for the convergence of the condition numbers in Section 1.6, for the approximation by generalized invertible matrices in Chapter 2, and for the asymptotic behavior of eigen- and singular values in Chapter 3. The applicability of the results of Chapters 1-3 to a concrete approximation method (A,~) depends essentially on the possibility of embedding (An) into a C*-subalgebra of ~- which can be described completely. Weare thus led in a natural way to the investigation of concrete C*-subalgebras of ~’, and this will be our subject in Chapter 4. The heart of this chapter is the construction of symbol mappingswhich, for instance, associate with every sequence in the C*-algebra generated by the Galerkin method for singular integral operators with continuous coefficients a certain triple of operators, in such a waythat the stability of a sequence, or its eigenvalue asymptotics, only depend on the symbol of the sequence. In the fifth chapter we consider these concrete examples once more, now in the light of the general representation theory of C*-algebras. Particularly we will point out that some features of the concrete algebras which seem to be incidental at the first glance, are actually consequencesof somequite general hypotheses on approximation algebras. The sixth chapter is devoted to a more special but fascinating property of many concrete algebras of approximation sequences, namely the existence of a Fredholmtheory with index calculus for the elements of the algebra. For this goal we introduce such notions as ’kernel dimension’ or ’cokernel dimension’ as well as ’index’ for approximation sequences (An) and then point out howthese notions are related to the asymptotics of the singular values of A,~ as n tends to infinity. In Chapter 7 we deal with a finer analysis of the eigenvalue distribution in case of self-adjoint sequences (An). Particularly, we present Arveson’s results culminating in a generalization of the Szeg5 limit theoremfor finite sections of self-adjoint Toeplitz operators to self-adjoint operators which are almost band operators with respect to the selected basis for the finite section method. The logical dependenceof the several chapters is as follows: Since this is a book about the interplay between functional analysis (operator theory, C*-algebras) on the one hand, and numerical analysis the other hand, but neither a book about functional nor numerical analysis, somebasic knowledgeof the latter is assumed(although all needed facts are cited exactly and are provided with references). The assumed material from functional analysis is completely covered by Cryer [39] (the view of which is very close to that we take), Douglas [50] (in which the use of Banach
CHAPTER O.
24
INTRODUCTION
5
Figure 5: Dependence of the contents algebra techniques in operator theory is explained), Reed/Simon[127], or Rudin [154], and for the numerical side of the coin we recommendGohberg/Feldman [64], Lebedev [101], or Tyrtyshnikov [175]. A presentation on a quite elementary level of the finite section methodfor Toeplitz operators (which serves as one main model throughout the present book) as well as related questions can be found in the recent text book BSttcher/Silbermann [27].
Chapter 1
The algebraic language numerical analysis
of
Numericalproblemsinvolving infinite dimensional operatorsrequire a formulationin terms of C*-algebras. WilliamArveson The goals of this chapter are to makethe readers familiar with the algebraic language and its dialect spoken in numerical analysis and to introduce two fundamental notions: the stability and the fractality of an approximation sequence. Further we present an approximation method which will accompany us throughout this book and which is without any doubt the most natural and most important approximation method for bounded linear operators on a separable Hilbert space: the finite section method.
1.1
Approximation
methods
Westart with a brief description of the basic subject of this book, approximation sequences, and proceed with two concrete approximation methods. The present and forthcoming section are the only places where we deal with operators acting on (arbitrary) Banach spaces. 25
26
CHAPTER 1.
1.1.1
Basic
THE LANGUAGE OF NUMERICAL ANALYSIS
definitions
Let X be a Banach space over the complex field C. Given a linear bounded operator A on X and an element f of X, consider the operator equation
Au= I.
(1.1)
For the approximate solution of this equation, we first choose appropriate closed subspaces Xn of X in which the approximate solutions un of (1.1) will be sought. In practice, the Xn usually have a finite dimension, but we will not require this at that point. Our only assumption for the moment is .that the Xn are ranges of certain projection operators Ln : X -+ Xn, and that these projections converge strongly (i.e. pointwise) to the identity operator: s-limn-~Ln = I. Recall that a projection is an operator L such that L2 = L. The strong convergence of Ln to I implies that the union t~=lXn is dense in X. Having fixed the subspaces Xn, we choose linear operators An : Xn ~ Xn and consider in place of (1.1) the equations Anun = Lnf,
n = 1,2,3,...
(1.2)
with their solutions sought in Xn = Im L,~. Of course, we have to assume some consistence between the operator A and its ’approximations’ An. The weakest requirement we will impose is that the sequence (An)nCX)=l is an approximation method for A in the following sense: Definition 1.1 A sequence (An) of operators An L(Im Ln) is an approximation method for A e L(X) if AnLn converges strongly to A Even if (An) is an approximation method for A, then we do not yet know anything about the solvability of the equations (1.2) and about the relations between (possible) solutions un of (1.2) and the (possible) solution (1.1). Thus, we distinguish the special class of applicable approximation methods. Definition 1.2 The approximation method (A,~) for A is applicable i] there exists a number no such that the equations (1.2) possess unique solutions Un ]or every n >_ no and every right hand side f ~ X and i] these solutions converge in the norm of X to a solution of (1.1). For an equivalent characterization of applicable approximation methods we still introduce the notion of stability of a sequence of operators.
1.1.
APPROXIMATION METHODS
27
Definition 1.3 A sequence (An) of operators there exists a numberno such that the operators ANare invertible for every n >_ no and if the norms of their inverses are uniformly bounded: sup ~__~n 0
The following b~ic result connects these notions. Theorem 1.4 (Polski) Let (Ln) be a sequence of projections which converge strongly to the identity operator, and let (A,) with A~ L(Im Ln) be an approximation method for the operator A e L(X). This method applicable if and only ff the operator A is invertible and the sequence (An) is stable. ProoL Let A be invertible and let (A~) be stable. Then, for all sufficiently large n and all x ~ X,
+ ~]Lnd-lx - A-~z~], which implies the strong convergence of A~Ln to A-~ and, thus, the applicability of the method (An). Let, conversely, (A~) be an applicable approximation method for Then, by definition, the operators A~ are invertible for large n, and the sequence (A~IL~) of their inverses is strongly convergent. So, the BanachSteinhaus theorem (= the uniform boundedness principle) entails the stability of the sequence (An). In order to check the invertibility of A recall first that the definition of an applicable methodrequires the solvability of the equation Au = f for every right h~d side f. Hence, A is onto. For the injectivity of A observe that, for all sufficiently large n and all x ~ X
Since the norms ]]A~Ln~] are uniformly bounded, one obtains the convergence A~LuAx ~ x for every x ~ X. If, in particul~, x ~ Ker A then necessarily x = 0. Thus, for a sequence (A~) to be an applicable approximation method for an operator A, three points are decisive: ¯ strong convergence of A,~L,~ to A,
28
CHAPTER 1.
THE LANGUAGE OF NUMERICAL ANALYSIS
¯ invertibility of A, ¯ stability of (An). In manyapplications, the strong convergence of A,~L,~ to A is more or less evident (see the following subsections), and the invertibility of operators belongs to the topics of operator theory which will concern us only in as muchas we will cite the needed results. So, our main attention will be paid to the stability problem. Let us still mention that, whenever(An) is an applicable approximation method for an operator A and (fn) is any sequence of elements fn E Im Ln which converges to f, then the equations Anun = fn are uniquely solvable for all sufficiently large n, and their solutions converge to the solution of Au = f. This follows from the estimate
the right hand side of which tends to zero as n ~ ~. 1.1.2
Projection
methods
Let X be a Banach space, A a bounded linear operator on X, and (L~) a sequence of projections converging strongly to the identity I ~ L(X). The idea of any projection method for the approximate solution of (1.1) is to choose a further sequence (R~) of projections which also converge strongly to the identity and which satisfy Im Rn = Im L~ (of course, one can take Rn = Ln also), and to choose A~ = R,~AL~ : ImL~ -~ ImL~ as approximate operators of A. That (RnALn) is indeed an approximation methodfor A is a consequence of assertion (a) of the following lemma, the proof of whichis left as an exercise. Lemma1.5 Let X be a Banach space and A, B, C, An, Cn be bounded linear operators on X. (a) If A,~ -+ A and Cn -+ C strongly, then A,~BC,~ -~ ABCstrongly. (b) If A~ -~ A and C~ --~ C* strongly and i~ B is compact, then IIAnBC,~ ABCII ~ 0 as n ~ o~. Assertion (b) of the preceding lemmaimplies a first stability result. Theorem 1.6 Let X be a Banach space, and let (Ln) and (Rn) be quences of projections on X with Im R~ =Im L,~ and L~ ~ I, L~ -~ I* and R,~ --~ I strongly as n --~ o~. If A is the operator aI + K with K compact on X and a ~ C~, then the projection method (RnALn) applies A if and only if A is invertible.
1.1.
29
APPROXIMATION METHODS
Proof. Weonly consider the case where X has infinite dimension. If A is invertible, then a ~ 0, and it is evident that the sequences (RnALn) (aLn + RnKLn), consisting of operators acting on Im Ln, and (RnAL,~ a(I - L,~)), consisting of operators on X, are stable or not only simultaneously. The stability of the latter sequence is an immediate consequence of the equality [[A - R~AL~- a(I - L~)[[
= IlaX + K -
aL~ - RnKL~- a(I -
L~)II = IlK - R~KL~II,
of Lemma1.5(b), and of a Neumannseries ~gument applied to the invertible operator A. The converse statement follows from Polski’s theorem. Example 1.7 Let X be the Lebesgue space LP[O, ~) with 1 < p < ~, I the identity operator on X, K the integral operator (Ku)(t)
=
k(t,s)u(s)ds,
[0,~ ),
and let A = aI + K with some a 6 C. If a ¢ 0 and K is compact, then A is a ~edholm operator of second kind. In particular, K is compact on LP[0, ~) if the kernel function k satisfies Ik(t,s)lqds
dt < ~
(1.3)
with l/p+ 1/q = 1, in which case K is a so-called Hille-Tamarkin operator For n = 1, 2,... let L~ stand for the operator on LP[0, co) of multiplication by the function Xl0.nl which is 1 on [0, n] and 0 on (n, oo). Evidently, L~ is a projection operator, and Ln -~ I, L~ -~ I* strongly as n -~ oo. By Theorem1.6, the projection method (L,~AL~) is applicable to A if and only if A is invertible. Hence, for invertible A, the equations aun(t) + k(t,s)u~(s)
= f (t ), t e
[O,n
(1.4)
are uniquely solvable for all sufficiently large n and all f E LP[O,oo), and their solutions converge in the normof LP[O, oo) to the solution of
au(t) + k(t,s)u(s) = ](t ), t e [0,~ In this example, the spaces Im Ln = L~[O, n] still have infinite dimension, and so a further discretization (e.g. by a quadrature rule) is needed solve (1.4) numerically.
30
CHAPTER 1.
THE LANGUAGE OF NUMERICAL ANALYSIS
Example 1.8 Weconsider the same operators as in Example 1.7, but now on the Hilbert space L2[0, c~) with standard inner product/., .). In the case p = 2, condition (1.3) reduces
~o
~ ~o°~ Ik(t, s)l 2 dsdt < c~,
(1.5)
and integral operators K with kernel functions k satisfying (1.5) are called Hilbert-Schmidt operators. The functions en(t) = e-t/2 ],~(t), n 0, where t fn(t) =e [~~jd ~n(t’ne--t ~ j refers to the n th Laguerre polynomial, constitute an orthonormal basis of the Hilbert space L2[0, c~). Thus, if we let Ln denote the orthogonal projection from L2[0, ~x~) onto the span of the functions co,el,... ,e,~_~, then L,~ = L~ -~ I as n --~ oo, and (LnAL,~) is an approximation methodfor A which is applicable if and only if A is invertible. In this setting, we look for an approximate solution un in the form A~) and the nth approximation equation for ~t n ~_ c(on) eo -~ ... -~ (~n_l~n_l, Au = f is just the linear n x n system n--1
~(Aei,ej)cl’O
= (f,
ei),
j=O,
1,...,n-1
i=O
with the unknowns -(~) . Here we chose f~ = Lnf as an approximation of t; i the right hand side. ¯ Example 1.9 Let ID = {z ~ C, Izl < 1} denote the unit disk, and consider the Hilbert space L2 (lI)) of all functions f on I~ with
Ilfll ~ = If(z)[2dA(z)
If(rei~)lerdrdt < c~.
The Bergman space A2(~) consists of all functions in L2(ll)) which analytic on I~. The Bergmanspace is a closed subspace of L2(I~), i.e. Hilbert space again. One can show that the functions kek(z) = kv~--~-~z with k = 0, 1, 2,... form an orthonormal basis of A~(I~). Let L~ denote the orthogonal projection from A2(~) onto the span of eo,e~,...,e,~. Further, for n _> 0 set ~,~ := exp(2ri/(n 1)), an choose p ~ (0, 1). Then the interpolation projection R,~ from A~(D) onto span{co,.., e~} is the operator which associates with every function f A~(ll)) a function Rnf e span{co,..., e,~} such that (R,~f)(p~k~) for k = O, 1,...,n. It is again obvious that Ln = L~ -+ I. Moreover, in this special setting one even has IIL~- n,~ll --~ 0 as n-+ (x)
(1.6),
1.1.
APPROXIMATION
METHODS
31
which implies that Rn -~ I. (Observe that (1.6) is a peculiarity of Bergmanspace projections Ln and Rn which, in general, is not typical for the relations between orthogonal and interpolation projections.) Consequently, if K is compact on A2(]~) and A aI+K with so me a ¯ C\{0} then the collocation method (RnALn) is an approximation method for A, and this methodapplies if and only if A is invertible. The solution u~ = ?-,i=0 ci ei of the equation R~AL~u~= R~f can be determined by solving the (n + 1) x (n + 1) linear system ~(Ae~)(p¢~)c~ ~) =
f(pe~), j = o, 1,...,n
for the unknowns’~) c~ 1.1.3
Finite
section
method
The finite section methodis a special projection methodwhich we will only consider on a Hilbert space. Let H be an infinite-dimensional separable Hilbert space with inner product (., ./and normIlxll~/= (x, x). For a given orthonormal basis (en)n>o of H, let P,~ denote the orthogonal projection oo from H onto span{eo,... ,e,~}. Thus, if x ¯ H and if x = ~k=o(X, ek)ek is the Fourier series expansion of x, then P~x = ~kn__o(X, ek)e~. Clearly, P~ = P,~ -~ I strongly. The approximation method (P~APn) is called the finite section method for A ¯ L(H) (with respect to the given basis To visualize the notion ’finite section method’ recall that, due to the Parseval identity [[xl[ 2 = ~°=0 [(x, ea)l 2, every element of H can be identified with the sequence of its Fourier coefficients. Moreprecisely: If 12 denotes the Hilbert space of all sequences (xk)~=o of complex numbers with norm[[(Xk)[I = (~-]~ Ixk[2) 1/2, then the mappingC: x ~, ((X, ek))~°=O is an isometry from H onto 12. Further, the operator equation Au = f on H is translated by this -1 Cu = Cf on 12: isometry into the equation CAC (Aeo,el)
(Ae~,el)
(u,e~)
__ (f,:e~)
.
(1.7)
The same correspondence transfers the approximate equations P~APnu~-Pnf into the equations Anun -- ]~ where , Ao = ((Aeo,eo)),
, ((Aeo,eo) A1 = (Aeo,e~)
(Ael,el)
32
CHAPTER 1.
THE LANGUAGE OF NUMERICAL ANALYSIS
and , , ( If, e0) f~ = ((f,eo)),fl = \ If,~1). Thus, the A~ are indeed the ’finite
sections’ of the system matrix -1 CAC
of(1.7). Theorem 1.10 Let H be an infinite-dimensional separable Hilbert space. (a) If A is an invertible operator on H, then there exists an orthonormal basis of H such that the finite section method with respect to this basis applies to A. (b) IrA = B+iS where B is positive definite and S is self-adjoint, then the finite section method with respec* to every orthonormal basis of H applies to A. Proof. A proof of (a) can be found in [64], Chapter II, Theorem4.1. For a proof of (b), let {ei} be an orthonormal basis of H, and let Pn denote the orthogonal projection from H onto span{e0,...,en}. For all x E H one has (PnAPnz, Pnx) = (APnx, Pnx) = (BPnx, Pnx) + i(SPnx, Since (BPnx, Pax) and (SPnx, P,~x) are real, this identity implies ](P~APnx, P,,x)] >_ I(ePnx, Pnx)[. Further, due to the positive definiteness of B, there is a C > 0 such that ](P~APnx, P~x)] >_ I[P,:ll z
for al l z E H,
which, via the Cauchy-Schwarzinequality, yields ]IPnAPnxl] ~ C]lPnxl] for all x e H. Hence, all matrices PnAP,~ are invertible, and the norms of their inverses are uniformly bounded by 1/C. ¯ Assertion (a) of the preceding theorem gives rise to at least two basic problems: ¯ Given an invertible operator A on H, determine a basis of H such that the finite section methodwith respect to this basis applies to A. ¯ Given an invertible operator A on H and a basis of H, does the finite section method with respect to this basis apply to A ? Wewill exclusively discuss the second question; mainly for the following reason: many operators (among them most of the concrete operators appearing in this book) possess their own ’natural’ basis, and any essential change of this basis woulddestroy the ’structure’ of the operator. A trivial exa~nple is provided by the diagonal matrices on l 2. Here is a less obvious example.
1.1.
APPROXIMATION
33
METHODS
Example 1.11 An operator S on a Hilbert space H is called a shi]t operator if it is an isometry and if (S*) j -~ 0 strongly as j -~ c~. Let K := Ker S*. One can show that SJK ± SkK whenever j ~ k and, moreover, that every x E H allows a unique representation as a sum ~-~j=o SJkJ with elements kj E K. Thereby, kj = Po(S*)Jx, where Po = I - SS* is the 2orthogonal projection from H onto K, and [[x[[ = ~’]~j~--o []kJ[[ Nowsuppose in addition that dimK= 1, let e0 be a unit vector in K, and set ej := SJeo. Then {ej}j~=0 is an orthonormal basis of H which is, in a sense, the ’natural’ one for S. The ’translation’ of S into an operator on -1, acts via (xo,xl,x2,...) 12, i.e. the operator CSC ~ (O, xo,xl,x2,...), which emphasizes the naturalness of this basis and also explains whyS is called a shift. Let us further mentionthat the ’translation’ of the operator polynomial ~-~-1_ k cj(S*) -j + ~=o c~Sj is the matrix ~Co
C--I
C--2
...
C--k
Cl
CO
C--I
¯¯
C~k+l
C2
Cl
CO
¯.
C-k+2
:
:
Ck
Ck-i
0
ck
C--k+l
: Ok-2
C--1
c~-1
which is the simplest exampleof what is called an infinite or a Toeplitz operator on 12.
Toeplitz matrix ¯
Further realizations of the finite section methodare the sequences (L,~AL,~) where L~ is as in Examples 1.8 and 1.9. If A is an invertible operator of the form aI ÷ K with K compact, then these finite section methods are even applicable. The applicability of the ’natural’ finite section methodto Toeplitz operators will be studied in the forthcoming Section 1.3. Let us consider one more example. Example 1.12 The operator
Ao
~
[0 1 0 0 0 0
1 0 0 0 0 0
0 0 0 1 0 0 :
0 0 1 0 0 0
0 0 0 0 0 1
0 .. 0 0 0 1 0
.. .. .. .. --
2) C L(l
34
CHAPTER 1.
THE LANGUAGE OF NUMERICAL ANALYSIS
is selfadjoint and unitary, but its finite sections PoAoPo,P2AoP2,P4AoP,~, ... (with respect to the standard basis (e~)~=o of/2) all contain a zero row and are therefore not invertible. Nowmodify Ao as follows: let ~ = (Ca), be a sequence of positive numbers with limit 0, and set ~o 1 0 0 A~ = 0 0 :
1 0 So 0 0 ~1 1 0 0 0 0 0 :
:
0 0 0 0 0 0 0 0 ¢2 1 1 ~2 :
:
¯ L(/2).
..
If we further assume that en _< 1/2 then, since 1 en
=
1 _ 1 < 2, max(1 + ~n, 1 - ca) 1 - cn -
the operator A~is invertible, and I]A[1 I] -< 2. Moreover,’all finite sections PnA~Pnof A~ are invertible, and ]](P,~A¢P~)-IPnl] _< 2 whenevern is odd, but the normsI](P,~A.~P,~)-IP,~I] are are at least 1/¢,~ in case n is even. Thus, although both A~ and all its finite sections are invertible, one has supn ]](P,~A~P,~)-~P,~I] = c~, and the finite section methodwith respect to (en) is not applicable to A~. Observe on the other hand, that the finite section method for Ao with respect to the basis (f~)~=0 with f2n = (e2n + e2,~+~)/x/~, f2n+~ = (e2,~ e2,~+~)/x/~ applies; the matrix representation of Ao with respect to this basis is the diagonal matrix diag (1, -1, 1, -1,...). It is its simplicity and universality which, distinguishes the f’mite section method from other approximation methods. Mainly for this reason, the investigation of the finite section methodfor several classes of operators has ever played a pioneering role in the development of (algebraic methods in) numerical analysis. Also in the present book, the finite section methodwill serve both as a source of inspiration and as a basic examplefor illustrating the abstract theory.
1.2 Banach algebras
and stability
As indicated in the introduction, we will study stability problems by translating them into invertibility problems in Banach algebras. The present section provides the fundamentals for this.
1.2. BANACH ALGEBRAS AND STABILITY 1.2.1
Algebras,
ideals
and
35
homomorphisms
An algebra is a complexlinear space B with an additional operation B x B -~ B, (a, b) ~ ab, called multiplication, which satisfies the following axioms for all a, b, c E B and A E C: (ab)c = a(bc) (associativity), (a+b) c = ac+bc, a(b+c) = ab+ac (distributivity), (Aa) b = a ()~b) = )~ An element e ~ B is called unit element or identity if ae = ea = a for all a ~ B. The unit element is unique if it exists. Algebras which possess a unit element are called unital. If B is a unital algebra with unit element e, then an element a ~ B is invertible if there is an element b E B such that ab = ba = e. -1. The element b is unique and will be denoted by a Obviously, (a-l) -I -- a, (ab) -1 -~. = b-la The set of all A ~ C such that e - Aa is not invertible is called the spectrum of a in B and is denoted by a~(a) or, simply, by a(a). A subset of an algebra whichis (with respect to the inherited operations) an algebra again, is called a subalgebra of the given algebra. A subalgebra ~ of an algebra B is an ideal if bk ~/~ and kb ~/~ for all b ~ B and k ~ ]C. The algebra B and the set (0} are the trivial ideals of B. Given an algebra B and an ideal ]C of B, one can form the quotient B//~, which is the set of all cosets of elements of/~ modulo~. There is a natural linear structure on B/~ which makes this quotient to a linear space. Moreover, provided with the multiplication (a + ]~)(b ÷/~) := ÷ ~, thi s lin ear spa ce becomes an algebra again, and e ÷/(: is the identity element of B/tg if e is the identity of B. Let B~, B2 be algebras. A homomorphism W from B1 into B2 is a mappingwhich reflects the algebraic structure, i.e. Wis a linear mapping, and W(ab) = W(a)W(b)for all a, b e B~. In case B1 and B~. are unital with unit elements el and e2, respectively, the homomorphismW: B1 -~ B2 is said to be unital if Wel = e2. A homomorphismwhich is invertible is an isomorphism, and algebras with an isomorphism between them are called isomorphic. Note that there is a close connection between ideals and homomorphisms: The kernel Ker W= {a E B~ : Wa = 0} of every homomorphism is an ideal of B] and, conversely, given an ideal ]C of an algebra ~, there is an algebra B2 as well as a homomorphismWfrom/~1 onto B~ such that E is the kernel of W. For, choose/~2 := B]//~, and let Wbe the canonical homomorphism W : B~ --~ B]//C, a ~-~ a + tC.
CHAPTER 1.
36
THE LANGUAGE OF NUMERICAL ANALYSIS
An algebra B is normedif it is a normed linear space, and if
Ilabll ~ Ilallllbll forall a,
beI~
(continuity of the multiplication). If B is unital with unit element e then we require in addition that lie[] = 1. If the underlying linear space of a normed algebra is a Banach space, then the algebra is called a Banach algebra. The perhaps most important property of a unital Banach algebra is that the spectrum of an element is a compact and non-empty subset of the complex plane. The natural substructures of Banach algebras are its closed subalgebras and closed ideals, and its natural morphisms are the continuous homomorphisms. Observe that the duality between ideals and homomorphisms discussed above involves a duality between closed ideals and continuous homomorphisms.Further, if B is a Banach algebra and K: is a closed ideal of B, then the quotient algebra becomes a Banach algebra on defining a normvia []a + K:]] := infke~: []a + k[]. Clearly, the canonical homomorphism from B onto B/]~ is continuous and has norm 1 in this case. A prominent example of a Banach algebra is the algebra L(X) of all bounded linear operators on some Banach space X. This algebra is unital, and its unit element is the identity operator I. If X is infinite-dimensional, then the set K(X) of the compact operators on X is an example of a nontrivial closed ideal of L(X), whereas the operators of finite rank form a non-trivial non-closed ideal. The quotient algebra L(X)/K(X) is called the Calkin algebra of X. An operator A e L(X) is invertible in L(X) if and only if Im A = X and Ker A = {0) (Banach’s theorem), whereas the coset A + K(X) is invertible in the Calkin algebra L(X)/K(X) if and only if A is a Fredholm operator, i.e. if the range Im A of A is closed, and if dim Ker A < oo and dim (X/Im A) < o~ (Calkin’s theorem). 1.2.2
Algebraization
of stability
Let X be a Banach space and (Ln) be a sequence of projections on X with strong limit I as n -~ c~. Let ~- refer to the set of all sequences A( ~o n)n=o of operators An E L(ImL,~) which are uniformly bounded: supn>0 [[AnLn[I < oo. The natural operations (An)+ (B~) := (A,~ + B,~),
(An)(Bn) := (AnB,~), A(An) := (AAn)
make 5r to an algebra. This algebra is unital with identity (I[ImL,), and can be normedon defining II(A~)ll:-- supn_>0IIA~L~ll.
37
1.2. BANACH ALGEBRAS AND STABILITY Proposition 1.13 9~ is a Banach algebra: Proof. Let ((A~m))n>0),~>o be a Cauchy sequence in ~-, that is, e > 0 there is an N such that supllA(~m)Ln-A(~k)L~ll~_¢
forall
k,m>_N.
(1.8)
n
In particular, (A(~m)),~_>0is a Cauchysequence for every fixed n and, thus, convergent. Set An~ := limm-~oA(~m). Letting m go to infinity in (1.8) one obtains sup~ IIA~Ln - A~)Lnll <_ ~ for all k _> N. This shows that the sequence (A~) belongs to ~ and t hat t he s equences (A(nk))n>o converge t o (A~) as k -~ ~o in the norm of v. ¯ Since every approximation method is a bounded sequence by the BanachSteinhaus theorem, we can regard every approximation method as an element of the algebra ~’. As indicated in the introduction, there are good reasons for identifying sequences in Y which differ in a finite numberof entries only. For this goal we introduce the set 6 of all sequences(Gn)in ~" with limn-~ooI IGn Ln I I = Clearly, 6 is an ideal in ~-. Proposition 1.14 6 is a closed ideal in ~. Proof. Let ((G(nm))n>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~
I1(0,... 0, Ano,Ano+a,...)ll sup IIAnLnll < limsupllAnLnll
Letting e go to zero yields the desired result.
¯
1.2. BANACH ALGEBRAS AND STABILITY 1.2.3
Small
39
perturbations
As a rule, the approximate operators An of a given operator A are affected with small errors. For example, the application of the finite section method to an operator A requires the computation of the scalar products (Aei, which in most cases can be done only approximately (by a quadrature rule, say). So, as a first application of Kozak’s theorem, we consider small perturbations of stable sequences. Notice that, by Proposition 1.16, a sequence (S~) e ~ is small if lim sup [[SnLn[[ is small. Theorem 1.17 Let (A,~) E ~ be a stable sequence and let (Sn) be a quence such that limsup IISninll < liminf IIA~linl1-1.
(1.9)
Then the sequence (An + Sn) is stable. Proof. If B is an arbitrary unital Banachalgebra and a E B is invertible, -1 then the element a + s is invertible for every s E B with [[s[[ < [[a-l[[ (Neumannseries). Let now (An) ~ bea s ta Sle seq uence. The n the coset (An) + 6 is invertible in ~’/~. The above remark, applied to the Banach algebra 9r/6, yields the invertibility of every coset (An) + (Sn) (a nd thus, the stability of every sequence (An + Sn)) if only II(Sn) +61l < 1/[I((An)
~- ~)--lll"
The inequalities (1.9) and (1.10) coincide due to Proposition 1.16.
(1.10) ¯
Let (Ln) and (Rn) be sequences of projections on a Banach space X as in Section 1.1.2. As a corollary to Theorem1.17 we obtain that the set of all operators A ~ L(X) for which the projection method (RnALn) applies is open in L(X). 1.2.4
Compact
perturbations
The Fredholm integral operators of second kind, A = I + K, introduced in Section 1.1.2, are the sum of a ’good’ operator (in this case, the identity) and a compact perturbation, K. Thus, the projection method (RnALn) for A is also the sum of a ’good’ sequence (in this case, the approximation method (RnILn) = (Ln) for the identity operator, which is obviously stable), and a ’compact’ perturbation, (RnKL,~). The goal of the present section is a perturbation theorem which describes the behavior of sequences under compact perturbations (Theorem 1.20 and its corollary below). The desired perturbation theorem does not hold for arbitrary approximation methods (see Theorem5.50 in Section 5.3.4 for one reason showing
40
CHAPTER 1.
THE LANGUAGE OF NUMERICAL ANALYSIS
why). So we have to start with introducing an appropriate subclass of approximation sequences. For, let again (Ln) be a sequence of projections on a Banach space X with strong limit I, but suppose moreover that the adjoint projections L~ also converge strongly to the identity operator I* on the dual X* of X. Clearly, this involves no further restriction if the Ln are orthogonal projections on a Hilbert space. In what follows, we will only consider sequences (An) of operators An E L(Im Ln) for which both sequences (AnLn) and (A~L~) are strongly convergent in X and X*, respectively. The set of all of these sequences will be denoted by .T C, and the strong limit of a sequence (An) ~ jrc by W(An). (We prefer this short notation to the more correct but cumbersomenotation W((An)n>_o).) Evidently, W(A~) = (W(An))* for all (A,~) ~ jrc. Theorem1.18 (a) The set jre is a closed subalgebra o] jr which contains the unit element of (b) The mapping W re -~L(X), (A, ~) ~-~ W(An) is a conti nuous unita homomorphism with norm 1. Proof. Strongly convergent sequences are uniformly bounded due to the Banach-Steinhaus theorem, thus, ~c is a subset of ~-. Recalling assertion (a) of Lemma 1.5 in Section 1.1.2 one easily checks that ~-c is a subalgebra of jr and that Wis a homomorphism. Further we conclude from Ln -~ I and L~ -~ I* that (IlimL~) ~ jrC and W(IlImLn) = and, co nsequently, jrc and Ware unital. Invoking the Banach-Steinhaus theorem once more, we get IIW(An)II <_liminf llAnLnll <_sup llAnLnll = II(An)ll,
(1.11)
which shows that W is continuous and has norm 1. It remains to verify the closedness of ~c in .T. Let ((A(~m))~>_o)m>o be a sequence of elements of ~-c which converges in the norm of jr to the sequence (A~) ~ ~-. Then ((An(m) )n>O)m>Ois a Cauchy sequence jr which, via (1.11), implies that (W(A(~m)))m>O is a Cauchy sequence in the Banach space L(X). Thus, the latter sequence converges to a certain operator A ~ L(X). Weclaim that A is just the strong limit of the sequence (A~Ln). Indeed, for all x E X and m, n _> 0, IIAx - A~ Lnxll <_IIAx - W(A(~m))xll + <_IIA - W(A(~’~))]I +I[m(~’~)n,~
1.2. BANACH ALGEBRAS AND STABILITY
41
Given ¢ > 0, choose ml and m2 such that IIA - W(A(~’~))I I < ¢ for all m >_ ml and II(A(, "0) - (A~)II~: = supllA~m)nn - A~L,~II < ¢ for all ra _> 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 <_ C IIAnLaxlI. Letting n go to infinity we obtain Ilxll < C I]W(An)xll for all x ~ X, i.e. the operator W(A,~) has a closed range and a trivial kernel by Lemma1.21. Repeating these arguments with the adjoint sequence (A~) in place of (An), we get that the kernel of W(A~) = W(An)* is trivial, too. This implies that the range of W(An) is all of X. Indeed, otherwise there would exist an x0 ~ X as well as a linear bounded functional .f ~ X* such that ](x0) = 1 and f = 0 Im W(An) (Hahn-Banach theorem). Clearly, ] belongs to the kernel W(An)*, which is a contradiction. Thus, Im W(An) = X, and W(An) is invertible. Our next goal is the invertibility of the coset (An) + ~c. As in the proof of Kozak’s theorem, let (Bn) ~ ~" be a sequence with Bn = ~ for al l sufficiently large n. Since we already knowthat (Bn) + G is the inverse the coset (An) + G, and since G _C Gc, it remains to check whether (Bn) belongs to Given y E X set x = W(A,~)-~y. Then, for all sufficiently large n, I[BnLny - W(An)-lyll <_ IIA’(~iLny - LnW(An)-lyll + I1(I - Ln)W(An)-~y[I <- I]A’~Lnll IlY - AnL,~W(An)-IYl] + II( I - Ln)W(A,~)-IYl] IIW(An)x - Aninxll I1(I -
=.]lA’~iinl[
which shows the strong convergence of (Bn) to W(A,~) -1. The strong convergence of (Bn)* can be seen analogously. Thus, indeed, (Ba) and -1. W(Bn) = W(An) Let now, conversely, the operator W(An)as well as the coset be invertible. Then there are sequences (Cn) ~ ~-c and (Gn) ~ ~ and compact operator K such that AnCn = ~]ImL~ -b LnKLn + Gn. Set Dn := Cn - LnW(An)-IKLn. The operator W(An)-~K is compact, hence (LnW(A,~)-~KLn) belongs to GC, and (Dn) lies in yc again. Further one has AnD,~
’b LnKLn + G,~ - AnLnW(An)-~ KLn = II~mL ~ + (L,~ - AnLnW(An)-~) KLn + The sequence (Ln - AnLnW(An)-~) converges strongly to 0 as n --+ and Lemma1.5 implies that even = IIlmL,,
II(gn-AnLnW(An)
-I)KII-~O
as
n~
44
CHAPTER 1.
THE LANGUAGE, OF NUMERICAL ANALYSIS
Consequently, AnDn= IIX~.L. + G’~with (G~) E ~, i.e. the coset (A,~) is invertible in ~’/~ from the right hand side. Its invertibility from the left side follows analogously. Thus, (A,~) is stable by Kozak’s theorem. Corollary 1.22 Let (An) ~ C bea s ta ble seq uence and K bea compact operator on X. Then the sequence (A,~ + L,~KL,~) is stable if and only the operator W(An) + K is invertible. For a proof observe that the cosets (An) + c and ( An +LnKL~) + C coincide, and apply Theorem 1.20. . Weare going to illustrate
the perturbation theorem by a few examples.
Example 1.23 Let X be a Banach space and let (L,~) and (Rn) be sequences of projections on X which are as in Section 1.1.2. The projection method (RnILn) = (Ln) for the identity operator is clearly stable. Consider the projection method (R~ALn) for the operator A = I + K with K compact. Since RnALn = Ln + RnKLn = Ln
+ LngLn-b
Cn
with IICnL,~II --+ 0 due to Lemma1.5, we can think of (R~AL,~) as compact perturbation of the stable sequence (Ln). Applying Theorem1.20 and its corollary to this special situation, we arrive at Theorem1.6 again. Example 1.24 Let H be a separable infinite-dimensional Hilbert space with orthonormal basis (el), let P,~ refer to the orthogonal projection from H onto the span of {e0,..., e,~}, and let A be the operator a(B + iS) + where B is positive definite, S is self-adjoint, K is compact, and a ~ ~2. Specification of Corollary 1.22 to this context yields: Corollary 1.25 The finite section method (PnAPn) applies to the operator A = a(B + iS) + K if and only if A is invertible. Proof. If A is invertible, then necessarily a ~ 0. From Theorem 1.10(b) we infer that the finite section method(P~(a(B + iS))Pn) for the operator a(B + iS) is stable, thus, (PnAPn) is a compact perturbation of a stable sequence. . It is surprising that also a certain converse of Corollary 1.25 is valid: Theorem 1.26 (Markus/Vainikko) Let A ~ L(H) be an invertible operator on a Hilbert space H. If the finite section method with respect to an arbitrary orthonormal basis of H applies to A, then A is necessarily of the form a(B + iS) + K with a, B, S, K as above.
1.3.
FINITE
SECTIONS OF TOEPLITZ OPERATORS
45 ¯
For a proof, see [64], Chapter II, Theorem5.1.
Further applications of the perturbation theoremare given in Sections 1.3.3 and 4.3.2 below.
1.3
Finite sections of Toeplitz operators with continuous generating function
Wehave already met Toeplitz operators as representations of polynomials of the shift operator with respect to the ’natural’ basis in Example1.11. Here we will consider Toeplitz operators which are generated by continuous functions. In contrast to the Toeplitz operators generated by polynomials, these operators are no longer band matrices. After recalling some basic facts on Toeplitz operators, we will apply the perturbation theorem to study the finite section method for Toeplitz operators, again with respect to the ’natural’ basis. One reason for the permanent interest in Toeplitz operators is that these operators constitute one of the best-studied and simplest classes of non-normal operators (and are thus beyond the classes of operators usually considered in courses on functional analysis). 1.3.1
Laurent,
Toeplitz
and Hankel
operators
Westart with the definition of Toeplitz operators with arbitrary bounded generating function. Let a E L~(~’), and denote by ak the k th Fourier coefficient of a: 1 ~02~a(ei~)e_ik Od~, k ~ Z. Then the Laurent operator L(a) on /2(Z), the Toeplitz operator T(a) on /2(Z+), and the Hankel operator H(a) on/2(Z+) are given via their matrix representations with respect to the standard bases of/2(Z) and/2(Z+)
L(a)
¯ ¯ ao
a-1
a-2
a-3
"*
¯ ¯ al
ao
a-1
a-2
"¯
¯ - a2
al
ao
a-1
a3
a2
al
ao
46
CHAPTER 1.
THE LANGUAGE OF NUMERICAL ANALYSIS
and
T(a)=
al a2 3
ao al a2
a-1 a0 al
a-2
...
a-1
’-"
ao
..-
a2 a3 U(a)=
a3
a4
4 a5
a4 a5 a6
a5 a6 a7
k respectively. Observe that in case a is the polynomial t ~ ~-~j=-k cjt j, the Toeplitz operator T(a) coincides with the operator considered in Example 1.11. Theorem 1.27 I] a E L~(~), then the Laurent operator L(a) is bounded on/2(Z), and I]L(a)ll- I]allo~. Proof. Let M(a) denote the operator of multiplication by a. Thought of as acting on the Lebesgue space L2(’I~), the operator M(a) is bounded, and IIM(a)ll = Ilall~. Further, the functions ek(eit) = (2~r)-l/2eikt, k e form an orthogonal basis of L2(31"). It is not hard to see that, actually, L(a) is the matrix representation of M(a) with respect to the basis (ek)aez from which the assertions immediately follow. ¯ For the boundedness of Toeplitz and Hankel operators one has the following results: Theorem 1.28 (a) (Brown/Halmos) I] a e L°°(~), then the Toeplitz operator T(a) is boundedon/2(~,+), IlT(a)lI -- I lal lo~(b) (Nehari) I1 a ~ L°¢(~), then the Hankel operator H(a) is bounded /2(Z+), and IIH(a)ll = dlStL~(T)(a,H Here, ~ stands for the algebra of all functions in L~(~") which possess analytic extensions into the exterior of ~. For a proof of Theorem1.28 see, for example, [26], Theorems 2.7 and 2.11. ¯ Toeplitz and Hankel operators can be viewed as the building stones of Laurent operators. For, we identify the Hilbert space/2(Z) with the direct sum/2(Z+) (9/2(Z+) via the mapping
Under this identification, ’block operator’
the Laurent operator L(a) corresponds to the
L(a) ~ ( H(a)T(5) H(5))T(a) ’
(1.12)
1.3. FINITE
SECTIONS
OF TOEPLITZ
47
OPERATORS
where h refers to the function 5(t) := a(1/t). ") (Observe that 5 is in L~(~ again, and that 5k ---- a-k for k E Z.) In the sequel we will need somealgebraic properties of Laurent, Toeplitz and Hankel operators. If a, b E L~(~’), then a + b and ab ") are in L°°(~l again, and one obviously has L(a) -t- L(b) = L(a A- b), T(a) ÷ T(b) = T(a ÷ b), U(a) ÷ U(b) Further, taking into account that L(a) is the matrix representation of the operator of multiplication by a, one gets L(a) L(b) = L(ab).
(1.13)
The situation is a little bit more involved for products of Toeplitz and Hankel operators. From (1.12), we obtain that L(a)L(b)
/’T(a) H(5)~ (T(~) ~ ik H(a) T(a) ]\ H(b)
{’T(5)T(~_) +H(5)H(b) T(h)H(~.) (1.14) = ~.H(a)T(b) + T(a)H(b) g(a)g(b) + as well as L(abl e--~ ~.H(ab) T(ab)
(1.15)
Due to (1.13), the matrices in (1.14) and (1.15) coincide, and comparison of their south-east resp. south-west corners yields T(ab) = T(a)T(b) + H(a)U(~), H(ab) = T(a)U(b) (1 .16) Similarly, if a ~ Lc°(~’), then the function ~(t) a(t ) is in L°°(~) again, and ~k = a_k for every k. Hence, L(a)* L(~), wh ence, by (1. 12), L(a)*
[ T(5) g(5)~* ~ \ H(a) T(a)
= \U(a)*
T(a)* = ~,U(~) T(~)
and, thus, T(a)*
T( ~),
U(a)* =
Westill mention a result which facilitates essentially.
H(~).
(1.17)
the work with Toeplitz operators
Theorem 1.29 (Coburn) Let a ~ L°°(T) (0 }. Then at lea st one o] t he spaces KerT(a) and/2/clos (ImT(a)) consists o] the zero element only. For a proof see [26], Theorem2.3s, or [27], Theorem1.10.
48 1.3.2
CHAPTER 1.
THE LANGUAGE OF NUMERICAL ANALYSIS
Invertibility ators
and Fredholmness
of Toeplitz
oper-
A linear and bounded operator A on a Banach space X is invertible in L(X) if (and only if) its kernel is {0} and its range is all of X (Banach’s theorem). Thus, an operator A is invertible if (and only if) both its kernel Ker A and its cokernel CokerA := X/clos(ImA) are linear spaces with dimension zero. Coburn’s theorem simplifies this condition essentially in case T(a) is a non-zero Toeplitz operator. Indeed, in this case it is sufficient to know that the difference indT(a) := dimKerT(a) di mCokerT(a) is zer o in order to guarantee the invertibility of T(a). In particular, a Fredholm Toeplitz operator is invertible wheneverits index vanishes. The following result describes the Fredholm properties of Toeplitz operators with continuous generating function (compare Figure 1.1). Theorem 1.30 Let a E C(’~). The Toeplitz operator T(a) is Fredholm on 12 if and only if O ~ a(~). In this case, indT(a) = -winda(~) where wind a(’l~) refers to the winding number of the curve a(~), provided with the orientation inherited by the usual counter-clockwise orientation of the unit circle, aroundthe origin. For a proof see [26], Theorem2.42, and compare also Section 4.1.3 where we will derive a more general result holding for Toeplitz operators with piecewise continuous generating function. Combining Theorem 1.30 with
a
ind T(a)
-= 0 ind T(a)
=
Figure 1.1: Fredholmness of Toeplitz operators Coburn’s theorem we get Theorem 1.31 Let a ~ C(~). The Toeplitz operator T(a) is invertible 12 if and only if 0 ¢ a(V) and if wind a(~’) = Figure 1.2 illustrates
this theorem.
1.3.
FINITE SECTIONS OF TOEPLITZ OPERATORS
T(a) not invertible
T(a) invertible
Figure 1.2: Invertibility 1.3.3
The finite
section
49
T(a) invertible
of Toeplitz operators
method
Weconsider the applicability of the finite section methodwith respect to the standard basis of 12, i.e. we let OO ~ Pn : ( Xk)k=O
(Xo,Xl,"
¯ . ,Xn-l,0,0,’"
")
Theorem 1.32 Let a E C(~). The finite section method (P,~T(a)P,~) plies to the Toeplitz operator T(a) if and only if T(a) is invertible. From Polski’s theorem we know that invertibility of T(a) is a necessary condition for applicability of the finite section method. In the case at hand this condition is also sufficient, whichis clearly the best-possible result. The proof of Theorem1.32 relies on the perturbation theorem from Section 1.2.4 and on the following elementary lemmata. Lemma1.33 If a ~ C(~), then the Hankel operator H(a) is compact. Proof. Approximate the continuous function a by a sequence (p,~) trigonometric polynomials in the supremumnorm. Then, by Theorem 1.27 and (1.12), [[H(a)- H(pn)[[ = [[H(a- Pn)[[ _~ [[L(a- Pn)[[ [[a-pn[[oo -+O, and the assertion follows from the (obvious) compactness of the Hankel operators H(pn) and from the closedness of the ideal of compact operators with respect to norm convergence. ¯ Lemma1.34 (Kozak) Let X be a linear space, P a projection on X, Q the complementary projection I - P, and let A be an invertible operator on X. Then the compression PAPIImpof A onto Im P is invertible if and only if the compression QA-1Q[ImQof A-1 onto ImQ is invertible, and one has (pAp)-IP
= pA-1p
_ pA-1Q(QA-IQ)-~QA-~p.
50
CHAPTER 1.
THE LANGUAGE OF NUMERICAL ANALYSIS
The proof is straightforward. Lemma1.35 Let a E L°°(~) and suppose T(a) is invertible. Then -1 finite . section method applies to the inverse Toeplitz operator T(a) Proof. Set Qn = I - Pn. Kozak’s formula implies that the sequence (PnT(a)-IP~) of operators on Im Pn is stable if and only if the sequence (Q~T(a)Q~) of operators acting on Im Qn is stable. Since the operators T(a) and QnT(a)Q~IImQnhave the same matrix representation with respect to the standard basis of 12, the stability of the latter sequence is obvious. ¯ Proof of Theorem 1.32. If the finite section method applies to T(a) then this operator is invertible (Polski’s theorem). Let, conversely, T(a) be an invertible operator. Then, by Theorem1.31, 0 ~ a(~’) and, hence, -1 exists and belongs to C(’F) again. From (1.16) we know I = T(aa -1) = T(a)T(a -1) + H(a)H(a-~). Invoking Theorem 1.31 once more we see that T(a-~) is invertible hence, T(a) = T(a-1) -1 - -~ H(a)H(a-I)T(a-~)
and,
whence PnT(a)Pn = PnT(a-i)-lPn
- PnH(a)H(a-i)T(a-1)-iPn.
The operator H(a)H(a-1)T(a-1) -1 is compact due to Lemma1.33, and the sequence (P~T(a-~)-~Pn) is stable by Lemma1.35, thus, the perturbation theorem yields the assertion. ¯ Nowconsider the finite section method for compactly perturbed Toeplitz operators A = T(a) + with a E C(~). If A i s invertible the n, cle arly, ind A = 0, and the invariance of the index under compact perturbations implies that ind T(a) = O. Then, by Coburn’s theorem, T(a) is invertible, and Theorem1.32 entails the stability of the sequence (P,~T(a)P,~). So, (P,~AP~) is actually a compact perturbation of a stable sequence, which yields, via the perturbation theorem, the following criterion. Corollary 1.36 Let a ~ C(~2) and K ~ L(l 2) compact. Then the finite section methodapplies to T(a) + K i] and only i] this operator is invertible. Wewill come back to this result from another point of view in the next section. Here we still mention two related applications.
1.3. FINITE
SECTIONS
OF TOEPLITZ
51
OPERATORS
Example 1.37 The finite section method for Laurent operators. For the approximation of Laurent operators we need finite sections of twosided infinite matrices. The appropriate orthogonal projections Rn on/2(Z) are Rn : (Xk)k6Z
(. n_2,xn-l,O ..,0,0, X,x
n,X-n-bl,-.,O,...).
Evidently, Rn -~ I strongly as n -~ oo and, thus, the finite section method (RnARn) is an approximation method for every operator A 6 L(/2(Z)). Corollary 1.38 If a 6 C(T), then the finite section method (RnL(a)R,~) applies to the Laurent operator L(a) if and only i] the Toeplitz operator T(a) is invertible. Proof. The matrices in the sequence (R,~L(a)Rn) coincide with those in the sequence (P2nT(a)P2n). Hence, both sequences are simultaneously stable or not. But stability of (P2nT(a)P2n)involves invertibility of T(a) (Polski’s theorem). Conversely, if the operator T(a) is invertible, then the sequence (P,~T(a)Pn) is stable by Theorem1.32 and, thus, its subsequence (P2,~T(a)P2,~) is stable. . Since Laurent operators are essentially nothing else but multiplication operators, the Laurent operator L(a) with a E C(T) is invertible if and only if 0 ¢ a(T) whereas, by Theorem1.31, the invertibility of the Toeplitz operator T(a) in addition requires that wind a = 0. Thus, one consequence of Corollary 1.38 is that the invertibility of the Laurent operator L(a) is not sufficient for the stability of the finite section method(RnL(a)Rn). Let us emphasize still another remarkable aspect: Whereis the ’center’ of the (doubly infinite) Laurent matrix
’.
0
1
4
2
0
0
0
0
0 0
1
4
0
1
2 4
0 2
0 ". 0 .
0
,
and which of its diagonals is the ’main diagonal’ ? Israel Gohberg’s answer is that ’it is every diagonal’s right to claim to be the main one’. So one might consider each of
A~’--")----
0 1 4
--o)
0
1
4
2
52
CHAPTI~R 1.
=
THE LANGUAGE OF NUMERICAL ANALYSIS
42000
4200
as reasonable approximation matrices, and the generating function of the associated Laurent operator is invertible in any case. But the four sequences (A(n/))n>o correspond to the finite section method for Toeplitz operators T(a(~)-) with generating functions a(~)(t) = t ~-3(t ÷ 4 + 2t-1). Since only one of these operators, namely T(a(3)), is invertible, we see (A(n3)) is the only stable approximation methodamongthe sequences (A(n/)). Consequently, there is a good reason to consider ..., 4, 4, ... as the main diagonal. Example 1.39 The finite section method for Toeplitz operators on the Bergman space. Let the notations be as in Example 1.9. The orthogonal projection P from L2(I~) onto its subspace A2(D) is given (Pf)(z)
f~
(1
.f(w)
dA(w)
Given a function a E L°~(I])), the Bergmanspace Toeplitz operator TS(a) is defined as T B(a) : Ae(~)-~A~(D), f~Paf. 2 This operator is bounded on A (~)). Let now a E L~ (It)) be a function which is continuous on D and possesses a continuous extension onto the closure clos I~ = 1~ t_J q~ of I~. We denote this extension by a again and write b for the restriction of a onto q~. One can prove that the matrix representation of TB(a) with respect to the basis {ek}k°°=o, ek(Z) : kV/~--~zk, is CTB(a)C -~ = T(b)
+
where T(b) is the usual (l 2-) Toeplitz operator and K is compact on l 2. So Corollary 1.36 and property (1.6) immediately involve Corollary 1.40 Let a ~ C(closl~). Then the invertibility of TS(a) necessary and sufficient both for the applicability of the finite section method (L,~TB(a)L,~) and of the collocation method (RnTS(a)nn).
1.4 C*-algebras To study stability study invertibility
of approximation
sequences
problems of approximation sequences is the same as to problems in the Banach algebra ~’/G of all bounded
1.4. C*-ALGEBRAS
OF APPROXIMATION
SEQUENCES
53
sequences modulo the zero sequences. In many concrete situations, one might desire to consider this invertibility problem in a suitable subalgebra of ~’/G rather than in ~-/6 itself. (Say, because one has a precise knowledge about this subalgebra from former investigations, or because one wants to employ something like Gelfand’s theory to study the invertibility problem, which only works in commutative algebras.) This simple idea involves someserious technical difficulties since, if is a closed subalgebra of a Banach algebra A, and b belongs to/3, then the spectrum of b in/3 can be strictly larger than the spectrum of b in In particular, the stability of a boundedsequence (An) is, in general, not equivalent to the invertibility of the coset (An) ÷ G in a certain subalgebra of ~-/G. In this section, we are going to introduce a special class of Banach algebras, one advantage of which is that it doesn’t matter whether invertibility problemsare studied in the algebra itself or in one of its subalgebras: viz. the C*-algebras. It would be, however, more than wrong to reduce the role of C*-algebras to that one of a useful tool for solving invertibility problems. It has rather become more and more clear in the last decades that C*-algebras provide a natural language for many fields in ’non-commutative’ mathematics and physics (which is comparable with the role of complex numbers in ’commutative’ mathematics). For getting an impression on these developments, have a look at Connes’ book [38]. Although we will apply C*-algebras on a quite elementary level only, we will nevertheless see that they can really do a lot of work for us.
1.4.1 C*-algebras, their ideals and homomorphisms Let /3 be a Banach algebra. A mapping a ~-~ a* of/3 into itself involution if, for all a, b E/3 and all (a*)*
=a,
(ha+#b)*
is an
=Xa*+~b* (ab) * =b’a
A Banach algebra/3 with an involution * is called a C*-algebra if Ila*all = ][al] 2 for all a e/3.
(1.18)
In C*-algebras, the involution is an isometry: Example 1.41 Let X be a compact subset of IRn (or, more general, a compact Hausdorff space) and let C(X) be the set of all continuous complexvalued functions on X. Provided with pointwise operations, the maximum norm, and the involution f*(t) = f(t), C(X) becomes a commutative and unital C*-algebra.
54
CHAPTER 1.
THE LANGUAGE OF NUMERICAL ANALYSIS
Example 1.42 If H is a Hilbert space, then the Banach algebra L(H) of the bounded linear operators on H is a C*-algebra with A ~4 A* (= the Hilbert space adjoint of A) as involution. Let us check axiom (1.18): A ¯ L(H),
IIAll2 --
sup((Ax, Ax):x ¯ H, Ilxll = 1} sup((A*gx, x) :x ¯ H, Ilxll 1) <_ sup(lld*dxll:x ¯ H, Ilxl] = 1} = IIA*AII, =
e. and conversely,IlA*dll <_ HA*IIlldll -- IIAII Example 1.43 Let H be a Hilbert space, (P,~) be a sequence of orthogonal projections on H which converge strongly to the identity, and let ~ be the associated Banach algebra of bounded approximation sequences. Then (A,~)* := (A~) defines an involution v which makes ~" t o aC*-al g,ebra. Since (Gn)* belongs to the ideal 6 of ~" again whenever (Gn) is in 6, see that ~ is also a C*-algebra. An ideal ]C of an involutive Banach algebra ‘4 is called symmetric or a *-ideal if k ¯ ]C implies k* ¯ ]C. If ]C is a *-ideal of .4 then
(a + ~:)*:= a*+~: defines an involution on the quotient algebra A/1C Finally, a homomorphism W: A1 --> .42 between involutive Banach algebras .41 and .42 is symmetric or a *-homomorphism, if W(a*) = W(a)* for every a ¯ .41. The following results summarize some basic facts about C*-algebras and show in particular, that working in C*-algebras is much more convenient than in general Banach algebras. All proofs can be found in the standard textbooks about C*-algebras. Theorem 1.44 Let .4 be a C*-algebra. (a) Every closed ideal of A is symmetric. (b) If ]C is a closed ideal of A, then the Banachalgebra A/]C is C* again. (c) If W is a *-homomorphism from A into a C*-algebra B, then W is continuous and ]IWII <_ 1. (d) If Wis moreover one-to-one, then it is an isometry. Parts of the following isomorphy theorems are well known from general ring theory; their important new aspect is that all occuring algebras are C*-algebras again, i.e. that Im Wis closed, ]C is an ideal of .4, and B + ]C is closed in Theorems1.45, 1.46 and 1.47, respectively.
1.4. C*-ALGE,
BRAS OF APPROXIMATION SEQUENCES
55
Theorem 1.45 (First isomorphy theorem) Let A and 13 be C*-algebras and W : A --~ 1~ a *-homomorphism. Then WA is a C*-subalgebra o] 13, and the C*-algebras A/Ker W and Im W = WA are *-isomorphic: A/Ker W--- Im W. Theorem 1.4{I (Second isomorphy theorem) Let A be a C*-algebra, ff a closed ideal of .4, and IC a closed ideal of ft. Then I~ is a closed ideal of A, and there is a natural *-isomorphy (A/IC)/(ff /E) ~ A/ft. Theorem 1.47 (Third isomorphy theorem) Let A be a C*-algebra, I3 a C*-subalgebraof A, and ]~ a closed ideal of A. Then B+I~ (= the algebraic sum) is a C*-subalgebra of A, and there is a natural *-isomorphy (B
Let us further recall two famous results due to Gelfand, Naimarkand Segal which state that the C*-algebras considered in Examples 1.41 and 1.42, together with their C*-subalgebr~s, do already exhaust the classes of all C*-algebras respective of all commutative C*-algebras. Theorem 1.48 (a) (General GNS-theorem) For every C*-algebra A, there exists a Hilbert space H such that .4 is *-isomorphic to a C*-subalgebraof L(H). (b) (Commutativecase) For every commutative C*-algebra A with identity, there exists a Hausdorff compact X such that A is *-isomorphic to C(X). For a more detailed explanation of assertion (b) we refer to Section 4.1. Finally, we mention two results concerning invertibility in C*-algebras. Theorem1.49 (Inverse closedness) Let A be a C*-algebra with identity e, and let 13 be a C*-subalgebra of .4 containing e. Then an element b E B is invertible in !~ if and only if it is invertible in ‘4. In other words, aA(b) al~(b) for every Thus, one can switch to appropriate C*-subalgebras of a given C*-algebra when dealing with invertibility problems. Theorem 1.50 (Semi-simplicity) Let ,4 be a unital C*-algebra and r ~ A be an element having the property that, whenever an element a ~ ‘4 is invertible, then a + r is also invertible. Thenr = O. Equivalently, the Jacobsonradical of a unital C*-algebra, i.e. the intersection of all maximalleft ideals of that algebra, consists of the zero element only.
56
1.4.2
CHAPTER 1.
THE LANGUAGE OF NUMERICAL ANALYSIS
The Toeplitz C*-algebra and the C*-algebra of the finite section method for Toeplitz operators
Weare going now to examine two concrete examples of C*-algebras. The Toeplitz algebra T(C) is the smallest closed subalgebra of L(/2) which contains all Toeplitz operators with continuous generating function. Since T(a)* T(a), th is al gebra is symmetric and henc e a C* -a lgebra. The formula (1.16) for the product of Toeplitz operators together with Lemma 1.33 indicate that T(C) does not only contain Toeplitz operators but also operators of the form T(a) ÷ with K compact. Wewil l sho w tha t the se operators already exhaust the algebra T(C). Theorem 1.51 T(C) {T(a) +
K : a E C(T), g
co mpact on
Proof. Let, for a moment, ‘4 abbreviate the set on the right hand side of the asserted equality. For the inclusion .4 C_ 7-(C) it is enoughto verify that every compact operator belongs to T(C). The reverse inclusion will follow once we have shown that A itself is an algebra (which is a consequence of formula (1.16) and Lemma1.33 again) and that A is closed. In the next step we show that the ideal K(/2) of the compact operators on 12 is contained in T(C). Let K ~ K(12). Since IIPrgPr - KII -+ 0 by Lemma1.5 and since 7"(C) is closed by definition, it remains to check whether PrKPr E T(C) for all r. Let V and V-1 denote the forward and backwardshift operators on 12, oo oo ~ (X~,X2,...), (0, the Xo,Xl,. .), representation V-1 : (X~)k=O (1.19) and Vlet: (Xk)k=O (kij)i,j=o ~ be matrix of P~KP~IImp ~ with respect r--1
to the standard basis of 12. A little
thought reveals that r--1
P~KPr = ~ kijViP~VJ_~ which gives the assertion because P1 = I - VV_I and V, V-1 are the Toeplitz operators T(a), T(a_~) with a(t) = t, a_~ (t) -~, res pectively. For the proof of the closedness of .4 we use the following formula (where Q, := I - P,), which holds for every A E L(/2): IIA + K(l~)ll
li m ]I QnAQnll
(1.20)
(its proof is left as an exercise). If K is compact, then IIQ~KQ~II-~ 0 due to Lemma1.5, and if A = T(a) is a Toeplitz operator then IIQnT(a)Qnll IIT(a)l I because T(A) and QnT(a)Q~I~n~Q.have the same matrix representation. Thus, I[T(a)[[ _< liT(a) + g[I (1.21)
1.4. C*-ALGEBRAS
OF APPROXIMATION SE~QUENCES
57
which even holds for arbitrary a ¯ LC°(~") and K ¯ K(12). From (1.21) we infer that, if (T(a (m)) + K(m))m>l is a norm convergent sequence operators in A, then the sequence (T(~(m))),~_>l is also convergent, and easily checks that its limit is a Toeplitz operator T(a) with a ¯ C(~’) again. This finally implies that (K(m))m>_lis a convergent sequence, and its limit is a compact operator K. Thus, limm_,oo(T(a(’~))÷ (’~)) =T(a)+ K ¯ ,4 ¯
Our next goal is a similar description of the smallest closed subalgebra of the algebra ~" which contains all sequences (PnT(a)Pn) with continuous functions a. Since (P,~T(a)P,~)* = (PnT(~)Pn), this algebra is symmetric and, hence, a C*-algebra. The desired description requires a formula for the product of two finite section sequences (PnT(a)Pn) and (PnT(b)Pn). To this end, introduce the reflection operators Rn:l 2-~1~, (xO, Xl,...)~(xn-l,x~-2,...,x~,xo, Lemma1.52 (Widom’s formula) PnT(ab)Pn = PnT(a)PnT(b)P,~
Let a,b ¯ L°°(~).
0,0,...). Then
q- PnH(a)H(~)Pn ÷ RnH(h)H(b)Rn.
Proof. From (1.16) we know that P~T(ab)P,~ = P,~T(a)T(b)P,~ + P,~H(a)H([~)P,~ = PnT(a)PnT(b)Pn + PnH(a)H(~)Pn + PnT(a)Q~T(b)Pn. It is elementary to check that RnT(a)V_~ P,~H(5), V~T(b)Rn = H(b)Pn, where V,~ := V~, V_,~ := (V_I) "~ and V, V-1 are the shift operators (1.19). Hence, PnT(a)Q,~T(b)P,~ -- R,~RnT(a)V-,~VuT(b)R,~R~ = RnH(5)H(b)Rn, which yields the assertion.
¯
If a and b are continuous functions, then H(a)H([~) and H(h)H(b) are compact (Lemma1.33). Thus, one might conjecture that S(C) consists all sequences (PnT(a)Pn + PnKPn+ RnLR,~) where a is continuous and K and L are compact. As the following theorem shows, this is almost correct. Theorem1.53 The algebra ~q(C) coincides with the set of all sequences (PnT(a)Pn
+ P~KPn + RnLR~ +
where a ¯ C(~), the operators K, L are compact on 12, and where (Gn)
58
CHAPTER 1.
THE LANGUAGE OF NUMERICAL ANALYSIS
Proof. Denote the set of all sequences of the specified form by 81 for a moment. Westart with verifying that the set $1 indeed forms an algebra. Let al, a2 be continuous functions and K1, K2, L1, L2 be compact operators, and set Q~ = I - Pn. A straightforward calculation employing Widom’s. formula as well as the identity RnT(a)R,~ = PnT(5)Pn with ~(t) := a(1/t) for every a e L°~(~") (1.22) yields (PnT(al)Pn + PnK~P~ + RnL1Rn) (PnT(a~)Pn + P~K~Pn + = P~T(ala2)P,~ + P~KP~ + R~LRn where K = K1K2 + K1T(a2) + T(al)K2 - H(al)H(~2), L = LIL2 + L1T(5~) + T(51)L2 H(5~)H(a2), Gn = P~KIR~L2R~ + R~LIRnK2Pn - PnT(a~)QnK~Pn -PnK1Q~T(a~)P~ - R~(R~T(al)R~ - T(a~))LeR~ -R~L1 (R~T(a~)Rn T(5~))R,~ -P~KIQ,~KuPn - RnL~Q,~L2R,~. The function ala2 is continuous, and the operators K and L are compact again, and Lemma1.5 involves that IIGn]] -~ 0 as n -~ oo, which gives our claim. Nowwe proceed in full analogy to the proof of Theorem 1.51: we show that all sequences (PnKPn+RnLRn+G,~)with K, L compact and (Gn) ~ belong to $(C), and then we verify the closedness of the set 81. Let V and V-1 denote the forward and backward shifts on 12 again. The arguments given in in the proof of Theorem1.51 yield that, to prove the inclusion (PnKPn) ~ $(C) for every compact K, it is enough to verify that (P,~ViP~V~_~P~)~ S(C) for every i,j >_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,~[[<_ liminf[[PnT(a)P,~[[ whencelira [[P,~T(a)Pn[[ = [[T(a)[[. (These arguments remain valid for finite sections of every operator A.)
64
CHAPTER 1.
THE LANGUAGE OF NUMERICAL ANALYSIS
Thus, the sequence of the condition numbers cond (P,~T(a)Pn) for an invertible Toeplitz operator with continuous generating function converges if and only if the sequence of the norms [[(PnT(a)Pn)-l[I converges. This convergenceis less obvious than that of the sequence I]PnT (a) Pn [], since the only thing we know about ((PnT(a)Pn)-1) is that this sequence belongs to the algebra ,S(C) again (inverse closedness). So we are led to the problem of verifying that the norms [IAnl] converge for any sequence (A,~) E S(C), and to compute their limit.
1.5.2
Convergence
of norms
Here is the desired result on the convergence of the norms. Theorem1.59 Let (A,~) ~ $(_C). Then the limit lim ][A~P,~[I exists, it is equal to max{l[W(A~)[[,[[W(A~)[[}.
and
Proof. From Corollary 1.58 we know that max{[lW(d~)[[,I[l~Z(An)[[} = I[(dn)
(1.28)
and Proposition 1.16 tells us that [[(An) + 61[ = limsup []AnPn[[. Finally,
(1.29)
by the Banach-Steinhaus theorem,
[]W(A~)][ ~ liminf []A~P~[[,
[[$(A~)[[ li minf[[R~A~R~[[
whence m~ {~[W(An)[], []~(A~)[I } ~ liminf []A~Pn~[.
¯
Thus, the Banach-Steinhaus theorem is powerful enough to finish the proof in shortest time but, in the form the proof stands, it seems not to explMn the reM re,on for the existence of the limits under consideration. So we add a second, somewhat longer and more detailed, proof of Theorem 1.59 (recMling some ~guments of the proof of the Banach-Steinhaus theorem), which indicates that the main point is the special form of the homomorphisms W and W: Since the operators W(A~) and ~(A~) are strong limits, every infinite subsequence (A~) of (A~) can reproduce them. Second proof. Again we have (1.28) and (1.29), and it remains to verify the existence of the limits. Assumethere is a sequence (A~) e 8(C) that lim inf [[A~P~[[ < lira sup []A~P~[[. Choose an infinite subsequence (A~)~0 of (A~) such that lim [[A~P~[[ exists as k ~ ~ and is equal to liminf [[A~Pn~[. The sequence ~ = (n ~)k=o ~
1.5.
ASYMPTOTIC BEHAVIOUR OF CONDITION NUMBERS
65
gives rise to a natural modification of the algebra ~: we let ~’n stand for the C*-algebra of all sequences (A,~,)k~__0with (A,~) r, and defi ne S(C) and Gn similarly. Obviously, if (An) S(C) (a nd th us (An~) ~ 8( ), then the strong limits Wn(An~) = s-lim k-~o~An~P,~ and 12Vn(An~) s-lim k~Rn, A~Rn~exist. A little thought shows that the stability of sequences in 8(C)n can be studied in the very same manner as that of sequences in S(C). What results is that the homomorphismsWn and ISdn can be glued together to produce a symbol mapping for the algebra 8(C)n/G n. The analogues of (1.28) and (1.29) limsupllA~,P~l I = max{lIWn(A~)ll, 1112dn(An~)ll}.
(1.30)
But evidently, Wn(An~) = W(An) and IZVn(An~) = ITV(An) for every sequence (An) S(C). Thus, pu tting (1 .28), (1 .29) an d (1 .30) to gether, we arrive at lim inf [[A°nPn[[
lirasupIIA°=Pnll -- max{llW(A°~)ll, max {lIWn(A°~)ll, Ill~n(A°n.)ll} =limsup liminf[]A°.P~II
which is a contradiction.
1.5.3
Condition numbers of finite operators
sections
of Toeplitz
An immediate consequence of Theorem1.59 is the following. Corollary 1.60 (a) If (An) ~ 8(C) is a stable sequence, then the lim cond An exists, and it is equal to max{llW(An)ll, III~(An)II} max{llW(An)-ill, which is just the condition numberof smb ((An) + ~). (b) In particular, a ~ C(~) and T(a) is invertible, lira cond PnT(a)Pn = cond T(a). To derive (b) from (a) one has to prove I]T(a)ll = I]T(fi)]l
IIT( a)-all = II T(a)-a[].
(1.31)
The validity of (1.31), even for arbitrary a E L°°(~’), is a consequence the identity T(a) = CT(a)* C (1.32)
66
CHAPTER 1.
THE LANGUAGE OF NUMERICAL ANALYSIS
where C is the anti-linear operator of conjugation C : 12 -~ 12, (Xk) ~-+ (’Zk). Let us mention that the explicit determination of the norm of T(a) -1 (or at least of an upper estimate for this norm) can be quite complicated in general (in contrast to the normof T(a) itself, which is available thanks to the Brown-Halmostheorem). The following theorem presents an estimate for liT(a) -1 [I in special cases. Let conv Mdenote the convex hull of a set MC_ C, and write dist (0, M) :-- inf{Iml,m ¯ M} for the distance between Mand the point 0. Theorem1.61 Let a ¯ L~°(’~) and d := dist (0, conva(’l~)) > 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]<_l]alloo }. Multiplying the latter set by A :----- d/[]al]~, we obtain a set which is contained in the disk {z¯C : Iz-11
with
r:=v/1-d2/llall~.
Hence, which implies the invertibility of T(a) and yields (via an estimate using the Neumannseries) the desired inequality (1.33).
1.6 Fractality
of approximation
methods
The topic of this section is another fundamental notion of numerical analysis - that of a fractal approximation method. Roughly speaking, fractality of a sequence (An) means that the knowledge of any infinite subsequence of (An) allows us to reconstruct the complete sequence (An) modulo a sequence or, in other words, the whole information about the coset (An)+~ is already stored in any subsequence of (An). As we shall see, the fractality of a sequence is responsible for the uniformity of certain limiting processes. At this place, we will only mention some basic properties of fractal sequences, and we will illustrate the applicability of this concept for proving once more the convergence of the condition numbers for stable sequences in S(C). Further applications (to the determination of the limiting set of the eigenvalues of the approximation operators, for instance) will be given in the forthcoming chapters.
1.6.
1.6.1
FRACTALITY
OF APPROXIMATION
67
METHODS
Fractal homomorphisms,fractal sequences
algebras,
fractal
It is not important in this section that the elements of the sequences under consideration are operators. So we will use slightly generalized definitions of the C*-algebras iT and G. Given unital C*-algebras Cn (n = 0, 1, 2, ...) with identity elements en, let iT stand for the set of all boundedsequences (c0,cl,c2,...) with cn E Cn, and let 6 refer to the set of all sequences (co, cl, c2, ...) in iT with [[cn[I -~ 0 as n -+ oo. Defining elementwiseoperations and an elementwise involution, and taking the supremumnorm, we make iT to a C*-algebra and 6 to a closed ideal of iT (compare Propositions 1.13 and 1.14). Clearly, with the special choice Cn = L(ImPn), we actually get the previous definitions of iT and 6. The algebras iT and 6 are also called the product and the restricted product of the C*-algebras Cn, and the quotient algebra iT/6 is sometimesreferred to as the ultraproduct of the Cn (and as the ultrapower of C in case Co = C1 .... = C). Given a strongly monotonically increasing sequence r/: N -+ N, let iT, and G, denote the product and the restricted product of the C*-algebras Cn(0), Cn(~), ..., respectively, and let R, stand for the restriction mapping Rn : iT -+ iTn, (an) (a n(n)). Th e mapping R nis a * -homomorphism from iT onto ~’n which moreover maps ~ onto 67. Given a C*-subalgebra A of iT, let A, refer to the image of .4 under Rn. By the first isomorphy theorem, A,~ is a C*-subalgebra of iTo. Definition 1.62 Let A be a C*-subalgebra of iT. (a) A *-homomorphism W : A -~ B o] A into a C*-algebra B is fractal if, for every strongly monotonically increasing sequence ~l, there is a *homomorphism Wn :n. Ao -~ I3 such that W = WnR (b) The algebra A is fractal, if the canonical homomorphism~r : A --~ A/(.4 f~ 6) is fractal. (c) A sequence (an) e iT fra ctal/] thesmallest C*-s ubalgebra of i T which contains (an) is ~ractal. Thus, given a subsequence (an(n)) of a sequence (an) which belongs fractal algebra .4, it is possible to reconstruct the original sequence (an) from its subsequence modulo sequences in .4 V) 6. This assumption is quite natural for sequences arising from discretization procedures. Example 1.63 Let Cn = L(ImPn), and let iTw refer sequences (An) ~ iT for which the strong limits
to the set of all
W(An) := ~-lim AnPn and ITV(An) := s-lim
RnAnRn
as well as the strong limits W(A~) and I/;V(A~) exist. The set iTw is
68
CHAPTER 1.
THE LANGUAGE OF NUMERICAL ANALYSIS
C*-subalgebra of ~ (compare the proof of Theorem 1.18(a)), and S(C) a C*-subalgebra of ~-w as we have checked in the proof of Theorem1.53. The *-homomorphismsW, l~: ~-w _~ L(I 2) prove to be fractal: given a strongly monotonically increasing sequence r~, we can define W~, lPd, : .TnW-~ L(l ~) via W,(A,(n)) s-l im A,( n)P,(n) andl~n( A,(,)) := s Then, obviously, W = W~R~and l~ = I~R,. Further, the algebra ~-w itself is not fractal, but its subalgebra $(C) is. These facts will become evident once we have established somecriteria for fractality of algebras (see Corollary 1.70 and the discussion before). Lemma1.64 Let A be a C*-subalgebra o] ~:, and let B be a C*-algebra. I] W: A ~ B is a ]ractal *-homomorphism, then A fq 6 c_ Ker W. Proof. Let (g,~) e A Cl 6. Given ~ > 0 there is an no such that for n > no. Consider the sequence r/(n) := n + no. Because Wis fractal,
=
=
=
and, consequently,
(recall that assertion.
IlWoll_<1 by Theorem1.44).
Letting e go to zero we get the ¯
Thus, for every fractal *-homomorphismW : A --~ B, the element W(a,~) does only depend on the coset rr(an) = (an) + A ¢3 LetW" denote the (correctly defined) quotient homomorphism W" : A/(A n 6) -~ I3, 7r(a,)
~+ W(a,).
(1.34)
Lemma 1.65 Let ~4 be a fractal C*-subalgebra of if: and 13 be a C*-algebra. A * -homomorphismW: A --~ 13 is fractal if and only if A fq 6 C Ker W. Proof. If Wis fractal, then A fq 6 C_ Ker Wby Lemma1.64. Let, conversely, A f’l 6 C_ Ker W, and define the quotient mapping W" by (1.34). Then, clearly, W= W’Tr.Further, since 7r is fractal by assumption, for every strongly monotonically sequence ~ there is a *-homomorphismr o such that 7r = ~roRo. Thus, which shows the fractality
of W.
The following two theorems provide equivalent characterizations subalgebras of ~-.
¯ of fractal
1.6. FRACTALITY
OF APPROXIMATION
METHODS
69
Theorem 1.66 A C*-subalgebra A of ~ is fractal if and only if the following implication holds for every element (an) E ‘4 and every strongly monotonically increasing sequence 7:
Proof. Let .4 be fractal, and assume there is a strongly monotonically increasing sequence y as well as an element (an) ~ A with Ru(an) ~ 6~, for which(an) is not in A n 6, i.e.
II~(a,~)ll =II(’~n)+‘4 ~611c>o with some constant C. Since R,(a,~) ~ 67, there is an no such that Ila,(n)ll <_ C/2 for n >_ no. Define a sequence # by #(n) ;-- ~(n + no). Then IIR~(an)ll <_ C/2. On the other hand, the fractality of A entails that
(1.35) which is a contradiction. Hence, (an) belongs to .4 A 6. For the reverse direction, let (an), (bn) E A be sequences with R~(an) R~(b~). By our hypotheses, then (an) - (bn) ~ A N 6, and so it is correct to define a *-homomorphismru via
Evidently, ~%/~ = r, which shows that r is a fractal A is a fractal algebra.
homomorphismand ¯
Corollary 1.67 (a) If .4 is a ]ractal C*-subalgebra of jr, then every C*subalgebraof ‘4 is also ]ractal. (b) A C*-subalgebraA of jr is fractal if and only if each of its elements fractal. Proof. (a) Let B be a C*-subalgebra of.4, and let (b,~) ~ B be an element with R,(bn) e 67. Since A is fractal, Theorem1.66 entails that (b,~) A N 6. Then, clearly, (bn) E B ~ 6, and invoking Theorem1.66 once more one gets the fractality of B. (b) If A is a fractal C*-subalgebraof ~’, then each of its elements is fractal due to assertion (a). For the reverse assertion we claim that, whenever is not fractal, then it contains a non-fractal sequence. Indeed, if .4 fails to be fractal, then, by Theorem1.66, there is an element (an) ~ ,4 as well as a sequence ~ such that R,(a~) ~ 6,, i.e. [la,(n)ll -~ 0 as n -~ ~, but (an) it .4~6, i.e. Ilan]l ~ 0 as n -~ ~x). Applying Theorem1.66 once again this shows that the C*-subalgebra of .4 which is generated by the sequence (an) cannot be fractal. Hence, the sequence (an) is not fractal. ¯
70
CHAPTER 1.
THE LANGUAGE OF NUMERICAL ANALYSIS
Corollary 1.68 I] A is a ]ractal C*-subalgebra of J:, then
(.4 n 6),. To motivate the following criterion for fractality,
recall Theorem1.55.
Theorem 1.69 Let A be a unital C*-subalgebra of i7:. Then .4 is fractal if and only if there exists a family (Wt}teT of unital and ]ractal* homomorphismsW~from A into unital C*-algebras Bt such that the following equivalence holds ]or every sequence (an) E A: The coset (an)+ is invertible in A/(An6) if and only if W~(a.) is invertible in Bt ]or every tET. Proof. If A is fractal, then the canonical homomorphism~r : A -~ AI(An 6) provides a ’family’ (~r} with the desired properties. Let, conversely, (W~}~eTbe a family of unital and fractal *- homomorphisms which is subject to the conditions of the theorem. Given a strongly monotonically increasing sequence ~/, define the operator Tv by
T, : A/(An0) -~ A,I(AnG),, (an) +An6~ R,(a~) One easily checks that this definition is correct and that T. is a *- homomorphismfrom A/(-4 N 6) onto A~/(-4 n 6)7. Weclaim that T, is an isomorphism. In accordance with Theorem1.57, it is sufficient to show that T. is a symbolmapping,i.e. that if (an) ~ .4, and if R, (a,~) + (An6), is invertible, then (an)÷-4N6 is invertible. So, let (an) be an element of.4 for which coset R~(an) + (.4 N 6)7 is invertible. Then there are a sequence (b~) e as well as sequences (gn), (hn) e A n 6 such
(1.36) (e~) referring to the identity element of ~. The h omomorphisms W ~ a re fractal by assumption, i.e. there are homomorphismsWt,, such that Wt = Wt,,R,. Applying Wt,~ to the equalities (1.36) we get Wt(an)Wt(bn) = Wt(e,~) + Wt(gn), Wt(b~)Wt(a,~)
= Wt(en)
Since Wt(en) is the identity element of Bt and Wt(g,~) = Wt(h~) (Lemma1.64), we conclude that all elements Wt(a,~) are invertible in Thus, the coset (an)+.4n6 is invertible, and T. is indeed a symbol mapping and, thus, a *-isomomorphism between .4/(.4 n 6) and A,/(.4 Let finally II, denote the canonical homomorphismfrom .47 onto the quotient algebra .47/(.4 N 6)7. Then, evidently, T~II.R. = ~, i.e. r is fractal (with ~. = T~-~I-I.), and we are done.
1.6. FRACTALITY
OF APPROXIMATION
METHODS
71
Let us return to Example1.63 once more. The se~luence (A,~) with A2n = 0 and A2n+l = diag(0,... 0, 1, 0,..., 0) (with the 1 standing in the center this diagonal matrix) belongs to the algebra ~-w, and W(An) s- lim AnPn = 0, I~d(An) = s -l im RnAnRn = s -l im AnPn = O . The sequence (An) does not belong to ~-w N G = G, but for the special choice y(n) = 2n one obtains that Rv(An) (A2n) ¯ ~. As a c onsequence of Theorem1.66, the algebra ~-w cannot be fractal. On the other hand, if (An) ¯ ,~(C) and if W(An) and I~V(An) are invertible operators, then the coset (An) + is invertible in S(C)/G. Since Wand 12d are fractal homomorphisms,Theorem1.69 implies the following result: Corollary 1.70 The algebra 8(C) is #actal. 1.6.2
Fractal
algebras,
and
convergence
of norms
Here we consider once again the problem whether limllanll exists for a sequence (an) E r. W e will p oint o ut t hat t he f ractality o f ( an) i ndeed guarantees the existence of this limit. Theorem1.71 Let A be a fractal C*-subalgebra of ~. (a) If (an) ~ .4 and ~ is a strongly monotonically increasing sequence,
(b) I] (an) ¯ .4, then the limit lim ]lanll exists andis equal to II(a,~) Proof. (a) The third isomorphy theorem for C*-algebras states that, is a C*-subalgebra of ~- and (bn) ~ B, then II(bn) + ~ll~:/g = I[(bn) + ~ll(’+g)/g = [l(bn) + B N ~lIB/(,n~).
(1.37)
Let now (an) ¯ A and (gn) ¯ ,4 ~ g. I](an) + ~[]:r/g
I](an) +-4~ 611A/(A~g) = I]~(an ÷gn)l]A/(A~g) = Ilr~R,(a~ +gn)IIA/(A~g) <- II(a,(n) + gv(n))%ollA,
(by (1.37)) (definition of (fractality of.4) (Theorem 1.44 (c)).
Taking the infimum over all (gn) ¯ .4 ~ ~, and invoking Corollary 1.68, obtain a II( n)+ll /o < I1( ,(n))n--0 +
72
CHAPTER 1.
THE LANGUAGE, OF NUMERICAL ANALYSIS
and a further application of (1.37) gives
II(a)
-
¯
(1.38)
On the other hand, the lira sup-formula for the normin 5c/6 (i.e. Proposition 1.16, which also holds in the present, slightly more general situation) yields II(a~(n)) + ~ll~:~/~ = limsup which together with (1.38) proves the assertion. (b) Assume(an) e is a s equence such tha t lim inf ~lanl~ < l imsup I~an~l. Then there is a strongly monotonically sequence ~ such that the limit lim ~lan(n)~l exists and is equal to liminf Ilanl~. By part (a) of this the~ rem and by the lim sup - formula again, we have limsup Ilan~ = ~(an) = lim sup which is a contradiction. As a simple consequence we get: Corollary 1.72 Let A be a ~actal C*-subalgebra o] ~, and let (an) e be a stable sequence. Then the limit limcondan = limllanll ]la~ll exists
to
ona +
What concerns the convergence of the norms S(C), we see now that our previous result (Theorem 1.59) is a simple combination of two basic observations: ¯ the fractality of S(C) (Corollary 1.70) which yields that the limit lim I~AnPnll exists and is equal to ~l(An) + ~ll for every sequence (An) e S(C) (Theorem 1.71), and ¯ the description of S(C)/6 as an algebra of ordered pairs (Corollary 1.58) which shows that ~l(g~) + 611 = ~lsmb((An) It is also clear that the decisive point in this approach is a precise knowledge on stability in S(C): it is both important to have a criterion for the stability of an arbitrary sequence in S(C) and that this criterion h~ the ’right form’, i.e. via a fractal mapping. In Chapter 4 we will derive simil~ stability criteria for sequences in more involved subNgebras of ~, and in all these cases, Theorem1.71 will automatically yield the convergence of the norms or of the condition numbers of the approximation operators.
1.6.
FRACTALITY
OF APPROXIMATION
METHODS
73
Notes and references Sections 1.1 - 1.4: The elementary theory of abstract projection methods can be found in the textbooks Gohberg/Feldman [64], PrSssdorf/Silbermann[122], [123], BSttcher/Silbermann [24], [26] and Hagen/Roch/Silbermann [77], for instance. Weespecially recommendBSttcher’s review papers [16, 18] as well as the recent textbooks BSttcher/Silbermann [27] and BSttcher/Grudsky [21] , which can, together with [64], also serve as excellent introductions to the theory of Toeplitz operators. As basic references for Banach and C*-algebras we mention Bonsall/Duncan [30], Bratteli/Robinson [32], Dixmier [49], Douglas [50] (which also can be read as textbook on Toeplitz operators), Davidson [44] (an approach to C*-algebras via examples), Kadison/Ringrose [91] (a monumentaltreatise in four volumes), and Pedersen [114]. Polski (1963) [117] was perhaps the first to realize the meaning of the stability of a projection method. A fundamental meta-theorem of numerical analysis is usually stated as consistence + stability -~ convergence, where consistence of a sequence (An) with an operator A means in our terminology that (An) is an approximation method for A. Polski’s Theorem 1.4 is the concrete specification of this meta-theorem we will employ on what follows. In the context of the numerical integration of ordinary differential equations, the phenomenonof a numerical instability was first observed by Todd (1950) [169] and later explained by Dahlquist (1956) [41] by establishing his celebrated root condition; see also the review paper Dahlquist [42]. The fact that stability of a sequence is equivalent to the invertibility af a coset of that sequence in a suitably constructed Banach algebra goes back to and was first employed by Kozak (1974) [95] who considered the finite section method for higher dimensional analogues of Toeplitz operators. The idea of a symbol was brought forth by Michlin in 1936 [105], [106], who associated with every two-dimensional singular integral operator a certain function the invertibility of which is responsible for the Fredholm property of the operator. Gohberg (1953) [62] showed then that Michlin’s symbol calculus actually coincides with the Gelfand transform (see Chapter 4 below) applied to the algebra generated by all singular integral operators with continuous coefficients, factored by the compact operators. Since that time, the distinguished role of a symbol calculus for several important classes of operators has been pointed out by several mathematicians, which in the sequel led to such highlights as the calculus of pseudo-differential operators due to Kohn/Nirenberg and HSrmander.
74
CHAPTER 1.
THE LANGUAGE OF NUMERICAL ANALYSIS
It was, nevertheless, quite surprising when it became obvious that one can also assign a symbol function to certain approximation methods such that the invertibility of the symbol is nowresponsible for the stability of the method. In 1983, BSttcher and Silbermann [25] succeeded in establishing a symbol calculus for certain approximation methods for quarter plane Toeplitz operators. Nowadays,symbol calculi are available for broad varieties of approximation methods for pseudodifferential operators, including Galerkin and collocation methods with polynomial or spline ansatz functions for singular integral and Wiener-Hopf operators. Here we only mention PrSssdorf/Schneider [121] and Hagen/Roch/Silbermann [77], where also further references are given. The Examples1.8, 1.9, 1.12 and 1.39 are taken from .[64], [28], [16] and [28], respectively, and the limsup formula (Proposition 1.16) is from [15], whereas we have learnt formula (1.20) for the essential aorm of an Hilbert space operator from W. Arveson. The elegant approach to the finite section methodfor Toeplitz operators presented in Section 1.3.3 is taken from [16], and the description of the algebra $(C) (Theorem 1.53) is from [25]. Sections 1.5 - 1.6-" Theorem 1.59 on the convergence of the norms of the approximation operators is due to BSttcher [15] and it was one of the ingredients needed to prove the convergence of the ~-pseudospectra of the finite sections of Toeplitz matrices (Sections 3.3.2 - 3.3.3 below). Theorem 1.61 and its proof are taken from [20], where muchmore delicate results on the asymptotic behaviour of condition numbersof finite sections of Toeplitz operators on/P-spaces are derived. The notion of fractality was introduced in Roch/Silbermann [147] in order to verify the uniformity of certain limiting processes. Most of the results in Section 1.6.1 are taken from [141].
Chapter 2
Regularization approximation
of methods Never computea generalized inverse of a matrix whose rank can change under small perturbations. This moral has dominated the literature and practice of generalized inverse computations. But there is also the second side of the coin whichsays that perhapswe are not asking the right question in this case. That is, if B approximates A, but rank B ¢ rank A, then B+ is not the matrix to seek as an approximation to A+. Instead we should develop ~ procedure to compute a variant B~ (of +) + which is a good approximationto the true A and which has in an approximate sense the properties o] a true generalized inverse. M. Zuhair Nashed
In this chapter we turn over to the second of the basic problems mentioned in the introduction: If (An) is an approximation method for the operator A, but A or the An are not invertible, do then the least square solutions of the equations A,~xn = Yn converge to the least square solution of Ax = y? We will see that this question indeed has a positive answer in some cases, but we will also point out that the posed question is the wrong one in some sense: The point is that the computation of least square solutions is instable with respect to small perturbations. So, even in case we can 75
CHAPTER 2.
76
REGULARIZATION
prove convergence, any perturbation be it ever so small can disturb this convergence drastically. A much better question seems to be the following more general one which was perhaps first raised by Moore and Nashed: Is it possible to replace the approximation operators A,~ by certain regularizations, A~ say, which also approximate A, for which the least square solutions of Anx,~ = Yn converge to the least square solution of Ax = y, and for which this convergence is in some sense stable? Wewill verify that this problem is equivalent to a generalized invertibility problem in the C*-algebra and, thus, accessible to algebraic techniques.
2.1 Stably regularizable
sequences
Westart with recalling some basic facts concerning generalized inverses and regularizations of matrices and linear operators. The main goal of this section is to define stably Moore-Penroseinvertible and stably regularizable approximation sequences, and to establish theorems of Polski type for these sequences. 2.1.1
Moore-Penrose matrices
inverses
and
regularizations
of
Let A E ~×~ be a non-invertible matrix. Then the equation Au = f is not solvable for every right hand side f, and if a solution exists, then it is not unique. It is appropriate to replace such linear systems by an equivalent extremal problem: determine vectors fi E Cn such that
(2.1) (Here and in what follows the norm of a vector resp. of a matrix is the Euclidean resp. the spectral norm.) It is knownfrom elementary analysis that the minimization problem (2.1) possesses solutions for arbitrary A and f, but it turns out that these solutions are not unique in general. Indeed, wheneverfi solves (2.1), then fi + k with k ~ Ker A is also a solution. What can be shownis that amongall vectors fi which solve (2.1) there is exactly one vector u+ solving a further minimization problem
Ilu+ll= min {1111:
satisfies
(2.1)}.
(2.2)
This vector is called the least square solution of Au = f. Wewill see in Theorem2.1 that the least square solution u+ depends linearly on f. The matrix A+ defined by A+f = u+ is the called the Moore-Penrose inverse
2.1. STABLY
REGULARIZABLE
SEQUENCES
77
of A. Clearly, if A is invertible, then u+ is the exact solution of Au = f, and A+ coincides with the commoninverse of A. Besides this constructive definition, there is also one way to introduce Moore-Penrose inverses axiomatically, and it is this way which we will go later in order to define Moore-Penrose inverses of operators on Hilbert spaces and of elements in C*-algebras. Theorem 2.1 Given a matrix A E Cn×n, there is a unique matrix B ~ Cnxn such that ABA = A, BAB = B, (AB)*
= AB and (BA)*
= (2.3
and this matrix is the Moore-Penroseinverse of A. This result remains valid for rectangular matrices, which, however, will not be considered here. For a proof of Theorem2.1 we will employ the singular value decomposition of a matrix. Recall that the singular values of a matrix A are the non-negative square roots of the eigenvalues of A*A. Theorem 2.2 (Singular value decomposition) nxn, Given a matrix A ~ C there are unitary matrices U and V such that A = U~V*, where ~ is the diagonal matrix diag (al, ¯ .., an) with the ordered singular values 0 al ~_ ... ~_ an of A as its entries. Note that the matrices U and V in the singular value decomposition of A are not unique: they are unique only up to a unitary operator which leaves every eigenspace of ~ invariant. So one rather should speak about a singular value decomposition of the matrix A. The proof of Theorem2.2 can be found in every textbook on linear algebra. Proof of Theorem 2.1. Let A = U~V*with F~ = diag (a~, ..., singular value decomposition of A, define ~ :=
01/ai
an) be
if = 0 if a~ ai>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 -
IIAu- fll = IlY, V*u- U*fll= IlY,(W*u - ~U*f)- (I which together with (2.4) yields
II~,(W*u - ~,f*f) - (I ~,)f*:ll = II(Z - Y,~,)U*flI. (2 Since E = E~E, and E~. is an orthogonal projection, the vector E(V*u ~.U*f) is orthogonal to (I - E~)U*f. Thus, as we have learned from Pythagoras, the equality (2.5) holds (or, equivalently, u solves (2.1)) only if EV*u = E~U*f. (2.6) Nowit is easy to see that Bf also solves (2.2). Indeed, if u is any solution of (2.1), then it satisfies (2.6), and we
IIBfll--IIY~V*fll = II~,V*ftl= II~V*fll= II~Y,V*ull _
¯
Corollary 2.3 The matrix AA+ is the orthogonal projection from C~~ onto Im A and I - A+Ais the orthogonal projection from Cn onto Ker A. Proof. The axioms (2.3) imply that the operators orthogonal projections and that further
AA+ and A+A are
Im A = Im AA+AC_ Im AA+ C_ Im A, KerA C_ Ker A+A C_ KerAA+A = KerA.
2.1. STABLY
REGULARIZABLE
SEQUENCES
Hence, ImA = ImAA+ and KerA = Ker A+A = Im(I - A+A).
79 ¯
A further immediate consequence of the equality A+ = V~U*is the identity IIA+II = max{l/,~ : ~ e a2(A) {0}} (2.7) where a2 (A) refers to the set of the singular values of In contrast to commoninvertibility, the mapping A ~ A+ is no longer continuous. Here is an elementary example: if A and A,~ are the 1 × 1 matrices (0) and (l/n), respectively, then IIAn - All --+ 0 but + - A+II = The concept of Moore-Penroseinversion allows us to define the condition number for an arbitrary (not necessarily invertible) matrix A condA := IIAll This number plays the same significant role for the solution of the minimization problems (2.1) and (2.2) as the usual condition number does the exact solution of Au = f. If u is the least square solution of Au = f, and u+Au is the least square solution of the equation A(u÷Au) = f+Af with perturbed right hand side then, in case AA+] ~ O, [[Au[[ < condA [[Af[[
Itull -
IIAA+.fII"
(2.8)
Seen with a computer’s eyes, an invertible matrix with large condition numberis as bad as a non-invertible matrix. Because of (2.7), the reasons for large norms of A-1 or A+ are very small positive singular values of A (whereas those singular values of A which are zero do not influence the norm of A+). This observation suggests to regularize ill-conditioned matrices A as follows: Find a singular value decomposition A = UEV*, choose a (small) cutting-off parameter ¢ > 0 and define ~.~ := diag (al.~,...,an,e) by 0 if ai < e ~i,e :----
ai
if
-ai > e.
The matrix Ae := U~,~V*, which is independent of the choice of U and V, is called the e-regularization of A. For small ~, it lies close to A: [IA-A~II = liE-E~II _< e, but its condition is better than those of A since IIA~+II < 1/~. The price one has to pay is a loss of accuracy: even if the matrix A and its ~-regularization A~ are close to each other, their Moore-Penrose inverses A+ and A~+ can have a large distance since Moore-Penrose inversion is not continuous. Let us finally mentionthat there are other possibilities of regularizing illconditioned matrices. The Tychonov regularization, for example, is based
80
CHAPTER 2.
REGULARIZATION
on the idea of shifting all singular values by a certain positive number~ to the right. For details we refer to [81]. 2.1.2
Moore-Penrose erators
inverses
and regularization
of op-
Let H be an infinite-dimensional Hilbert space with inner product (., .). An operator A E L(H) is said to be Moore-Penroseinvertible if there is an operator B ~ L(H) satisfying together with A the axioms (2.3). If such an operator B exists, then it is unique (see the proof of Theorem2.1), and we denote it by A+ again. Clearly, Corollary 2.3 remains valid also in this context. For infinite-dimensional spaces H there exist (in contrast to the finitedimensional setting) operators which are not Moore-Penrose invertible. An example is provided by the operator A = diag(1, 1/2, 1/3,...) E L(/2). (The only possible candidate for the Moore-Penrose inverse of A is the operator B = diag (1,2, 3,...), which is unbounded.) Here are two criteria for the Moore-Penroseinvertibility of an operator. Theorem 2.4 Let H be a Hilbert space. An operator A ~ L(H) is MoorePenroseinvertible if and only if it is normally solvable (i.e. if and only if its range is closed). + Proof. If A is Moore-Penrose invertible then the ranges of A and AA coincide by (the infinite-dimensional analogue of) Corollary 2.3. Since + AA is an orthogonal projection, it has a closed range. Thus, Im A is closed, too. Let, conversely, Im A be closed, and denote by HKand HI the orthogonal complements of Ker A and Im A in H. The restriction of A to HK is an invertible operator from HKonto Im A, and the inverse B of this restriction is bounded since Im A is closed and, hence, a Banach space. Weextend B to a bounded operator on all of H by setting B = 0 on the complementHx of Im A. It is easy to check that this extension is actually the Moore-Penrose inverse of A. ¯ A more detailed examination of Moore-Penrose invertible Example 2.16 below.
operators is in
Theorem 2.5 Let H be a Hilbert space. An operator A ~ L(H) is MoorePenroseinvertible if and only if A*Ais invertible or if 0 is an isolated point in the spectrum of A*A. In this case, [IA+II = sup{l/a:
a ~ a2(A) {0}}/
2.1. STABLY
REGULARIZABLE
SEQUENCES
81
Wewill prove this result in a more general situation in Section 2.2. If A E L(H) is Moore-Penroseinvertible, then we call A+f the least square solution of the equation Au = f. Thus, in the case of infinite dimension, least-square solutions are exclusively defined for normally solvable operator equations. Our next goal is to explain how to construct e-regularizations for operators which act on an infinite-dimensional Hilbert space. ’This requires some basic knowledge from the spectral theory of self-adjoint operators, which we will recall first. In what follows, we will exclusively apply the v-regularization to the approximation operators An, not to the operator A ~ L(H) itself. Thus, everyone who is only interested in (the practically relevant) approximation sequences (An) with finite matrices A,~ can ignore the remainder of this section. Let, for a moment, A be a bounded linear operator acting on a Banach space X, and let E be a closed subset of the complex plane. A closed linear subspace Mof X is called A-invariant if AMC_ M, and an A-invariant subspace Mof X is said to be a E-spectral subspace of A if (i) a(AIM) C_ ~, and (ii)
N C_ Mfor every A-invariant subspace N of X with a(AIN) C_ E.
Observe that a E-spectral subspace of an operator A ~ L(H) does not necessarily exist. On the contrary, one knowsexamples of linear operators on Banach spaces without any non-trivial invariant subspace, and it is still an open problem whether every bounded linear operator on a Hilbert space has a non-trivial invariant subspace. But if a E-spectral subspace of A ~ L(X) exists, then it is uniquely determined. The following result states that for self-adjoint operators on Hilbert spaces there is even an abundance of E-spectral subspaces. Theorem 2.6 Let H be a Hilbert space and A ~ L(H) be a self-adjoint operator. Given numberss, t ~ ~ with s < t, the Is, t]-spectral subspace o] A exists, and it is equal to Ker (A - sI) clos Im~s,~ ¯ K er(A - tI) where as,~
= ~ (t-
z)(s-
and Fs,~ is the curve in Figure 2.1.
z)(A-
-l dz
82
CHAPTER 2.
0
REGULARIZATION
t
-i
Figure 2.1: Integration path in Theorem2.6 If s or t belong to the spectrum of A, then the integral in the definition of ~8,t exists as a Cauchyprincipal value: ¯
-l 1 dz fr (t-z)(s-z)(A-~I)
where F~s,t is that part of Ps,t which lies outside the disks {z : and {z : [z For a proof of this and the following results we refer to [65], Chapter V. There the interested reader will also find a detailed study of the operators The following result is a consequence of Theorem2.6. Let [re(A), M(A)] denote the smallest closed interval which contains the spectrum of the selfadjoint operator A E L(H). Theorem 2.7 Let H be a Hilbert space and A ~ L(H) be a sel]-adjoint operator. Given a numbert ~ ~, the (-oc, t]-spectral subspace o] A exists, and it is equal to L := closIm ~t ® Ker (A - tI) where f~t
= ~ (t-
z)(A-
-l dz
t
and Ft is the rectangle with vertices t:ki and t-1-[t-m(A)[+i. Moreover, ((A- tI)x,x I <_ 0 whenever x ~ L, and ((A- tI)x,x) >_ 0 if x ±. Given a fixed self-adjoint operator A ~ L(H), let E(t) refer to the orthogonal projection from H onto the (-oc, t]-spectral subspace of A, and E(t-O) to the orthogonal projection onto the smallest closed subspace of H which contains Im E(s) for all s < t. Theorem 2.8 Let A ~ L(H) be a self-adjoint operator and s, t ~ ~ with s ~_ t. Then the Is, t]-spectral subspace L of A is equal to the range of
2.1. STABLY
REGULARIZABLE
83
SEQUENCES
E(t) - E(s - 0), a(AIL) N (s,t) = a(A) g~ (s,t). Thus, the spectrum of the restriction of A to its Is, t]-spectral subspace is essentially (with the possible exception of the boundaries s and t) equal the part of the spectrumof A whichlies in Is, t]. This is actually the result we need to define the e-regularization of a self-adjoint operator. Let us still mentionthat the (-oc, t]-spectral subspaces lead directly to a central result of functional analysis: the spectral theoremfor self-adjoint operators. Namely, the family {E(t)}teR forms what is called a partition of the identity on the interval Ira(A), M(A)]in the following sense: (i) Im E(s) C_ Im E(t) for all s _< t, (ii) Im E(s) = ¢~t>sImE(t), (iii)
E(t) : if t <m(A)
(iv) E(t) = if t > M(A) Property (i) is equivalent to E(s) = E(t)E(s) = E(s)E(t) for all s _< t, i.e. any two elements of {E(t)} commutewith each other, and property (ii) equivalent to the strong continuity of the partition {E(t)} from the right hand side: E(s)x li mt%sE(t)x for al l x E H. The following spectral theorem states that a self-adjoint operator does not only give rise to a partition of identity, but conversely, that every selfadjoint operator can be rediscovered from its partition of identity. This is a far reaching generalization of the fact that every self-adjoint matrix can be diagonalized. Theorem 2.9 (Spectral theorem) Let A ~ L(H) be a self-adjoint and {E(t)} the associated partition of identity. Then A
:
fM(A)+O
operator
z dE(z),
dm(A)-O
the integral taken in the Stieltjes sense. Nowwe are in a position to define the e-regularization self-adjoint non-negative operator A ~ L(H) by
(with ~ > 0) of
A~ := (I - E(~)) A = A (I - E(s)). Taking into account Theorems 2.5 and 2.8 one easily checks that A~ is Moore-Penrose invertible and that lid- A~I[ = ][AE(~)I[ _< ~ and []A~+II _< 1/e.
(2.9)
84
CHAPTER 2.
REGULARIZATION
One cannot exclude the equality sign in the latter estimate since Theorem 2.8 says nothing about the boundary e of the interval (-~x~,~]. In order to introduce the ~-regularization of an arbitrary operator A E L(H) we recall the notion of a partial isometry as well as the polar decomposition of an operator (which has its origins in writing a complex number as ur where lul = 1 and r _> 0). A partial isometry is an operator U e L(H) such that I]Uxll = [Ixll ±. for all xe(KerV) An equivalent characterization of partial isometries is the equality UU*U = U, which in particular shows that every partial isometry U is MoorePenrose invertible and that U+ = U*. Recall further that an operator R ~ L(H) is non-negative if (Ax, x) >_ for al l x ~ X. Theorem 2.10 (Polar decomposition) For every A ~ L(H), there exists a partial isometry U as well as a non-negative operator R such that A = UR and closImR = (KerU) ±. Both operators U and R are unique, and ~. A*A = R If A is an operator with polar decomposition URwhere R is Moore-Penrose += invertible, then A is Moore-Penrose invertible, too, and A+ = R+U *. R+U For checking this, take into account that U*Uacts on closIm R as the identity operator. Let nowA ~ L(H) be a not necessarily self-adjoint operator with polar decomposition A = UR, and let ¢ > 0. Then we define the ~-regularization A~ of A by A~ = UR~where R~ is the ~-regularization of R as defined above. Theorem 2.11 Let A ~ L(H) and e > O. Then the e-regularized operator Ae is Moore-Penroseinvertible, [IA- Ae[I <_ e and [[(Ae)+[I _< l/e, and moreover, A~Ae = R~. Proof. Westart with showing that A is Moore-Penrose invertible with Moore-Penroseinverse (RE)+ U*. Let (E(t)) denote the partition of identity associated with the self-adjoint operator R. Then A~A+~A~ = UR~(R~)+U*UR~---
UR~(R~)+U*UR(I- E(~)).
Since U*Uacts on closIm R as the identity operator, the right hand side of this equality coincides with UR~(R~)+R(I - E(~)) = UR~R+~R~ = UR~ The other axioms of Moore-Penrose invertibility Further we have
can be checked similarly.
II(A~)+II = II(R~)+U*II < II(R~)+II _< ~/¢,
2.1. STABLY
as well as
REGULARIZABLE
SEQUENCES
85
II(A~- A)*(A~ - I = [](Re - R)U*U(R -IR 2IIR-RelI I]E(¢)RU*URE(¢)II = I]E(¢)ReE(¢)II (2.1o)
A~A~ = R~U*UR~ = (I-
E(¢))RU*UR(I-
Z(¢))
e~
(2.11)
which finishes the proof. 2.1.3
Stably
regularizable
approximation
sequences
Let again H be a Hilbert space and (P,) be a sequence of orthogonal projections which converges strongly to the identity operator. Let further A E L(H) and An e L(Im P,) be operators such that AnP, and A~P, converge strongly to A and A*, respectively. Whatwe are interested in in this chapter is situations where (AN) is not an applicable approximation method for A. According to Polski’s theorem, the reasons for non-applicability might be (i) that A is not invertible, (ii) that infinitely manyof the An are not invertible, (iii) that the An are invertible for all sufficiently large n, but sup IIA~I]I = The wayout is obvious: in cases (i) or (ii) one will replace the usual inverses of A or An by their Moore-Penrose inverses (if they exist), and then one has to consider the problem whether A~+ -~ A+ strongly
as n -+ oo
(2.12)
(i.e. whether the least square solutions of Anun = In converge tothe least square solution of Au = f for every f). In case (iii) one will choose a cutting-off parameter ¢ > 0 and replace the operators An by their ¢regularizations A,~,~. Theorem2.11 guarantees that the operators A,~,~ are Moore-Penroseinvertible and that sup, IIA~+,~II < oo. The question in this case is whether A~+,~ -~ A+ strongly as n -~ o~ (2.13) (i.e. whether the least square solutions of the C-regularized equations A,~,~un,~ = fn converge to the least square solution of Au = f for every f). In addition we will require that IIAn,~ - A,~II -~ 0 in order to
86
CHAPTER 2.
REGULARIZATION
guarantee that (A~,~) is an approximation method for the same operator
as(An). Clearly, the approach via regularization can alsos be applied in cases (i) and (ii) or if sup ]IA~+II = oo and, as we shall see later, it yields in sense more satisfactory results and proves to be more natural in these cases than (2.12). For both problems (2.12) and (2.13), there are theorems of Polski type. Theorem2.12 Let H, Pn, A, A,~ be as above, and suppose that AnPn --~ A and A~Pn --~ A* strongly as n -~ oc. Then the following assertions are equivalent: (a) The equations Anu,~ = Pnf possess least square solutions un for all right handsides f and all sufficiently large n, and the sequence (Un) of these solutions converges in the norm of H. (In other words: the operators An are Moore-Penroseinvertible for all sufficiently large n, and the sequence (A+nPn)of their Moore-Penroseinverses converges strongly.) (b) The operators A,~ are Moore-Penroseinvertible for all sufficiently large n, and supn IIA+~P~II< If one of these conditions is satisfied, then A is Moore-Pcnroseinvertible, and A+nPn-~ A+ strongly as n ~ o~. Theorem 2.13 Let H, P,~, A, A,~ be as above, and suppose that AnPn -+ A and A~P,~ --~ A* strongly as n --~ oo. Then the following assertions are equivalent: (a) There is a parameter ~ > 0 such that ]IAn,~ - Anll -+ 0 as n -+ oo and such that the least square solutions of the regularized equations A,~,~u~,~ P,~f form a convergent sequence for every right hand side f E H. (b) There is a parameter ~ > 0 such that IIA~,e - Anll -~ 0 as n -~ oo and
supIId+,Pnll< If one of these conditions is satisfied, and A+~,~P,~~ A+ strongly as n
then A is Moore-Penroseinvertible,
Observe that in case H has infinite dimension, not every operator in L(H) is Moore-Penroseinvertible, and so we have to require the existence of least square solutions in Theorem2.12. If the An are finite matrices, one can obviously drop this assumption. On the other hand, e-regularized operators are Moore-Penrose invertible in any case, and so the regularized equations An,~u,~,~ = Pnf are always least square solvable. To prove Theorems 2.12 and 2.13 we still need a few more results on Moore-Penrose inverses in C*-algebras, which will be presented in Section 2.2.1. The analogy between Theorems 2.12 and 2.13 on the one hand and Polski’s theorem for stable approximation sequences on the other hand
2.1.
STABLY REGULARIZABLE SEQUENCES
87
suggests to call a boundedsequence (An) of operators Moore-Penrosestable if it satisfies condition (b) of Theorem2.12 and stably regularizable if it is subject to condition (b) of Theorem 2.13. Thus, Theorems 2.12 and 2.13 reveal that it is again a (generalized) stability property of the sequence (An) which is essential for the (generalized) applicability of the approximation method (An). To have a few trivial examples, consider the n x n matrices (i) An = diag(1,1,...,1,1), (ii) An = diag(0,1,...,1,1), (iii) An = diag (l/n, 1,..., 1, 1), (iv) An = diag (l/n, 1/(n - 1),...,
1/2, 1).
The sequence (An) is stable in case (i), Moore-Penrose stable (but stable) in case (ii), stably regularizable (but not Moore-Penrosestable) case (iii), and not stably regularizable in case (iv). At the same time, these elementary examples already provide a good picture of the behaviour of the singular values of (generalized) stable approximation sequences (An) (compare Figure 2.2): Theorem 2.14 A bounded sequence (An) o] operators An E L(Hn) (a) stable if and only if there is a numberd > 0 such that a2(An) (b) Moore-Penrosestable if and only if there is a number d > 0 such that a2(An) C_ {0} U [d, ~), (c) stably regularizable if and only if there are numbersd > 0 and ¢n >_ with limCn= 0 such that a2(An) C_ [0, en] U [d, ~x)) for all n.
a2(A,~) V/’/’,-"/ d
(An) stable
Moore -Penrose
stable d a2(An)
(An) stably regularizable
0
d
Figure 2.2: Location of the singular values Proof. Assertions (a) and (b) are immediate consequences of Theorem 2.5. For a proof of (c), let first (An) be a stably regularizable sequence,
88
CHAPTER 2.
REGULARIZATION
and let An = UnRnbe the polar decomposition of An. Observe that, A~An = R2~ by Theorem 2.10, a2(An)
= V/a(A~An)
since
= ~ = a(Rn)
and, analogously by Theorem 2.11, a2(An,~) a(Rn,~) for ev ery ~ > 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} <_.lid respective inf{ a : a E a2(An,~) \ {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-<2Ila- a~ll. Proof. Wewill prove that g can be found in alg (e, a), the smallest closed subalgebra of B which contains e and a. Since alg (e, a) is a commutative C*-algebra, it is *-isomorphic to the algebra C(X) with a certain compact X by the Gelfand-Naimark theorem, and every element b ~ alg (e, a) can be identified with a continuous function/~ on X. Set s := Ila -aell < 1/4. Since the mappingb ~-~ ~ preserves involutions, we conclude that 5 is a real-valued function, and the isometry of the mapping b ~/~ yields that
Ila - a211-- 115- 521t=~ I~(x)- ~(x)2] =s < 1/4. Hence, the compact X decomposes into two both open and compact sets X0 := { x e X: 5(x) E [1/2 - ~/VqTT~, 1/2 - VqT~- s] Xl:={xeX:
5(x)
e[1/2+~,1/2+~]},
which implies that the function 15 : X-+C ’
x~ { 01 if if
x~Xo x~X1
is continuous on X. Let p E alg (e, a) be the element which corresponds the function 15. One easily checks (by operating with functions instead of elements) that p is a projection in B and that .Ila-pll--maxla(x)-p(x)l x~X
< s--Ila-aZll --
Hence, choosing g :-- a - p, we get the assertion.
¯
94
CHAPTER 2.
REGULARIZATION
Let B be a C*-algebra and ~ be a closed two-sided ideal of/3. One says that lC lifts projections if, for every coset a + lCE /3/1~ which is a projection in /3//C, there is a representative a + k of the coset a + ~ which is projection in/3. Observe that not every closed ideal of a C*-algebra lifts projections, and that the ’solution’ a + k of the lifting problem is not necessarily unique. Elementary examples can be found already among the algebras of continuous functions. Proposition 2.21 The ideal ~ of 3: lifts
projections.
Proof. Let (An) + ~ be a projection in ~. Since every self-adjoint coset contains (at least one) self-adjoint representative, we can suppose (An) be self-adjoint in 9v, i.e. we have A~= Anfor all n, and IIAn - A2nll -~ 0. If n is sufficiently large, say n _> 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 <_ 2 IIAn - A~II. If we define Gn = -An for n < no, then A,~ + Gn becomes a projection for every n and, hence, (An + Gn) is a projection in r. Finally, t he s equences (An) and ( An) + belong to the same coset modulo6 because IIG,~II _~ 2 IIA,~ - A~II --~ 0 as n --~ (x).
¯
Given a sequence (An) ~ ~- such that (An)+~ is a Moore-Penrose invertible coset in ~-/~, we denote by II(An) tile set of all projections in ~" which are representatives of the Moore-Penrose projection ((An)+ n associated with (An) + ~. By Proposition 2.21, H(An) is not empty. The following result states that the set H(An) always contains an (essentially unique) sequence (Hn) such that A~AnIIn = [inA~An for all n. This observation will be employed later on in order to diagonalize the matrices A~Anand [I n simultaneously. Theorem 2.22 Let (An) ~ J:, and suppose (An) + 6 to be Moore-Penrose invertible in J:/~. Then there is a sequence (1-In) ~ H(An)such that Hn alg (A~An, IIHn) (= the smallest closed subalgebra o] L(Hn) which contains A~Anand the identity IIHn) ]or all n. The sequence ([In) is unique in the and ]ollowing sense: I] Hi(n), (II~) IX(An) E IIn,~ n2 ~al g(A~An,IIHn) ]o all n, then [i~ = II~ ]or all sufficiently large n. Proof. Let (Rn) be a sequence in II(An), i.e. the Rn are projections, (Rn) + ~ = ((An) + ~)II. By Theorem 2.15 (c), (Rn) + ~ ~ alg~:/6((A~A~) (I [H.) + 6). Thus, one can find polynomials Q~, Q~,... in one variable such that liQ~((A~An) + ~) - ((Rn) ~)[[:r/~ "-
~ 0 ask -
and
2.2. ALGEBRAIC
CHARACTERIZATION
95
With the limsup formula for the norm in ~’/G (Proposition 1.16) this gives lim limsup IIQk(A~A~) - Rnll = which together with the obvious estimate 0 ~ diStL(H.)(Rn, alg (A~An, I~H.)) ~ ~[Qk(A~An)yields lim~ diStL(H~)(Rn, alg (A~An, I[H~)) : Lete~ denote the distance between the operator Rn and the algebra alg (A~An, I[H.), and let B~ be operators in this algebra such that I]B~ - R~[] < 2en. These operators can be chosen to be selhadjoint; indeed, if [[Bn - R~[[ < 2~, then ~B~ - Rn~l = ~B~ - R~ < 2~, whence
II(B~+ B~)/~- R~II~ (IIB~- R~II+ IIB~- R~II)/~ ~. ~rther
one h~
B~ - B. = n.(~.
- R.) + (B. - R.)n.
+ (~. = - (S.
which gives
Choose n0 such that 6~u + 4s~ < 1/4 for ~l n ~ no. Then B~ is an almost projection in sense of Proposition 2.20 for n ~ n0, and the same proposition yields the existence of selhadjoint operators G. in alg (A~A., I]~.) such that Bn + Gn is a projection and
Set Hn := I~g. for n < n0 and Ha := Bn + G. for n ~.no. The sequence (Hu) is a projection in ~ by construction, every H. belongs alg (A~An, IJg.), and the estimate
IIR.- ~.11~ IIR.- B.II+ I1¢-II< 1~.+ for n ~ no shows that (Rn) and (Hn) belong to the s~e coset modulo Hence, (Ha) e H(An). For a proof of the uniqueness, let (H~), (H~) be two sequences of 2 jections which are subject to the conditions of the theorem. Then 1H.H n = 2 1 H.H., hence (H~ - H~)2 is a projection for every n. On the other hand, ~$(H~- H~)e]] ~ ~JH~ - H~]Je ~ 0 as n ~ ~, and the only projection with norm less than one is the zero operator. Consequently, H1n -Hn2 is nilpotent for all sufficiently large n. Since H~ - H~is self-adjoint, this gives H~= H~
CHAPTER 2.
96
REGULARIZATION
for all large n.
¯
Let SII(An) denote the set of all sequences (1-In) in II(An) such that belongs to alg(A~A,~,IIg., ) for every n. By Theorem 2.22, SII(An) ~ As we have just observed, any two sequences in SH(An) can differ only in finitely manyelements. Proof of Theorem 2.19. To start with, let (An) + G be Moore-Penrose invertible and (H,~) e SH(An). Then, by Theorem 2.15(c), ((A~,An) G)((Hn) + ~) = 0, and the coset ((A~An) + ~) (( IIn) + ~)is inv ertible. Equivalently, IIA~A~IIn ][ ~ 0 as n ~ oo, and the sequence (A~A,~+II,~)n~__I is stable. Choose d such that
lid > 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 en
all
n_>no.
(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 <_ en -4 0 as n -4 ~ by (2.23), and the operator A~An+ l-In is invertible and satisfies the estimate II(A~An + IIn)-lll _< min{1,1/d} by (2.24). On defining Hn = 0 for n < no, we thus obtain a sequence (Ha) which is a projection in v s uch t hat 11 ((A~An) ~)((Hn) + ~)II = 0 (( A~An ) + ~) +(( Ha)+ ~) is in vertib Theorem2.15(c) yields the Moore-Penroseinvertibility ¯
As a consequence of this proof, one has Corollary 2.23 Let (An) ~ J: be a stably regularizable sequence with a(A~An) C_ [0, en] U [d,~x~) where d > 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
CHAPTER 2.
REGULARIZATION
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
CHAPTER 2.
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 <_ Iz* - zk] + Iz~ - m~)l < 2/k showing that z* belongs to lim inf Mn.
¯
The following result relates the two notions introduced in Definition 3.1. Theorem 3.3 Let S be a metric space, T be a compact subset of the complex plane, and let M be a set ]unction on S taking values in the closed subsets of T. Then M is upper semi-continuous at s ~ S if and only if limsup~_~oM(s,~) C_ M(s) for every sequence (s,~) C_ S tending
3.1.
SET SEQUENCES
107
Proof. Let M be upper semi-continuous at s E S and U be an open neighborhood of M(s) in sC. Then, for every sequence (s,~) tending to s, the sets M(s,~) are contained in U for sufficiently large n, hence, lim sup M(s,~) C_ clos U. Thus, the partial limiting set is contained in the intersection of the closures of all open neighbourhoods of M(s) in C. Because M(s) is compact, this intersection coincides with M(s) itself. Let, conversely, lim sup M(s,~) C_ M(s) for every sequence sn -~ s, but assume Mto be not upper semi-continuous. Then there is an open subset U of C which contains M(s), and there is a sequence (sn) tending to s such that M(sn) ~_ for al l n. Choose poi nts mn ~ M(s n) \ U These points lie in T, and since T is compact, the sequence (mn) possesses a convergent subsequence (m,~) with limit m. Clearly, m ~ limsupM(sn) but, on the other hand, m ¢ U which implies that m ~. M(s). This contradiction proves the reverse implication. ¯ Given a subset M of C we denote its radius sup{[m[ "m ~ M} by radM. If M is the spectrum of some operator A, then rada(A) is nothing than the spectral radius p(A). Proposition 3.4 Let (Mn) be a set sequence such that supn rad M,~ < co. Then lim sup rad Mn<_ rad (lim sup Mn) and rad (lim inf M~)_< lim inf rad Mn. Proof. For the first assertion, assume that lim sup rad Mn> rad (lim sup Mn).
(3.1)
Let ( n~)k=l be a sequence of points mn~ ~ Mn~such that [m,~ I converges to lim sup rad M,~. The boundedness condition ensures that (m,~) has convergent subsequence with limit m. Thus, m ~ limsup M,~ on the one hand, and [rn I = lim sup rad M~on the other hand, which contradicts (3.1). To prove the second assertion, given ~ > 0 choose m E lim inf Mnsuch that tad (liminf Mn) <_ [m[ + ~. Then, for every Sequence (ran) of points m,~ ~ Mn which converges to m, [m[ +~ = lim [m,~ I +~ _< liminfrad(Mn)
+~.
Letting ~ go to zero in rad (liminf M,~) _< liminfrad(Mn) + ~ yields assertion.
.
Weconclude this subsection by an alternative characterization of the partial and the uniform limiting set. For, denote by Ni the set of all infinite subsets I of the natural numberssuch that N \ I is infinite, too.
108
CHAPTER 3.
Proposition
3.5 Let (
n)n=l
APPROXIMATION
OF SPECTRA
be a set sequence. Then
lim sup M,~ = ~IkClOS (U,~_>kMn) and liminf Mn = ~31e~clos(UnelM,~). Proof. Let m E lim sup Mn. Then there are a monotonically increasing subsequence ( n~)~=1 of N as well as points m~ E t su ch th at m~--~ m as l --~ cx~. Fix k ~ N and choose n~o > k. Then, for every n~ > nt0, ml ~ (-Jn>kMn, hence, m = limmt ~ clos(U,>kM,). Since k is arbitrary, this proves the inclusion Conversely, let m ~ NkClOs (U,>kM,~). Then m E clos (U,~>kMn) every k, hence, one can find points mk ~ Un>_kMn such that [m- ink[ 1/2 k. Set rhl := ml, and choose a subsequence (rhk) of (ink) as follows: if nk is the smallest number n with (rh~) ~ M,~, then set ~hk+l := mn~+~. What results is an infinite subsequence (nk) _C N and points ~hk such that rhk -+ m, whence m ~ lim sup Mn. The proof of the second assertion is similar. ¯ 3.1.2
Coincidence set
of the
partial
and uniform
limiting
Let C~ denote the set of all non-empty and compact subsets of the complex plane. A criterion for the coincidence of the partial and uniform limiting sets of a set sequence taking values in C~ can be given in terms of the C convergence . of that sequence with respect to the Hausdorff metric on C The Hausdorff distance of two elements A and B of CC is defined as h(A, B):= max{amea~dist(a, B), r~ dist(b, where dist(a, B) = minbeBla - b]. The function h is actually a metric Cc . Wedenote limits with respect to this metric by h-lim. Proposition 3.6 Let (M~) be a set sequence taking values in c. Then lim sup Mn and liminf Mn coincide if and only if the sequence (Mn) convergent as a sequence on the metric space (Cc, h). In that case, lim sup M~= lim inf M~= h-lim Mn. Proof. Let (M,~) be a sequence which converges in C, h)to a s et M. W claim that lim sup MnC_ M C_ lim inf M=.For the first inclusion suppose that (av(~)) with an(n) ~ Mn(n) is a convergent sequence with limit a ¢ M. Then dist (an(n) , M) is greater than some positive constant for all n, which is impossible. Further, for every a ~ Mwe have by definition dist (a, Mn) -~ 0 as n tending to infinity, which shows that a actually belongs to lim inf M,~.
3.1. SET SEQUENCES
109
For the reverse implication we check that max dist
(an,
limsupMn) -~ 0 and
a~ EM,~
max dist
aElim inf
Mn
(a, Mn) -~ O. (3.2)
Then, if lim sup Mn= lim inf Mn=: Mfor some set sequence, we get max{ max dist (an, M), maxdist (a, Mn)} -} 0, an ~ Mn
a~ M
i.e.
h-lim Mn = M. Suppose the first assertion of (3.2) to be wrong. Then there are 6 > 0, a subsequence y of the natural numbers, and elements a?(n) ¯ Mn(~) such that dist (an(n), limsupMn) _> 6. The sequence (an(n)) convergent subsequence with limit a, say. By definition, a belongs to the partial limiting set, whereasdist (a, lim sup M,~)_> 6 by construction. Similarly, if the second assertion of (3.2) would be wrong, then one could find an 6 > 0, a subsequence y of the naturals, as well as a point a ¯ liminf Mnsuch that dist (a, Mr(n)) >_ 6. In this case, a cannot belong to the uniform limiting set of the Mn, which is a contradiction, too. ¯ The only substantial result of this section is the following. Theorem 3.7 Every bounded set sequence (Mn) in C possesses a subsequence (M~) such that lim sup Mn~ °. = lira inf M~ Thus, every bounded subset of Cc is relatively compact with respect to the Hausdorff metric. Proof. Without loss, suppose that UnM,~is contained in the square Q0,1 := [0, 1] x [0, 1]. Given N _> 0, subdivide [0, 1] into 2g intervals of length 2-N, which involves a subdivision of Qo,1 into 4N congruent squares. Enumerate these squares by QN,1, QN,2,...,QN,4N in any way and, given N > 0 and 1 _< k _< 4N, define Q(4~_i)/3+~ := Qg,k. What results is a sequence of squares Q~(=[0, 1] x [0, 1]), Q2, Q3,... such that diam Qn = y/-~2-N for all (4 ~ - 1)/3 < n _< (4 N+I - 1)/3 and {-J(4~v--1)/3
Q,~= [-.)l
Set Mn~ := Mnfor all n, and suppose that a subsequence (Mnk) of (Mn) is given for some k _> 1. Then construct a new subsequence (M~) k+~ (Mn) as follows: If (lim sup M~\ lim inf k) ¢~Q} ~ ~, the n choose and k) ~ Qk, and let (M~TM) be a fix an element z k ¯ (limsupM~k \ liminf M~
110
CHAPTER 3.
APPROXIMATION
OF SPECTRA
subsequence of (M~k) such that zk belongs to lira inf Mnk+l. Otherwise set k+l := M~ k for all n. Finally, for k _> 0 and all n, set M(nk) := 4~+’-1)/3 Mn M(~ n) and M~ := M(n . If (Mn(~)) is an arbitrary subsequence of (M,~), lim inf MnC_ lim inf Mn(,~) C_ lim sup Mn(n ) C_ lim sup Mn. Thus, since (M~) is a subsequence of every sequence (Mn(k)), and limiting sets are closed, one has k) C_ lim inf M~C_ lim sup M~C_ CI~ lim sup M(~ k) . (3.3) clos Uklim inf M~ Nowlet a E nk>0 limsup Mn(k). Then, for every N, a belongs to some N) r’IQN,k is not empty. Hence, square Q~v,k and, by construction, lim inf M(~ dist (a, liminf Mn(N)) <_ diam -~, Qg,k = X/’~2 which implies that a belongs to clos Uk>Olim inf M(~k). Together with (3.3), this yields the coincidence of lim sup M~and lim inf M~. ¯ Weconclude with the observation that uniform limiting sets are more uniform that their definition suggests. Proposition i}.8 Let (Mn) be a bounded sequence of subsets of the complex plane with limsupM,~ = liminfMn, and let U ~ C be an open set which contains lim inf M,~. Then M,~C_ U for all suJ:ficiently large n. Proof. Suppose there exists an open neighborhood U of lim inf M,~ as well as a sequence (n~) C_ N such that M,~ ~ZU for allk. Choose points m~ ~ istence of a partial limsupMm hence, by hand, m* ¢ U due to
Mnk \ U. The boundedness of UMnimplies the exlimit m* of the sequence (mk). By definition, m* ~ hypothesis, m* ~ liminf M,~ C_ U. On the other the choice of the sequence (m~). Contradiction.
3.2 Spectra and their
limiting
sets
Here we deal with the limiting sets of spectra of approximation operators and with their relation with the spectrum of the limit operator. Clearly, if (A,~) is an approximation methodfor an operator A, then lim inf a(A,~) lim sup a(A,~). For the other possible inclusions we consider a few examples.
3.2. SPECTRA AND THEIR LIMITING
SETS
111
Example 3.9 Weconsider approximation sequences (An) for an operator nxn we obt ain an approxiA E L(/2). With An = diag(1,1,...,1,0) mation sequence for the identity operator A = I, where {0, 1} = liminfa(An)
= limsupa(An) q~ a(A) {1},
whereas the choice diag(1,1,...,1,1) An = diag(1,1,..,1,0)
ifniseven ifnisodd
yields {1} = liminf a(An) _C limsup a(An) = {0, Example 3.10 Let
/°1° )
1 0 n×n
Then, evidently, a(An) {0} an d hence, li m sup a(An) -- liminf a(An) = {0}. On the other hand, the sequence (An) converges strongly to the shift operator
V:~2_~12,(zo, z~, :c~, ...) ~ (0, zo, z~,...), and since V is nothing but the Toeplitz operator with generating function a(t) = t, we conclude from Theorem 1.31 that the spectrum of V is the closed unit disc lI~ t~ ~. Thus, ~ U ~£ = a(V) li m sup a( An) = li m inf a(An) = {0}. ¯ Example 3.11 Let An be as in the previous example, and let Gn be the n x n matrix having 1In in its north-eastern corner and zeros at all other places. Then the eigenvalues A of An + Gn satisfy the equality An = 1In and are, hence, equal to (l/n) 1/n exp(2~rij/n) where j = 0, 1,..., n - 1. Thus, the spectra of An + Gn approach the unit circle in the sense that lim sup a(An + Gn) = lim inf a(An + G~) = q2. Example 3.12 Let K be a compact operator on/2, and let PnKPnrefer to the finite sections of K with respect to a fixed basis of 12. In this setting, it is well known(see also the following subsection) that lim sup a(I + PnKPn) = lim inf a(I + PnKPn) = a(I + K).
112
CHAPTER 3.
APPROXIMATION
OF SPECTRA
Thus, in general lira inf a(A~) can be a proper subset of lim sup a(An), and none of the inclusions a(A) C_ limsupa(An) or liminf a(An) C_ a(A) need hold. Moreover, limiting sets of spectra can change drastically already under perturbations which tend to zero in the norm (Example 3.11). the other hand, in some instances one even has continuity of the mapping a ~-+ a(a) in the sense that limsupa(A,~) = a(limAn).
3.2.1
Limiting quences
sets
of spectra
of norm convergent
se-
Weare going to examine limiting sets of spectra of approximation operators in more detail and start with the simplest case, the norm convergent approximation sequences. Theorem 3.13 Let .4 be a Banach algebra with identity function a ~-~ a(a) is upper semi-continuous. Proof. Wefirst
e. Then the set
show that limsupa(a,~)
C_ a(a) if [[an - all ~
o,
(3.4)
Indeed, let s E lim sup a(an), and let s,~ with s~ ~ a(a=~) be a sequence which converges to s. Then an~ -- Sn~e converges to a -- se in the norm of A. Assume s ~[ a(a). Then a - se were invertible, and a Neumannseries argumentwouldimply the invertibility of a,~ - s,~ e for sufficiently large k. This contradiction proves (3.4), and it remains to apply Theorem3.3 with S := {a e A : I]all _< r} and T := {z e C : Izl _< r} for some sufficiently large r to get the assertion. ¯ Observe that the inclusion in (3.4) is proper in general. A famous example by Kakutani moreover shows that there exist limpotent operators (i.e. operators with spectrum strictly larger than {0} which are the uniform limit of operators with spectrum {0}) already in the C*-algebra L(/2). For completeness we present Kakutani’s construction here, where we follow [130]. Recall for this examplethat the spectral radius of an element a of a unital Banach algebra can be computed via #(a)
li m ~,
(3.5)
which is a very remarkable formula since it connects algebraic (p(a)) and metric (lla’~ll) quantities of
3.2.
SPECTRA AND THEIR LIMITING
SETS
113
Kakutani’s limpotent operator. Let (em)~=l be an orthonormal basis of l ~ and define numbers a,~ := e-k where m = 2k(2/+ 1) and k,l _> Let A denote the weighted shift operator on/2, Aem
=
amem+l , m = 1,2,
....
The operator A is bounded on 12, and IIAII = suplaml. From Anel = al... an en+l one further concludes that IIAnll > ala2.., aN. By the definition of the am, one has ~--1
*-j-~) ala2...a2,-1
= exp(- Z j2 j--1
which implies that ~-1 (ala2...
a2,_1)
1/(2’-1)
> ((ala2...
a21_1)1/2’-’)
2 = (exp
(--
2. ~-~ j2-J)) j=l
If we set /~ := ~j=~ j2-J, then the formula (3.4) for the spectral radius yields p(A) >_ -~. This s hows in particular t hat a(A) cannot c onsist o f the zero element only. Nowdefine operators Ak for k k 0 by Ak em :=
0 if m = 2k(2/+ 1) amem+1 if m~2k(2/+l).
Then A~~+~ = 0, hence a(Ak) = {0}, and furthermore
{
e-aem+l (A - Ak) em := 0
if m=2~(2/+1) if m # 2k(2/+ 1)
which yields ]IA - Ak]l = e -~, hence A = lim Ak.
¯
There are two special instances being relevant to our purposes where equality of lim sup a(A,~) and a(lim An) can be guaranteed: Theorem 3.14 (a) Let .4 be a unital Banach algebra and (an) a sequence o] elements o~ .4 which converge to an element a ~ A with totally disconnected spectrum, or (b) let .4 be a unital C*-algebra and (an) a sequence o~ normal elements ‘4 which converge to an element a ~ ‘4. Then, in both cases, lim sup a(an) = a(a). ’Recall that a metric space X is totally disconnected if every singleton (x} C_ X is open in X and that an element a of a C*-algebra is normalif a*a = aa*. The proof of Theorem3.14(a) is based on the following lemmawhich is its owninterest.
114
CHAPTER 3.
APPROXIMATION
OF SPECTRA
Lemma3.15 Let A be a unital Banach algebra and (an) C_ A a convergent sequence with limit a E ~4. Then every open neighbourhood o] every connected componento] a(a) contains points ]rom a(an) if only n is su~ciently large. For proofs of Lemma3.15 and Theorem3.14 see [113] or [130], Chapter 1, Section 6. As an application we will derive the result cited in Example 3.12 on the convergence of the spectra of the finite sections of a Fredholmintegral equation of second kind. Lemma3.16 Let H be a Hilbert space, (Pn) be a sequence of orthogonal projections on H converging strongly to the identity operator on H, and K be a compact operator on H. Then lim sup a(I + PnKPn)-- lim inf a(I PnKPn) = a(I + K). Proof. The inclusion lim sup a(I + PnKPn)C_ a(I + is a c onsequence of Theorem 3.3 because I1(I + PngPn) - (I ÷ I --~ 0 asn -+ c~. T he reverse inclusion follows from Lemma3.15 since the spectrum of a compact operator consists of at most countably manypoints the only possible cluster point of which is 0 E C. The equality lim sup a(I ÷ PnKPn)li m inf(I ÷ PnKPn)will be established in a more general context in Section 3.2.3. ¯ 3.2.2
Limiting
sets
of spectra:
the
general
case
Our next goal is the asymptotic behaviour of the spectra a(An) for arbitrary bounded sequences (An) of operators, or, slightly more general, elements of certain C*-algebras. That is, we let (Cn) be again a family C*-algebras with identities en and let ~ resp. G refer to their product resp. their restricted product. Let (an) ~ ~-. It turns out that the partial limiting set limsupa(an) related to a new stability notion which might be called ’spectral’ stability (in contrast to the conventional ’normal’ stability introduced in Section 1.1.1). A sequence (an) spectrally sta ble if its entr ies an a re invertible for all sufficiently large n and if the spectral radii p(a~1) of their inverses are uniformly bounded (whereas conventional stability requires uniform boundedness of the norms Ila~lll). Clearly, every conventionally stable sequence is also spectrally stable. Theorem 3.17 Let (an) ~ J~. Then s E C~ belongs to the partial limiting set limsupa(an) if and only if the sequence (an - Sen) is not spectrally stable.
3.2.
SPECTRA AND THEIR LIMITING
SETS
115
Proof. Let (an - sen) be a spectrally stable sequence, i.e. sup p((an - sen) -1) =: m < oc for a certain no. Then, for all n _> no, sup
It[
= sup It[ -1 -1, ----tEa(an--se,~)
( inf [tD tEa(an
--sen)
whence l/m<
inf Itl= ist(s - I--d t~(~n)-s
inf
It-s
Hence, s cannot belong to lim sup a(an). For the reverse direction suppose (an - sen) is a sequence which fails to be spectrally stable. Then either there is an infinite subsequence (an~ sen~ ) whichconsists of non-invertible elements only, or all elements an - Sen with sufficiently large n are invertible, but p((an~ - sen~)-1) -~ oo as k -~ o~ for some subsequence. In the first case one has s ¯ a(an~) for every k and, thus, s ¯ limsupa(an). In the second case one can find numbers tn~ ¯ a(an~) such that Itn~ - s1-1 --~ oo resp. Itn~ - s I -~ 0 as k -+ oc which also implies that s ¯ lim sup a(an). Thus, the determination of the partial limiting set lim sup a(an) requires to investigate the spectral stability of the sequences (an - sen). This problem proves to be muchmore involved than the investigation of normal stability. Even in the (seemingly well understood) case of the finite section method for Toeplitz operators with continuous generating function one is able to determine lim sup a(PnT(a)Pn) only in case a is a polynomial. There is a result by Schmidtand Spitzer ([156] or [27]) whichsays that, in this special situation, lim sup a(PnT(a)Pn)is the union of so-called analytic arcs which mimics - as a skeleton - the shape of a(T(a)). Sometypical pictures are shownin Figure 3.1 where -- resp .... mark the ’boundary’ a(T) of a(T(a)) resp. the partial limiting set lim sup a(PnT(a)Pn). These difficulties disappear if we restrict our attention to sequences for which conventional and spectral stability coincide. Corollary 3.18 If (an) ¯ i7: is a sequence of normal elements, then lim sup a(an) = a~/g((an) Proof. The spectral radius and the norm of a normal element of a C*algebra coincide. Hence, the sequence (an -- sen) is spectrally stable if and only if it is conventionally stable which, by Theorem3.17 and Kozak’s
116
CHAPTER 3.
APPROXIMATION
OF SPECTRA
Figure.3.1: Limiting sets of eigenvalues of Toeplitz matrices theorem, implies that s E limsupa(a~) if and only if s ~ ay/g((a,~) + Weconjecture that this assertion remains valid if the sequence (a,~) is only supposed to be essentially normal in the sense that Ilana~ - a~anll --~ 0 as n -~ oo. In case the an are n × n matrices (i.e. when C = C~×~), then Lin’s theorem tells us that the sequence (an) is essentially normal if and only if a~ = b,~ + g,~ where the b,~ are normal matrices and IIg,~ll -+ 0 as n -~ oo (see [104], Chapter 19). Even in this case the conjecture is open. Let us return once more to the context of in general non-normal sequences. Examples 3.10 and 3.11 indicate that, for every sequence (an) 9v, the partial limiting set limsupa(an) is a subset of a~/g((an) + 6) that one might get different parts of the latter spectrum if one considers the limiting sets of perturbed sequences (an + g,~) with (g,~) ~ 6 as well. The following result showsthat one will indeed obtain all of a:~/g ((an) + if one takes into account all perturbations of (an) by a zero sequence, least in case the a,~ are n by n matrices. Theorem 3.19 Let C,~ = C, n×n and (An) ~ J~. Then U(a,)cg limsup a(A~ + G,~) a~:/g((A,~) + Proof. For the inclusion _C, let A ¢( a:~/g ((A,~)+ 6). Then (An- AIn) (with In referring to the identity matrix) is a conventionally stable sequence,
3.2.
SPECTRA AND THEIR LIMITING
SETS
117
hence, all sequences (An + Gn - Mn) with (G,~) E 6 are conventionally stable, too. But every conventionally stable sequence is spectrally stable, whence, by Theorem 3.17, Thus, ‘k does not belong to the left hand side of the asserted equality. For the reverse inclusion, let ‘k E a:~/g((An) + 6), i.e. suppose the sequence (An - ‘kin) fails to be conventionally stable. Then, either there an infinite subsequence (nk) of the naturals such that none of the matrices An~- ‘kln~ is invertible, in which case (An - ,kin) is not spectrally stable by definition and, hence, ‘k belongs to the partial limiting set lim sup a(An) due to Theorem3.17, or the matrices An - ,kin are invertible for all sufficiently large n, but limsup In the latter case there is a subsequence (n~) of N as well as vectors xn~ E Cn~ with IIx,~ll = 1 such that for k =- l, 2, ... . Set Yn~:= (An~ -‘kIn~)--lXn~ (which is clearly non-zero), and define linear mappings G~ :(~ -+Cn~, 2. Z~--~(z, yn~)Xn~/llyn~[[ Then,
llan~ll =sup[
I II~,~II
(An~- an~- ‘k-~,~)y,,~ = (An~ - ‘kIn~)yn~ - an~Y,~ = ~,~- ~n~ This shows that the sequence (An - Gn - Mn)is not spectrally stable and, consequently, ‘k belongs to limsupa(A, - G,~) for that special (Gn). Applied to the finite section method for Toeplitz operators, this theorem in combination with Corollary 1.36 says that t~(a,~)~g limsupa(P,~(T(a) + g)Pn + Gn) = a(T(a) for every continuous function a and every compact operator K. 3.2.3
The case
of fractal
sequences
The following result provides a condition for the equality of partial and uniform limiting set of spectra. Recall in this connection that, due to Theorem3.7, every sequence in ~" possesses a subsequence for which the partial and the uniform limiting set of the spectra coincide.
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Theorem3.20 Let J~ be as in the previous section, and let .A be a fractal C*-subalgebraof ~: which contains the identity. (a) A sequence (an) E Mis conventionally stable i] and only i] it possesses a conventionally stable (infinite) subsequence. (b) If (a,~) ~ ,4 is normal, then lim sup a(an) = lim inf a(an). (c) If (an) ~ M is normal, then the limit lira p(an) exists and is equal
+ 6). Proof. (a) Let the notations be as in Section 1.6.1. Suppose~ is a strongly monotonically increasing sequence such that the sequence (an(N)) = Rn(a~ ) is stable. Clearly, ~/can be chosenin such a waythat all an(,~) are invertible. Then, by the inverse closedness of An in 9vn, there is a sequence (b,~) ~ such that R~(a,~) . R~(bn) = R~(bn) " Rn(an) =
(3.6)
The canonical homomorphism r : A -> A/ ( A [q 6) is fractal by hypothesis, i.e. ~r = ~rnRv with a certain homomorphism r n. Applying rn to (3.6) one gets the invertibility of n(an) = (an) + A in A/(A ~ g), whencethe stability of (an) follows. (b) Assume, there is a t ~ limsupa(a,~) \ liminfa(an). Then one can a 5 > 0 as well as a strongly monotonically increasing sequence ~ such that dist (t, a(an(~))) >_ 5 for all n. The normality of the an(,~ ) guarantees that (an(, 0 - ten(,~)) is a conventionally stable sequence (with the norms of the inverses being bounded above by 1/5). Then, by part (a), the quence (a,~ - ten) itself is conventionally stable. Hence, t f[ a((an) + 6) limsup a(a~) by Corollary 3.18 which is a contradiction. Assertion (c) is an immediate consequence of part (b) and Proposition ¯
As an application, consider once more the finite section methodfor Toeplitz operators, i.e. specify ~- and 6 appropriately and choose .A -- $(C), which is a fractal algebra by Corollary 1.70. Theorem 3.21 Let (An) ~ $(C) be a normal sequence. liminfa(An) = limsupa(A,~) = a(W(A,~)) t~ a(I;V(A,~)) and lira p(A,~) = max{p(W(A,~)), p(l~d(A~))}. The proof follows immediately from the previous theorem and Corollary 1.58. If, in particular, a E C is real-valued and K ~ K(/2) is self-adjoint then, by Corollary 1.36, liminf
a(Pn(T(a) + K)P~) = limsupa(Pn(T(a)
+ K)Pn) =
3.3. PSEUDOSPECTRA AND THEIR LIMITING
SETS
119
3.2.4 Limiting sets of singular values Let B be a unital C*-algebra and a E B. The set a2(a) of the singular values of a is defined as {A E If{ + : A2 E a(a*a)}. Since the determination of the singular values is equivalent to the determination of the spectrum of a self-adjoint element, the results of the preceding section have the following evident analogues for the singular value sets of arbitrary bounded sequences. The notations are as in Section 3.2.3. Theorem3.22 /~ (an) ~ ~’, then limsupa2(an) = a2((an) Theorem3.23 If A is a fractal C*-subalgebra of ~: containing the identity, and if (an) e .4, then lim sup a2(an) = lim inf a2(a,~), and the sequence (rad a2(an)) converges to rad ((an) + Theorem3.24 /~f (An) e $(C), lim inf a2 (An) = lim sup a2 (An) = (W(An)) t9 a~ (If V(An)) and limrad a2(An) = max {tad a2(W(An)), rad a2(l~(An))}.
3.3 Pseudospectra
and their
limiting
sets
In contrast to commonspectra, the s-pseudospectra which we are going to introduce now are distinguished by a much nicer asymptotic behavior, which allows to approximate the ~-pseudospectrum of an operator A by ~-pseudospectra of approximation operators An also in cases where A is non-normal. 3.3.1
e-invertibility
A computer working with finite accuracy cannot distinguish between a noninvertible matrix and an invertible matrix the inverse of which has a very large norm. This suggests the following definition reflecting finite accuracy. Definition 3.25 Let I3 be a Banach algebra with identity e and let e be a positive constant. An element a ~ 13 is e-invertible if it is invertible and if Ila-lll < lie. The e-pseudospectruma(~)(a) o] a consists of all )~ ~ which a - he is not e-invertible. The usual invertibility and the usual spectrum follow from their e- counterparts by letting e formally go to zero. To be more precise: an element
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a E B is invertible if and only if it is ¢-invertible for all sufficiently small ¢, (e) (a). and a(a) = V~>0a In analogy to the usual invertibility, one can showthat the ¢-invertible elements of a Banach algebra form an open set, and that ¢-pseudospectra are compact and non-empty subsets of the complex plane. As the following theorem shows, the upper semi-continuity of spectra also has its analogue for pseudospectra. Theorem 3.26 Let 13 be a unital Banach algebra and ¢ > O. (a) If (an) C_ B is a norm-convergent sequence, then limsupa (~)(an) C_ a(~) (lim an). (b) The mapping13 ~ a ~-~ (~) (a) C_ Cis upper semi-continuous. Proof. (a) Set a := liman. Given h ii msupa(~)(an), choose a sequence (nk) as well as numbers hn~ ~ a(e)(an~) such that hn~ ~ h as k -~ oo. Thus [l(a,,~ - hn~e)- (a - he)l[ -~ 0 as k -~ ~, (3.7) where e denotes the identity element of B. If there are infinitely manyof the an~ - h~ e which are not invertible then a - he cannot be invertible (the set of the invertible elements is open), thus h a(a) C a(~) (a So suppose that there are only finitely many of the an~ - hn~ e which are not invertible. Then (a,~ - ~n~e)-1 exists for all sufficiently large k, and [](a~- ~,~e)-~ll >_ 1/~. (3.8) If nowa - he is invertible, then (3.7) and the continuity of the inversion imply -~-(a-he)-~ll-~0 II(a~-h,~e) as k--~o~ which in conjunction with (3.8) and the continuity of the norm shows that II(a-)~e)-~ll 1/~, i.e . tha t h b elongs to a(~)(a). If a-his not i nvertible, then evidently h is also in a(~) (a). Assertion (b) follows via Theorem ¯ Nowwe move from general Banach to C*-algebras. In this setting, the following equivalent description of pseudospectra.
we have
Theorem 3.27 Let B be a unital C*-algebra and ~ > O. Then, ]or every a ~ B, the ~-pseudospectrum o] a is equal to a(~) (a) = [.Jpel3,
Ilpll~_¢
a(a+p).
(3.9)
Proof. Abbreviate a(~)(a) to S~. and t~peB, Ilpll<_~ a(a +p) to $2. Our first claim is the inclusion S~ C_ $1.
3.3. PSEUDOSPECTRA AND THEIR LIMITING
SETS
121
Given t E $2, there is a p in B with IlPll -~ z such that a +p - te is not invertible. If t lies in a(a), then it also belongs to $1, and we are done. So suppose t ~ a(a). Then a - te is invertible, and the identity
a + p - te = (a - te)(~ + (a - t~)-lp) yields that e + (a - te)-lp cannot be invertible. Hence, I[(a - te)-lp[I >_ (otherwise invertibility would follow via Neumann’sseries). Because 1 _< [[(a-te)-~p[I <_ [[(a-te)-~[[ ¯ [[p[[, one has II(a-te)-~ll
_> 1/llpll
>
i.e. t E s~ in this case, too. For the reverse inclusion $1 C_ $2 assume there is a t E $1 such that a + p - te is invertible for all p E B with Ilpll _<e. Choosingp = 0 we get the invertibility of a-te -~ and, hence, of a*-~e and, choosing p = 1(a*-~e) with a certain complex number,~ which satisfies
o < I~l _<elll(a*-~e)-~ll, we obtain the invertibility
(3.10)
of
a- te + A(a* - ~e)-~ = A(a- te)(~e + (a- te)-~(a * - -1) ~e) (observe that Ilpll < e in both cases). Hence, ½e + (a - te)-l(a * -~e) -~ is invertible for all A satisfying (3.10). This gives p((a - te)-~(a * - ~e)-1) < I](a* -
~e)-lll/e,
whence, by the self-adjointness of (a- te) -~ (a* -~e) -~ (implying the coincidence of spectral radius and norm),
II(a - te)-~ll2 = II(a -
te)-~(a * -
ie)-~ll < II(a*- ~e)-~ll/e.
-~ II, whichyieldsII(a-te)-lll < 1/e Finallyone has II(a-te)-~ll= II(a*-~e) in contrast to t ~ S~. This contradiction verifies the inclusion S~ Observe that $2 _C $1 in the Banach algebra case, too, whereas the reverse inclusion fails for Banachalgebras in general. Theorem 3.27 also offers a-way for the numerical computation of pseudospectra in case B = C~x’~ by randomly choosing n x n matrices p ~xn with. IlPll -< e and then plotting the eigenvalues of a + p for a ~ C
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Observe furthermore that if B is a unital Banach algebra and A a closed subalgebra of B which contains the identity and which is inverse closed with respect to commoninvertibility, then it is also inverse closed with respect to e-invertibitity, and a(~)(a)=a(~)(a)
for every
In particular, this equality holds if B is a unital C*-algebra and ,4 a C*subalgebra of B which contains the identity element. The remaining part of this subsection is devoted to a further characterization of e-invertibility of linear boundedoperators A on a Hilbert space H. Banach’s theorem on the inverse operator says that A is invertible if and only if Ker A = {0} and Im A = H. In what follows we are going to introduce the e-kernel KereA as well as the e-range ImeA of A E L(H) which share many of their properties with the commonkernel and range of A. Especially we will prove an e-version of Banach’s theorem: A is einvertible if and only if KereA = {0} and Im~A= H. These considerations will not be needed in the sequel and can be ignored without loss. Westart with defining the e-kernel and the e-range for self-adjoint nonnegative operators. As already remarked, an operator A _> 0 is invertible if and only if re(A) := inf{A : ,~ E a(A)} > 0, and in this case IIA-111 re(A) -~. Thus, A _> 0 is e-invertible if and only if m(A) > e. This suggests the following definition: The e-kernel KereA of A is the (-oo, el-spectral subspace of A. Thus, the orthogonal projection from H onto KereA is nothing but the spectral projection E(e) where {E(t)}teR is the partition of identity associated with A (compare Section 2.1.2). By Theorem2.7, (3.11)
Ker~A= clos Im fte (9 Ker (A - eI). : In order to motivate the definition of the e-range of A, recall that (KerA) ± = clos for every self-adjoint
Im A and (ImA) ± = KerA
(3.12)
operator A ~ L(H), and that
(M+N) ± = M±NN± and
(M~IN) ± = clos(M
±+N±)
(3.13)
for arbitrary closed subspaces M, N of H. Thus, taking the orthogonal complement of the decomposition (3.11) yields ± (Ker~A)
= = =
(clos Im f~e ¯ Ker(A- ± eI)) (clos Im f~E)±.r3 (Ker (A- el)) ± (by (3.13)) Kerf~e r3 clos Im (A-el) (by (3.12))
3.3.
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SETS
123
whichjustifies to call Im~A:= Ker fl~ N Im (A- eI) the e-range of A. Observe that Im~Ais not necessarily a closed subspace of H. Proposition 3.28 Let H be a Hilbert space, A E L(H), A >_ O, and e > Then the ]ollowing analogues o] (3.12) hold: (Ker~A) ± = closIm~A
and (Im~A) ± = KereA.
Proof. For every linear (not necessarily (M±)± -- clos M. Hence,
closed) subspace of H one has
clos Im~A---- clos (Ker ~ N Im (A - el)) = ((Ker ~ (A- e I)) ±, which is equal to (clos ((Ker fl~)±÷(Im (A-eI))±) ± by (3.13) and coincides with (clos (clos Im fl~ + Ker (A - el))) ± (3.14) by (3.12). Since the subspaces clos Im f~ and Ker (A - el) are orthogonal to each other by (3.11), their sum is closed, and hence, (3.14) is equal (closIm f~ @Ker (A - el)) ± ±. = (Ker~ A) The second assertion is a simple consequence of the first one.
¯
For introducing the e-kernel and the e-range of an arbitrary operator A E L(H), let A = URstand for the polar decomposition of A (compare Theorem 2.10). Then one has KerA -- KerR and Im A -- UIm R for the commonkernels and ranges (for the first equality observe that if Ax = 0 then Rx = 0 because 0 = (Ax, Ax) = (RU*URx, x) = (R2x, x) = (Rx, and so the following definition makessense. Definition 3.29 Let H be a Hilbert space, A ~ L(H), and e > O. Then the e-kernel Ker~Aand the e-range Im~Ao] A are defined by KereA = KereR and ImeA = UImeR, where A = UR is the polar decomposition o] A and Ker~R and ImeR are defined as above.
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Theorem 3.30 Let H be Hilbert space, A E L(H), and ~ > O. Then A ~-invertible i] and only if Ker~A = {0} and ImeA = H. Proof. Weprove that the following assertions are equivalent: (i) KereA ----- {0} and ImeA= (ii) Ker~R = (0}, ImeR = H, and U is unitary. (iii) is ¢-invertible, and U isunita ry. (iv) is ¢-invertible. Westart with the equivalence (i) ¢~ (ii). If KereA = {0} and Im~A= H then, by definition, KereR = {0} and UImeR = H. Since Ker~R is the (-cx),¢]-spectral subspace of R and ¢ > 0, we have Ker R C_ Ker~R and, hence, Kerr = {0}. Since R is self-adjoint, this implies by (3.12) that clos Im R = H, and since R and U are specified in such a way that clos Im R = (Ker U)± (see Theorem 2.10), we conclude that KerU = {0}. Moreover, as we have already seen, Im U = H, hence, U is unitary. This finally yields ImeR=U*ImeA= H. The reverse implication (ii) :¢ (i) is evident. For the implication (ii) ~ (iii), suppose R to be a self-adjoint and nonnegative operator with Ker~R = {0} and Im~R = H. The kernel condition implies via the Spectral Theorem 2.7 that ((R- el)x, x) >_ for ev ery x E (Ker~R)± = H, whence (Rx, x) >_ ¢ Ilxll 2 and thus IIRxll
_>ellxll
for every x e H.
(3.15)
By Lemma1.21, this gives that Kerr = {0} and ImR is closed, which, due to the self-adjointness of R and by (3.12), involves the invertibility of R. Setting x = R-ly in (3.15) we obtain [lYll > ¢[IR-lYll resp. IIR-lyll <_ (1/¢)llyll for all y ~ H, i.e. IIR-111 _< 1/¢. It remains show that I]R-111 cannot be equal to 1/¢. Assumethe contrary, that is, IIR-111 = 1/¢. Because R-1 is non-negative, one then has p(R-1) = 1/e and 1/~ ~ a(R-~), hence, R -¢I cannot be invertible. On the other hand, the hypotheses Ker~R = clos Im f~e @Ker (R- el) = {0}, Im~R = Kerf~ fq Im (R- el)
=
imply Ker (R - ¢I) = {0} and Im (R - eI) = H, i.e. the invertibility of R - ¢I. This contradiction shows that IIR-111 < 1/~, i.e. that R is ¢-invertible. Let, conversely, R be a non-negative ¢-invertible operator. Then, as we have already remarked, ¢ < re(R) := min{,k : ,k ~ a(R)}, which immediately implies that the (-cx~, el-spectral subspace of R (i.e. Ker~R)consists
3.3.
PSEUDOSPECTRA AND THEIR LIMITING
SETS
125
of the zero element only. By Theorem 2.7 we then have Ker~R = clos Im f~ @ Ker (R-~I)
= {0},
which shows that f~ is the zero operator. Consequently, Im~R= Ker ~ t3 Im (R - ~I) = H C~ Im (R - ~I), and Im (R- ¢I) = H because ~ m(R). Hence, R- ¢I is inv ertible. Thu s, (ii) and (iii) are equivalent. Finally, we have to check whether (iii) ¢* (iv). Observe that if A is invertible and A = URis the polar decomposition of A, then U is unitary and R is invertible, and conversely. Further, for invertible A one has [[A-I[[ 2 = [[(A-1)*A-I[[ = [[UR-1R-1U*[[= 2, [[R-2[[ = I[R-I[[ thus, A and R are ¢-invertible only simultaneously. 3.3.2
Limiting
sets
¯
of pseudospectra
Let again (Cn) be a family of unital C*-algebras with product ~ and restricted product G. The main result of this section identifies the limes superior of the ~-pseudospectra of the elements of a sequence (an) E with the ~-pseudospectrum of the coset (an) + in ~/~. Theorem 3.31 Let (an) E ~ and ~ > O. Then lim sup a~) (an)
=
+
Recall that in case ¢ -- 0, i.e. for common spectra, this result does not hold in general. The proof of Theorem3.31 is based on the following result. Theorem3.32 (Daniluk) Let 13 be a C*-algebra with identity e, let a ~ B, and suppose a - Ae to be invertible for all A in some open subset U of the complex plane. /f [[(a- Ae)-ll] <_ C for all A e U, then [[(a - Ae)-ll[ < for all A ~ U. In other words: the analytic function U --~ B, A ~ (a - Ae)-1 satisfies the maximum principle. This is a surprising result since - in contrast to complex-valued analytic functions - the maximumprinciple for operatorvalued analytic functions does not hold in general as the simple example
°1)
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OF SPECTRA
reveals. For a proof of Theorem3.32 in a more general context see Section 3.3.4. Proof of Theorem 3.31. Let ~ E a(~g((an) -t- Theneithe r (an hen) ÷ G is not invertible, or the inverse of this coset exists, but II((an
~e~)+ 6)-1II > 1/~. In the first case, the sequence (an - £en) fails to be stable, and follows easily the existence of an infinite subsequence (nk) of N such that ~ ~ a(~)(an~) for every k. In particular, ~ e limsupa(e)(an) in this case. In the second case, we infer from Daniluk’s theorem that, in every open neighbourhoodU of A, there is a ~0 such that II((an Aoen) + ~)-1 II > 1/~ (Otherwise we would have II((an-~oen)+6)-lll <_ 1/~ for all ~0 e U involves via Theorem3.32 that II((an- Aoen)÷6)-111 < 1/~ for all ~o ~ U including ~0 = A.) Thus, for all sufficiently large k, there are numbersAk with £k --~ ~ as k -~ ~ such that
II((a~Ak e~) +G)-~I[ > 1/(e -
l/ k).
By Proposition 1.16, this is equivalent to the inequality lim sup II(an- )~ken)-~ll >_1/(e- Ilk) (the invertibility of an- )~ken for all sufficiently large n being a consequence of a Neumannseries argument). Since 1/e < 1/(e- l/k), there are numbers nk tending to infinity as k -+ cx~ such that II(a~ - ~ken~)-lll _> 1/~ for all in other words, )~k ~ (~) (an~) for every sufficiently l arge k, which i mplies that ,k = lim ,~ E lim sup a(~) (a~). For the reverse inclusion, let A ~ limsupa(e)(a,~), but assume in the contrary that ,~ ¢ e2/~((an) -(~ + ~). Thus, (a,~ - Ae~) + G is invertible ~r/6 and II((an - )~en) 6)- 11= 1/e- 25< 1/e with s ome 5> 0. By Polski’s theorem, the an - ;~en are invertible for sufficiently large n, and by Proposition 1.16 again, lim sup [[(an - Aen)-ll[ = 1/¢ whenceII(an - Aen)-~[I < 1/e - 5 for n sufficiently large, say n _> no. If n > no and [A - #1 < e~(1/e - 5)-1 then one gets, using Neumann’sseries again, II(a~ -,e~)-lll
< -
a~II( 1 - I,~ -/t[ [[(an -- &en)-lll
3.3.
PSEUDOSPECTRA AND THEIR LIMITING
SETS
127
1 - e5(1/e - 6) -1 (1/e - 6) and hence ~ ¢ aCe) (an) for all/~ belonging to a certain open neighbourhood of A and for all sufficiently large n. But then A cannot belong to the partial limiting set of the e-pseudospectra a(¢) (an), which is a contradiction. For a more general result, see Section 3.3.4. 3.3.3
The case
of fractal
sequences
In presence of a fractal sequence (an), one will expect equality between the partial and the uniform limiting sets of the pseudospectra of the an and, indeed, this equality is true as we will see now. Weagree upon calling a sequence (a,~) E c ~-stable i f t he coset ( an) +G is ~-invertible in .~/ Theorem 3.33 Let A be a fractal C*-subalgebra of yr which contains the identity of J:, and let ~ > O. (a) A sequence (an) ~ ‘4 is e-stable if and only if it possesses an e-stable subsequence. (b) If (an) ~ ‘4, then limsupa(e)(an) liminf a(e )(an). (c) If (an) ~ .4, then limrada(e)(an) exists and is equal to rada(e)((an) Proof. (a) Let the notations be as in Section 1.6.1. Suppose~/is a strongly monotonically increasing sequence such that (an(n)) = Rn(an) is an ~-stable subsequence of (an). Then the sequence (an) is stable by Theorem3.20(a), and Theorem 1.71 implies that 1/~ >
II((an(n )) + G)-lll/a = II((an)
i.e. (an) is ~-stable. (b) Assumea numbert belongs to lim sup (~) (an) \ li ra in f a( ~) (a n). Th there is a subsequence~/of N such that t ¢ lim sup a(~) (an(,~)). By Theorem 3.31, the sequence (an(n) te n(n)) is ~-stable, whence by assertion (a) the e-stability of the complete sequence (a~ - ten) follows. Again by Theorem 3.31 , this implies t ¢ lira sup a(~) (an), which contradicts the assumption. Assertion (c) is an immediate consequence of part (b) and Proposition As an application, we consider once more the finite section method for Toeplitz operators, i.e. we specify 2- and G accordingly and let .4 = S(C), which is a fractal algebra by Corollary 1.70.
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Theorem 3.34 Let (An) e S(C) and ~ > O. lira sup a(~) (An) li m inf a (~) (An) -- (~)(W(An )) (~ a (~) (IYV(An)) and lim tad a(~) (An) = max{tad (~) (W(A,~)), t(~) (I2V (An)) In particular, i] a E C, then limsupa(~)(PnT(a)Pn) and
= liminfa(~)(PnT(a)Pn)
lim rad a (~) (PnT(a)Pn) = rad a (~) (T(a)). Proof. The first assertion is a consequence of Theorem3.33 and Corollary 1.58, and the second assertion follows from identity (1.32) which implies that IIT(5- A)-lll = I]T(a- .~)-11[ for all A e C \ a(T(a)). Beautiful plots of pseudospectra of several operators (including Toeplitz operators) as well as of limiting sets of pseudospectra can be seen in [129], [172], [171] and in [27], Chapter 3. In the proof of the previous theorem, we employed the obvious fact that, for every ordered pair (A, B) viewed of as an element of the algebra L(l 2) x L(12), one has a(~)(A, B) = a(~)(A)t~a(~)(B). An analogous result is valid for finitely manyoperators. As a corollary of Theorem5.39 below we will derive a generalization to the case of infinitely manyoperators. 3.3.4
Pseudospectra
of
operator
polynomials
Weare now going to extend the results of the previous subsections to the case of operator polynomials. Let B be a C*-algebra with identity e, and let b0, bl .... ,bm elements of B. Then, for every complex number )~, the expression L(A) = bo + b~A +... + bmAm is well-defined, and L is (not quite correctly) called an operator polynomial with coefficients in B. The spectrum of L is the set a(L) of all A ~ C for which L(A) is not invertible; similarly, for ~ > 0, the ~-pseudospectrumof L is the set a(~) (L) of all ~ ~ C such that either L(A) is not invertible II(L(A))-lll >_ 1/~. Evidently, if L()~) -- b - )~e, then the spectrum ~-pseudospectrum of the polynomial L coincide with the commonspectrum and ~-pseudospectrum of the element b ~ B, respectively. The following theorem is the analogue of Theorem3.27, and it can be proved in the very same way. So, we omit the proof.
3.3. PSEUDOSPECTRA AND THEIR LIMITING
SETS
129
Theorem 3.35 Let B be a unital C*-algebra and ~ > O. Then, for every operator polynomial L with coe~cients in B, a(e) (L) = t3peB a(L+p). ’ i1~11_<~, Clearly, L+p refers to the polynomial (L+p)( A) = (bo+p)+blA+. . TM. Let now C1, C2,... be a family of unital C*-algebras. Denote by ~ and G their product and their resticted product, respectively, and let (a~)), (a~l)), ..., (a~m)) be sequences in ~. Wesuppose that m > 0 and that the sequence (a~m)) is stable. For every n, consider the polynomial with coefficients in Cn given by TM .a~’)A + ... Ln(A) := a~°) + + a~)~
(3.16)
As for the commonspectra, we would like to identify the limiting set of the pseudospectra of the polynomials L~ with the pseudospectrum of something in the quotient algebra ~/G. It turns out that this is indeed possible, and that the something is the polynomial n(x) := ((a~)) + G) ’) ) + G)X +... + ((a~ m)) + G)$m (3. 17). with coefficients in ~/~. Theorem3.36 Let m > O, let (a~)), (a~)), ..., ~)) be sequences in ~ with (a~m)) being stable, and let L~ and L be the polynomials (3.16) and (3.17). Then, ]or eve~ e > lim supa(~)~ rc. ,~=a~}g(L)" For the proof, we need a generalization we will present and prove now.
of Daniluk’s Theorem3.32, which
Theorem 3.37 Let B be a C*-algebra with identity e, let m > 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~
130
CHAPTER 3.
APPROXIMATION
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 <_~ r2~IIXT-~-~Y flI~ <_~. M211fll j>o Let e > 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
132
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
+...
m lain
~ ::C~
with a constant C depending on the polynomial Q and the radius r only. Letting e go to zero in 1 = IIAII <_ Ilaoa~lao[I " Ila~ama~IA[I <_ Cllaoa~aolle
3.3.
PSEUDOSPECTRA AND THEIR LIMITING
SETS
we arrive at a contradiction.
133 ¯
Having this result at our disposal, we can prove Theorem3.36 by repeating the arguments of the proof of Theorem3.31. Also the proof of the following result proceeds completely analogously to that of its counterpart for commonspectra, Theorem 3.33. Theorem3.38 Let ,4 be a fractal C*-subalgebra of jr which contains the identity, let ~ > O, and let the polynomials Ln be as in (3.16) with the additional requirement that all coefficient sequences (a(n j)) belong to A. Then lim sup a(~) (Ln) = lim inf (~) (Ln). Weconclude by applying these results to our standard example, the finite section method for Toeplitz operators. Let f0, fl, ..., f,~ be continuous functions and suppose that the Toeplitz operator T(f,~) is invertible. Consider the matrix resp. operator polynomials "~, Ln(A) := T~(f0) + Tn(f~))~ +... + Tn(f,~)~ P(A) := m. T(fo) + T(f~))~ +... + T(f,~)A Theorem 3.39 Let the functions fj and the polynomials Ln and P be as above. Then, for every ~ > 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) <_Ilbll, where e is Euler’s constant 2.71... (which is muchstronger than the estimate 0 <_p(b) <_IIbll ]or the spectral radius). (c) For all a, b E 13 one has AN(a ÷ b) C_ AN(a) ÷ AN(b) (which counterpart ]or spectra). (d) If ff is a closed ideal o] B, then AN~/j(b+ if) = ~ AN~ (b+ j) ]or every b e B (which also Jails for spectra in general). Nowwe turn over to spatial numerical ranges. The basic and classical result is the following theorem. Theorem 3.44 (Toeplitz/Hausdorff/Stone) of every operator A ~ L(H) is convex.
The spatial
numerical range
For a proof see [791 or [45]. Thus, there are two notions of numerical ranges for operators A on a Hilbert space H: the spatial one, whenA is considered as an operator, and the algebraic one, when A is viewed as an element of the C*-algebra L(H). The relation between these two notions is as follows: Theorem 3.45 Let H be a Hilbert
space and A ~ L(H). Then
ANL(H) (A) = clos SNH(A).
CHAPTER 3.
136 3.4.2
Limiting
sets
of
APPROXIMATION
numerical
OF SPECTRA
ranges
The main goal of this subsection is a result which connects the partial limiting set of the numerical ranges of an approximation sequence with the numerical range of the coset (An) modulo zero sequences. Again we will not only deal with operators, but with elements of arbitrary C*algebras, i.e. we let (Cn) be a family of unital C*-algebras with product and restricted product 6. Theorem 3.46 For every sequence (an) E 3: one has conv lim sup ANc~(an) = AN~=/g((an) Observe that the partial limiting set of a sequence of convex sets not be convex again, which explains the conv operator on the left side. For the proof of Theorem 3.46 we need two auxiliary results: describing an elementary convexity property of sequences of convex and one characterizing the numerical range of an element of 3:.
need hand one sets
Lemma3.47 Let ( Mk ) be a monotonically decreasing sequence of compact subsets of C, i.e. MkD_ Mk+1 for all k. Then conv Ok Mk : ~k cony Mk. Proof. Since V~kMk_C ¢~k conv Mkand the intersection of convex sets is convex again, the inclusion conv ~k Mk C_ g~k cony Mkis evident. Conversely, let m E g~kconv Mk. Then, for every k, there are elements m~k) and m~k) in Mkas well as non-negative numbers A~k) , A~k) with ),~k) A~k) = 1 such that (k) .(k) m = A.(k) (3.22) 1 m1 -k A2 m2 ¯ The compactness and the monotonicity of the Mk entail the existence of an infinite subsequence N1 of N such that (m~k))ker~l is a convergent subsequence of (m~k))keW with limit ml, of an infinite subsequence N2 of such that (m~k))ker% is a convergent subsequence of (m~k))keN, with limit m2, and of an infinite subsequence N3 of N2 such that (A~k))keN3is a convergent subsequence of (A~k))keN2 with limit A~. One easily checks that rn~,m2 ~ VlkMk and that A1 _> 0 and A2. := 1 - ),1 _> 0. Thus, letting k ~ N3 go to infinity in (3.22), one obtains m = Alm~ + A2m2, whence m ~ conv V~k Mk. ¯ Theorem 3.48 For every sequence (an) ~ 3: one has AN~:((an)) = conv clos Uk ANc~ (a~).
3.4.
NUMERICAL RANGES AND THEIR LIMITING
SETS
137
Proof. Let m ¯ ANc~(ak) for some fixed k, and let ¢ be a state of Ck with ¢(a~) = m. Then co ~ ¢(bk) ¢~ : ~: -~ C, (b,,),~= 1 is a state of 5v, and ¢¢((a,~)) = ¢(ak) ---- m. Thus, m ¯ AN~:’((an)), which implies that UkANc~(ak) C_ ANy((an)). Since ANy((an)) is a closed convex set by Proposition 3.40, we arrive at the inclusion conv clos ANc~(ak) C_ Ag:r((a,~)). For the reverse inclusion, we think of Ck as a C*-algebra of linear bounded operators on some Hilbert space Hk (which is possible by the GNS-construction, Theorem 1.48). Let ~H~ refer to the orthogonal sum of the Hilbert spaces Hk, i.e. the elements of (~Hk are the sequences (Xk) of vectors Xk ¯ Hk satisfying ~o=1 [[Xk[[ 2 < OO, and (~Hk is made into a Hilbert space on defining element-wise operations as well as an inner product by
:= (xk, Evidently, the elements of 9v can be identified with diagonal operators acting on the orthogonal sum @Hkin an obvious manner. For simplicity, we denote the diagonal operator which corresponds to the sequence (a~) by (ak) again. From the inverse closedness (Proposition 3.41) and Theorem 3.45 in. fer that Agy((an)) = ANL($H~)((an)) = closSNsg~((a,~)).
(3.23)
Thus, given m ¯ ANy((an)) and e > 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.
ifif nn<-k - l>_k
and, hence, AN.r/6((a,~) + 6) f~=lconvclos t2, ~>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 $(PC) we are interested
in is contained in ~-w.
As a guide how to construct the algebra ~w can serve the algebra ~-c with its ideal ~c introduced in Section 1.2.4. Recall that, in order to make to an ideal, we had to diminish the algebra ~ by requiring the existence of certain strong limits. In analogy, we let ~-w stand for the collection of all sequences (An) ~ J: for which there exist two operators W(An) and l~d(An) in L(I 2) such that AnPn --~ W(An), A~Pn --~ W(An)*, RnAnRn -~ IfV(An),
RnA~Rn --+ IfV(An)*
in the sense of the strong convergence. Then one has the following counterpart to Theorems 1.18 and 1.19. Theorem 4.22 (a) :~w is a C*-subalgebra o] J: which ~ontains the identity clement o] ~. (b) The mappings W : :W - ~ L(12), ( An) ~s- lim,~-~ooA,~Pn, an d IT ~rw _~ L(12), (An) ~ s-limn-,~oRnAnRn, are *-homomorphisms. (c) 6Wis a closed ideal o] :~w. Proof. The proof is quite similar to those of Theorems1.18 and 1.19; so we will indicate a few points only. The inclusion ~w C ~w was shown in the proof of Theorem1.53, where we moreover verified that W(PnKP,~+R,~LRn+Gn)
= K, ITV(PnKPn+RnLRn+Gn)
(4 .9
whenever K and L are compact and HG,~[[ tends to zero. The identities (4.9) together with the estimates
IIW(An)l] <_liminfliA,~P,~ll _~supIlA,~P,~ll = II(A,~)ll~, II~(An)ll_
4.2. FINITE
SECTIONS-
161
PC FUNCTIONS
Westill check one half of the ideal property. (PnKPn + RnLRn + Gn) ~ W t hen
If (An) ~ ~-w and
An(PngPn + RnLRn + Gn) = AnPnKPn + RnRnAnRnnRn + AnGn = Pn(AnPn - W(An))KP~ + R,(P~A,Rn - I;V(An))LRn + AnGn + PnW(An)KPn + RnITV(An)LRn. (4.10) Since AnPn - W(An) ~ an d RnAnRn - 17V(An) ~ 0, we ded uce fro m Lemma1.5 (b) that I](AnPn - W(An))KII -’~ and I$(RnAnRn - IT V(An))LI] -~ as n --~ oo. Thus, the first three terms on the right hand side of (4.10) go zero in the norm, and the last two terms are of the form PnKPn+ RnLRn with K and L compact. . So problem (A) is solved. The following result, which can be regarded an analogue of the perturbation theorem 1.20, solves problem (B). Theorem 4.23 (Lifting theorem) A sequence (An) ~ W is sta ble if and only if the operators W(An)and ITV(An) are invertible L(/2), andif t he coset (An) + W i s i nvertible i n t he q uotient a lgebra Yrw / Gw. Wewill also say that the ideal 6w is lifted by the homomorphismsWand l~ r. For more general versions of the lifting theorem we refer to Section 5.3.2. Proof. Let (An) be a stable sequence. Then, by Kozak’s theorem, the coset (An) + is invertible in ~’/ ~ and, hen ce (in verse clo sedness of C*algebras) in yrw/G. Let (Bn) be a sequence in ~-w such that (An)(Bn) (Pn) + (Gn), (S n)(An) = (P
(4.11)
with certain sequences (Gn), (Hn) fi ~. Applying the unital *- homomorphisms Wand l~ to both sides of the identities (4.11) we get W(An)W(Bn)
= W(B,)W(An)
= W(Pn)
17V(An)W(Bn) = 17V(B,,)ITV(An) = 17V(Pn) i.e. W(An) and 17V(An) are invertible. Similarly, applying the canonical homomorphismfrom 5vw onto .TW/6W to (4.11), one obtains the invertibility of the coset (An) + ~w.
CHAPTER 4.
162
STABILITY
ANALYSIS
Let, conversely, (Bn) + Wbe t he i nverse o f ( An) +W,let W(An) and l]d(An) be invertible operators, and abbreviate the operator W(An)-1 W(Bn) to M and the operator I]d(A,~) -1 - l~(Bn) to N. Weclaim that (Bn + PnMP,~ + R,~NRn) +
(4.12)
is the inverse of (An) + in .Tw/6. Indeed, ((An) + 6)((Bn + PnMPn + RnNRn) = (P,~ + Pn(W(A,~B,,) - I)Pn + R,~(I;V(A,~Bn) + AnPnMPn + A,~RnNRn) + 6).
(4.13)
Since (Bn) is an inverse of (An) modulo W, t here a re c ompact operators K and L as well as a zero sequence (Gn) such that (AnBn) = (Pn + PnKPn + RnLRn + Gn). Applying the homomorphismsWand 12d to this equality, the compact operators K and L: K = W(A,~Bn)-I,
we can identify
n = I~V(AnBn)-I.
Thus, the operators M = W(dn)-~(I N = 12d(An)-l(I
- W(AnBn))
= -W(An)-’K,
I; V(AnBn) = -l ~F(An)-~g
are compact and, since AnPn - W(An) -). strongly as n -+ cx), we conclude that
and RnAnRn - I~V(An) -- + 0
AnPnMPn = PnW(An)MPn + Pn(AnPn = Pn(I-w(anBn))Pn
- W(An))MPn (4.14)
with (Gn) E and, an alogously, AnRnNR,~ = R,~(I - 12d(AnB,~))Rn
(4.15)
with (Hn) E 6. Inserting (4.14) and (4.15) into (4.13) shows that the (4.12) is an inverse of (An) ÷ 6 from the right hand side. The invertibility from the left hand side follows similarly. Let us finally remark that the inclusion S(PC) C_ W is evident sin ce PnT(a)Pn --~ T(a) and RnPnT(a)PnRn = PnT(~)Pn T(5) even for arbitrary a ~ L~ (’l~), which solves problem (C) mentioned before.
4.2.
FINITE SECTIONS - PC FUNCTIONS
163
4.2.2 Application of the local principle The lifting theorem allows us to reduce invertibility problems in S(PC)/G to invertibility problems in S(PC)/GW, which implies an essential simplification since the ideal 6w has been introduced in such a way that the cosets (PnT(f)Pn) ÷ with a c ontinuous fun ction f b elong to thecent er of S(PC)/6 W. The so-forced non-triviality of the center of W $(PC)/6 opens the way of localizing the invertibility problem in S(PC)/GW. In complete analogy to the localization in T(PC)/K(I 2) we have Proposition 4.24 The setC of all cosets (P,~T(f)P~)+6 w with f E C(T) is a C*-subalgebra of the center of $(PC)/~ W which is *-isomorphic to C(V), the isomorphism being (PnT(f)P,~) + ~w ~.~ Proof. Widom’s formula (Lemma 1.52) and the compactness of Hankel operators with continuous generating function (Lemma1.33) involve that C is a symmetric algebra in the center of $(PC)/GW and that the mapping ~r : C(T) S( PC)/6 W, f ~ (P ~T(f)P~) +
W
is a *-homomorphism. It remains to check that the kernel of ~r is trivial. Weclaim that liT(f)
g(/2)]]T(pc)/g(t2)
~_II( PnT(f)P,~) + G IIw$( PC)/gw (4.1 6)
for every f e C(T). Given e > 0, one can find compact operators K and L as well as a sequence (G~) in 6 such that $](P~T(f)P~+PnKP~+RnLRn+G~)I]~: <_ ]](P~T(f)P~)+6w]]~w/gw Application of the homomorphismWto the sequence on the left hand side yields
liT(f) ÷ KILL(,2)< I](P~T(f)Pn) ÷ GwI]~w/~w (recall that ]]W]] = 1) and, consequently, liT(f)
+ g(l:)llL(t2)/g(t:)
<_ II(PnT(f)Pn) + Gwll~w/~w
This estimate holds for every e > 0, which gives our claim (4.16). Combining (4.16) with Proposition 4.14 one obtains
$1fll~ < ,,II(P,~T(f)P,~)÷ GWll.~,lg~ showingthat the kernel of 7r is trivial and, hence, that 7r is a *-isomorphism and C is a closed subalgebra of $(PC)/~ w. ¯
164
CHAPTER4.
STABILITY ANALYSIS
As a consequence of the preceding proposition, the maximalideals of C are in one-to-one correspondence with the points of the unit circle: Tgx ~{(PnT(f)Pn)+~w
: f ~C(~),
f(x)=O}
EM(C).
(4.17)
Given x E T denote, in accordance with the local principle, by .7, the smallest closed ideal of 8(PC)/~W which contains the maximal ideal (4.17), and ¯ abbreviate the quotient algebra ($(PC)/6w)/~7, to $, and the canonical homomorphismfrom 8(PC) onto $~ to a2~. Further, let X, be the piecewise constant function introduced in Section 4.1.3. Proposition 4.25 Every local algebra S, is singly generated, and the coset q~x(PnT(x,)Pn) is a generator Proof. As in the proof of Proposition 4.15, one can verify that a2,(P~T(a)Pn) = a(x + O)+=(PnT(x=)Pn) + a(x - O)O=(PnT(1 for every piecewise continuous function a. For a complete description of ,S~ via the Gelfand-Naimark theorem for singly generated C*-algebras we still have to determine the spectrum of the generator of Proposition
4.26 as.(&.(PnT(x.)Pn))
Proof. We start
= [0, 1].
with showing that a((P,~T(x,)P~ ) + g) = [0, 1].
(4.1s)
Let A ~ C. If (PnT(x~)Pn)-A(Pn)+G = (PnT(x,-A)Pn)+G is an invertible coset, i.e. if the finite section methodapplies to T(X, - A), then T(X~-A)is invertible by Polski’s theorem, which involves that A ~ C\[0, 1]. Let, conversely, A ~ C\ [0, 1]. If Im ). = 0, then X~- A is either positive or negative definite, hence T(X, - A) is either positive or negative definite, which yields the applicability of the finite section methodto T(X~ - ~) due to Theorem1.10(b). In case Im A ~ 0, write X~ - A --
Im A --\1+i\ i
ImA
Then T(X~-)O= --~--ImA (I+iT\(X~ImA A)):Re
4.2.
FINITE SECTIONS-
165
PC FUNCTIONS
with the self-adjoint operatorT[x’-~eM -~ Im which implies convergence of the finite section methodfor T(X~ - )0 again via Theorem1.10(b). This proves our claim (4.18). The local principle now gives as. (~(PnT(x~)P,~)) [0, 1] foreveryx E q~.Forthereverse inclusion, assumethereexistsan x and a A ~ [0, 1] whichdoesnot belongto as, (~(PnT(X~)Pn)). We will see thatthenA cannotbelongto the spectrum of ~r~(T(x±)) in ~ (see Proposition 4.18), whichis a contradiction. If ~(PnT(x~ A)Pn) is invertible in S~,then there existsequen ces (Bn),(In)and (Jn)$(PC) with ¢~( In) = (J ~n) = such 0 (P,~T(x~-A)Pn)B,~ = P,~÷In,Bn(PnT(x~-A)P,~) = (4.19) Clearly, the operators W(Bn), W(In) and W(J,~) belong to the algebra T(PC). Further, due to the definition of the ideals ~w and :7~, the sequence (In) can be approximated in ~ as closely as desired by sequences of the form J
~,(A~))(P,~T(.f(~))P,~)
+ (P,~KPn + R,~LR,~
(4.20)
where (A~)) ,. .q(PC), f( J) ~ C(with f (J)(x ) = O, Kand Lar e compact, and (Gn) ~ G. The strong limit of the sequence (4.20) as n -+ oo ~ W(A~))T(.f 0)) + K, the operators W(A~)) belonging to the Toeplitz algebra T(PC). Hence,
for every sequence of the form (4.20), which implies that ~r,(W(I,~)) and, analogously, ~r~(W(J,~)) = Applying the homomorphisms W and ~r~ to (4.19) we thus obtain ~r~(T(xz A))-~r,(W(Bn)) = r, (W(B,~)) . i.e. the coset ~r~(T(x, A)) is invertible in tion 4.18.
T~,which contradicts Prop osi¯
Wewill now complete the proof of Theorem 4.20 by verifying that the Fredholmness of the operator W(A,~) already implies the invertibility of the coset (An) + Gwfor arbitrary sequences (A,~) S(PC). After th is, Theorem4.20 follows immediately from the lifting theorem. Weshall even prove a little bit more, namely:
166
CHAPTER4.
STABILITY ANALYSIS and $(PC)/6 w are *-
Theorem 4.27 The C*-algebras T(PC)/K(12) isomorphic, the isomorphism being given by ~: 8(PC)/~ w -+ 7-(PC)/g(12),
(A,~)
+ ~w ~ W(An)
Proof. It is easy to see that the coset W(A,~)+K(/2) depends on the coset of (A,~) moduloGwonly; so the definition of ~ is correct, and it is also clear that ~ is a *-homomorphism.Wewill show that ~ is even an isometry. Consider cosets of the form I
(An)
J
+ ~w = E (PnT(aij)Pn) +
~w
(4.21)
i----1 j--1
where I, J E Z+ and aij ~ PC. The local principle in connection with the description of the local algebras 8~ yields the following invertibility condition for the coset (4.21): This coset is invertible if and only there no x on "~" such that the function I
[0,
1]-~
J
(4.22)
C, t~-~II(aij(x+O)t+aii(x-O)(1-t)) i=1 j----1
has a zero on [0, 1]. On the other hand, one has I
~((dn)
J
+ ~w) = EII T(aij)
÷
and comparing the invertibility criterion (4.22) for (An) Wwit h the criterion (4.8) for the coset ~((An) ~w), we see that (An) ÷ 6W and ~((An) W)are invertible or n ot onlysimultaneously. Hence , aS(PC)/gw
((An) W)= ¢r T(pC)/K(12)(~((An )
÷ 6W ))
(4.23)
for all cosets of the form (4.21). Nowobserve that, if (A,~) Wis of the W form (4.21), then (An)*(An)+6W is of the same form. But (A,~)*(A,~)+6 is self-adjoint, and the normof a self-adjoint element coi.ncides with its spectral radius. Thus (4.23) yields that
II(A,~)2GW ll = II(A,~)*(A,,) ÷ ~Wll -- 1[5((An)*(An) ~W )ll --II~((an) ÷ G2 for all sequences (An) in a dense subset of S(PC). Since the norm is continuous, the equality ll(A~)÷~w[I : II~((A~)÷~W)ll holds for arbitrary sequences (An) in $(PC), i.e. ~ is an isometry. ¯
4.2. FINITE 4.2.3
SECTIONS - PC FUNCTIONS
167
Galerkin methods with spline for singular integral equations
construction
As a first application of the stability criterion derived for the finite section method for Toeplitz operators, we consider a simple spline Galerkin method for the approximate solution of the singular integral equation (aI+bS)u
= ], ] L2(I~+),
(4.24)
where a and b are complex constants and S is the operator of singular integration against ~+, 1 fo ~° u(s)
=
t
the integral being understood as a Cauchyprincipal value integral. +) For n = 1, 2,..., let S,~ refer to the smallest closed subspace of L2(I~ which contains all functions X[k/n,(k+l)/n) with k = 0, 1,..., where X[~,~) denotes the characteristic ]unction of the interval [a, fl), 1 if t~[a,/~) X[o,,~) (t) = 0 if t ~’ [a,/~), and write Ln for the orthogonal projection from L2(~+) onto the spline space Sn. It is elementary to check that L,~ --+ I strongly as n To solve (4.24) approximately by the spline-Galerkin method, one seeks approximate solutions u~ E S~ of (4.24) by requiring that L,~ (aI + bS) Un = Lnf. The Galerkin method (Ln(aI + bS)ls~) is an approximation method for aI + bS because L,~ --+ I, and what we will examine in what follows is the stability of the sequence (L,~(aI It proves to be advantegeous to translate this sequence into a sequence of operators acting on l 2. That this is indeed possible follows from the identity
~=0
xkX[k/n’(k+l)/n)
L2C]P.+) = ~ I1(~)~°=011’~’
which holds for arbitrary functions ~ XkX[k/n,(k+l)/n trary sequences (x~) E ~. Thus, t he mappings En : 12 "-~ Sn,
(Xk)
) ~ Sn
r-~ ~ E XkX[k/n,(k+l)/n),
resp. for arbi-
168
CHAPTER4.
STABILITY ANALYSIS 1
are isometries, and E_,~E,~ = I(E L(/2)) and E~E-n = I]s~. Consequently, the sequence (Ln(aI + bS)ls~) is stable if and only if the sequence (E_nLn(aI+bS)E,~)of operators on l 2 is sta.ble. Let us consider these operators in detail. For the entries of the infinite matrix E-nLn(aI + bS)E,~ =: Pjk )j,k-=O one finds p(-) jk
~
n. ((aI + bS)X[k/n,(k+~)/n),
~+((aI
+ bS)x[o,1))(t +j -- k)
X[O,1)t)
dr.
This simple identity has remarkable consequences. First, the operators E_,~Ln(aI ÷ bS)E~ are independent of n, thus, the sequence (E_~L,~(aI bS)E~) is actually constant and, therefore, it is stable if and only if its ’generating’ operator, E_~L~(aI + bS)E~ say, is invertible. The second observation is that ,(1) depends on the difference j-k only. Thus, E_~L~(aI+ bS)EI is a bounded Toeplitz operator on 12. A somewhat tedious derivation yields that the generating function c of this Toeplitz operator is given by c(e ~) = a+ba(y), y [0 ,1), (4.25) where
o(y)
2sin ny
sgn (m + 1/2) rn~Z
(y
(see, e.g. [77], Section 2.2.3, for details). The function a is monotonically increasing from -1 to 1 on [0, 1], hence, c is a piecewise continuous function on ~ which has its only discontinuity at 1 ~ ~ (where it jumps from a- b to a+b), and which is continuous only in case b = 0. Theorem4.12 yields: Theorem 4.28 The Galerkin method (Ln(aI+bS)ls~) is stable i] and iIO ¢ [a = b,a + b]. Similar results hold for related approxi~nation methods such as spline collocation or qualocation methods. The only point which requires a modification is the function a (but in any case this function remains piecewise continuous with a jump at 1 ~ ~). For a detailed analysis of spline approximation methods for singular integral operators (even with piecewise continuous coefficients) we refer to [77], [120], [123] and [155].
4.3.
FINITE
SECTIONS-
169
QUASICONTINUOUS FUNCTIONS
Let us consider once more the singular integral equation with constant coefficients, but nowover the interval [0, 1], (aI + bS[o,1])u
], f E L2[0,1],
(4.26)
where
~(s--A) es,t e [0,11.
(sI°’ll~’)(t)
Weseek approximate solutions un of (4.26) in the spline space Sn[0, 1] S,~ N L2 [0, 1] by solving the equations Ln(aI + bS[o,1]) Un = Lnf, n = 1,2, ... where nowL~ refers to the orthogonal projection from L2[0, 1] onto S~[0, 1]. Translating the sequence (Ln(aI bSt0,1])ls~[o,,]) in to a sequence ofoperators acting on 12 in the same manner as before, one gets that (Ln(aI bSto,1])ls~to,x]) is a stable sequenceif and only if the sequence(P,~T(c)P,~) with c given by (4.25) is stable, i.e. if and only if the finite section method applies to the Toeplitz operator T(c). Corollary 4.21 yields Theorem4.29 The Galerkin method (Ln(aI and only if 0 f[ [a - b, a + hi.
4.3
bSto,,])ls.[o,1]) is sta ble if
Finite sections of Toeplitz operators generated by a quasicontinuous function
In the present section, we are going to consider the finite section method for someclasses of Toeplitz operators with oscillating generating function. The in a sense simplest class of oscillating functions is that of the quasicontinuous functions which we are going to introduce first. 4.3.1
Quasicontinuous
functions
Let Ha stand for the subset of La(T) which consists of all functions with vanishing negative Fourier coefficients, i.e. an = 2--~
a(e it)
e -int
dt = 0 if n = -1,-2,....
The sets Ha and Ha (consisting of all functions with vanishing positive Fourier coefficients) form closed (but non-symmetric) subalgebras La(T). Functions in a ( resp. H a) possess a nalytic c ontinuations i nto the interior (resp. the exterior) of the unit disk.
170
CHAPTER 4.
STABILITY
ANALYSIS
H°%functions enter the scene when examining the compactness of Hankel operators. Theorem 4.30 (Hartman) Let a ¯ L~(~). The Hankel operator H(a) (resp. H(5)) is compact on 12 i/and only i] a ¯ C + H°° (resp. a C + H~). For a proof see [26], Theorem2.54. The following result is a simple consequence of Theorem4.30 and of (1.16). Theorem 4.31 Both C + H~ and C + H°° are closed subalgebras o/L°~(~).
(non-symmetric)
C + H°~ is the simplest example of a closed algebra which properly lies between H°~ and Lc~, i.e. of a so-called Douglas algebra. For the treatment of the finite section methodfor the Toeplitz operator T(a) it is desirable (however, not necessary) to have available not only the compactness of the Hankel operator H(a), but also that of H(5) (compare Section 4.2). motivates to introduce the class QC := (C + c~) N(C+ H°~ ) the elements of which are called quasicontinuous/unctions. Clearly, QCis a C*-subalgebraof L°° (’1~). Since H°~ V~ H~ consists of the constants only, one might ask whether non-continuous but quasicontinuous functions exist at all. The answer is ’yes’ as the following theorem shows. Moreover, there is even an abundance of quasicontinuous functions: in fact, QCis - in contrast to C - no longer a separable algebra (which, amongother things, involves that the maximal ideal space of the commutative C*-algebra QCis not metrizable). Theorem4.32 Let xn, y,~, Zn be real-valued continuous ]unctions on the unit circle such that (i) xn + iyn is in °~ f or all n >_1, (ii) suptevEn°~=lIz, (t)l (iii) En°°__Isup,t_l,>e ]za(t)] < oc /or all e >
(iv) En%l Ilynll (v) E %1Ilxn and let an ¯ C with lanl <_ 1. Then the series a(t) := ~n=lanzn(t) converges pointwise on ~ as well as uniformly on {t ¯ ~£ : ]t - 1] >_ ~} for every ¢ > O, and its sum a belongs to QC.
4.3.
FINITE
SECTIONS-
QUASICONTINUOUS FUNCTIONS
171
¯ Proof. The convergence of the series is a consequence of (ii) and (iii). Moreover, (ii) and (iii) yield that a is bounded on q~ and continuous on q~ \ { 1 }. Writea as a ----
E an(Zn n=l
--
Xn) q- E an(xn
-t-
iyn)
n=l
--
i
E anYn.
n=l
Due to (iv) and (v), the series ~-~an(zn -xn) and ~-]~anyn converge uniformly to certain continuous functions. Thus, the series ~ an(Xn q-iy,~) converges pointwise on q~ and uniformly on compact subsets of "1~ \ {1} to a certain bounded and continuous function on T \ {1}. Weare going to show that this function belongs to H~. Indeed, for every e > 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,~<_l+r2~_2rncoss_
2
n+2r._<
2(l+rn)
we conclude that
IIX.lt _<ma{
In 2 2 ln(1 - r~)’ 1} _<
(4.29)
If, moreover,[s I _> 1In2, then 1 2 1 sin2~-~_< (r.-cos~--ff)
1 1 +sin2~ l+r2n--2rncos-~ <_ l+r2n--2rncoss
and, consequently, IX~(eis)l
1 -< In (1 - r.) max -~--, {ln2
-lnsin~}
n2 _< C-In(n+1)
(4.30)
4.3.
FINITE
SECTIONS-
QUASICONTINUOUS FUNCTIONS
173
with a certain constant C. Let Dn be a continuous function on ~" with values in [0, 1] such that 2, 1In 2, 2In
Dn(e~S) = if Isl _>
and set Zn := DnXn. Estimate (4.30) involves that -
2n
+1)
(4.31)
Nowdefine functions xn, Yn, Zn aS rotations of
~) zn(e~s):= X~(e~(~-~/")),W(e z.(e~’) :=Z.(~(~-~/")). These functions satisfy (i) - (v) in Theorem4.32. Indeed, (v) is sequence of (4.31), and (iv) follows from (4.28). Further, the support of zn = zn(e i~) is contained in [6/n - 2In 2,6/n ÷ 2/n2], and the pairwise disjointness of these intervals together with the inequalities IIxnll~o _~ 1 (which hold by (4.29)) yields (ii). Moreover,given ~ > 0, there are only finitely many functions Zn the supports of which are contained in {t E ~l" : It - 11 >_ ~), 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< ~<1
form an open neighborhood base of the point (t, 1) E Z, and the sets ((re -i~,t]x[0,~))U((te
-~,t]x[~,
1]) with 0<~<
do the same for the point (t, 0) E Z. For details we refer to [26], 4.88, and compare also Section 4.1.3. Roughly speaking, this topology cuts off the zylinder along each of its fibers {t} x (0, 1): points which belong to one and the same fiber and are close in the standard topology, are also close in the hull-kernel topology, whereaspoints belonging to different fibers are far awayfrom each other in the hull-kernel topology, even if they are close with respect to the Euclidean metric (Figure 5.2).
s
strangers
Figure 5.2: Spectrum of T(PC)
Example 5: Algebras of singular
integral
operators
Let P : /2(~) _~/2(~) denote the projection operator
(..., x_~,x-l, x0, xl, ~,...) ~ (..., 0, 0, x0, xl, ~,...), set Q := I - P, and write L(a) := (ai_j)i,c~=_oo for the Laurent operator generated by the (Fourier coefficients of the) function a ~ L~°(T). consider the smallest closed subalgebra Z(C) of L(/2(Z)) which contains all paired Laurent operators L(a)P + L(b)Q with continuous coefficients a and b. From Section 4.4.1 we recall that these operators are just the matrix representations of singular integral operators on L2 (T) (which also justifies the notation for the algebra). Every operator in Z(C) is of the form L(a)P + L(b)Q with uniqu ely deter mined a, b ~ C(T) and K
218
CHAPTER 5.
REPRESENTATION
THEORY
The algebra Z(C) is a C*-algebra which contains the ideal of the compact operators on /2(~), and the quotient algebra :~(C)/K(12(~)) is * isomorphic to C(X) where X is the disjoint union of two circles, say T0 and ~1. The same reasoning as in the preceding examples yields that every irreducible representation of Z(C) is unitarily equivalent either to the identical mappinge of Z(C) into L(12(Z)) or to one of the following mappings: L(a)P + L(b)Q + K ~ a(t),
L(a)P + L(b)Q + (5 .4
with t E "I~ ~ To and t E ~ ~ ~?1, respectively. So one can think of Spec :[(C) ~- Prim Z(C) as an annulus the boundary of which is viewed as X, whereas its interior is identified with the single point ~. In what follows, it is more advantegeous to think of the spectrum of Z(C) as the cylinder ~ × [0, 1], where the set ~ x (0, 1) corresponds to ~, and where the points in the circles ~ × {0} ~ To and T × {1} ~ ~1 correspond to the representations (5.4) (Figure 5.3).
~0~~ Figure 5.3: Spectrum of Z(C)
Example 6: Finite
sections
algebras for Toeplitz
operators
Let ~- stand for the algebra of all bounded sequences (AN) of matrices A,~ ~ L(C~), and let 6 refer to its ideal of all sequences tending to zero in the norm. Further, let again $(C) denote the smallest closed subalgebra of ~ which contains all sequences (PnT(a)Pn) where a is continuous. From Section 1.4.2 we know that S(C) contains all sequences of the form (PnKPn + W~LWn+ Gn) with K and L compact and (Gn) ~ 6, that these sequences form an ideal in $(C), and that every sequence in S(C) can uniquely written in the form (P,,T(a)P~ + PnKP~÷ W~LWn÷ G~) with
a ~ C(~r). The algebra ~q(C) contains the ideal 6; so the quotient algebra 9~ $(C)/6 is well defined, and what we are interested in is the spectrum of
5.1.
RF, PRESENTATIONS
219
this quotient algebra. For this goal, we introduce the sets 31 := {(PngPn)+~,
co mpact}, 32 := {(WnLWn)+~, L
compact},
which are contained in 91 and form closed ideals of this algebra. Their sum 31 + 32 is a closed ideal 3 of 91. Observe that this ideal actually coincides with the ideal 6w/6where Gwis the ideal of S(C) introduced in Section 4.2.1. If ~r is an irreducible representation of 91, then either and, in the case (ii), (ii.1)
(i)
3C_ger~r
or (ii)
3~gern,
we have moreoverat least one of the following: 31 q~ Ker ~r or (ii.2)
32
~
Ker ~r.
In the case (i), ~r/3 is an irreducible representation of the quotient algebra 91/3, which is commutative and *-isomorphic to In the case (ii), both 31 and 32 are elementary ideals which are isomorphic to the ideal K(12), the isomorphisms being given by R1 : (PnKPn) + ~ ~+ an d R2 : ( WnLWn) + 6 ~ respectively. Suppose (ii.1) holds. Then, by Theorem5.7, the restriction of ~r to 31 coincides with R1 and, conversely, ~r is the (up to unitary equivalence) unique extension of R1 to an irreducible representation of 91. On the other hand, the mapping 92 -~ L(l~),
(An) + ~ s-l im AnP~
(5.5)
extends R1. Thus, ~r is unitarily equivalent to the mapping(5.5) which is, in earlier notations, just the quotient mappingWIG. Similarly, in the case (ii.2) the representation ~r is unitarily equivalent to the mapping This describes Spec 91 completely: there are two infinite-dimensional representations, W/6 and l/~d/6, as well as a family of one-dimensional representations which is parametrized be the points of the circle (PnT(a)Pn + P~KP~ + W~LW~) + 6 ~ a(t),
(5.6)
and every irreducible representation of 92 is unitarily equivalent to one of these representations. One can think of Spec 91 ~ Prim 91 as a sphere whose northern and southern hemispheres (both without the equator) are identified with and I~z/~, respectively, and whose points of the equator are corresponding to the one-dimensional representations (5.6) (Figure 5.4). The spectrum of the algebra of the finite section method for Toeplitz operators with piecewise continuous generating function allows a similar description.
220
CHAPTER 5.
REPRESENTATION
THEORY
Figure 5.4: Spectrum of Example 7: Finite tors
sections
algebras
for singular
integral
opera-
Let Rn denote the projection operator acting on/2(Z) by the rule
(..., z_2,z_l, ~0,xl, x~,...) ~+(..., 0, 0, z-n, ..., xn-1,0, 0,...), and consider the finite section method (Rn(L(a)P + L(b)Q)Rn) for the paired Laurent operator L(a)P + L(b)Q with continuous coefficients a and b. Let further T’(C) stand for the smallest closed subalgebra of the C*algebra ~" of all bounded sequences (An) of matrices An e L(Im Rn) which contains all sequences (Rn(L(a)P + L(b)Q)Rn) with a, b E C(T). On identifying the Hilbert space 12 (Z) with the orthogonal sum 12 (Z /2(Z+) in the obvious way, we can think of the finite section operator Rn(L(a)P + L(b)Q)Rn the matrix ( PnH(b)Pn PnT(b)Pn
PnT(a)Pn PnH(a)Pn)
(5.7)
and we shall make use of these identifications throughout what follows. In particular, the operator RnL(f)Rn with continuous f corresponds to the matrix PnT(])Pn PnH(])P,~ (5.8) PnH(f)Pn PnT(I)Pn and taking into account the compactness of Hankel operators with continuous generating functions, one easily finds that the commutator of the matrices (5.7) and (5.8)
(
PnK~Pn PnK2I P. ( PnKn Pn +’WnL-~ Wn PnK22Pn + WnL~ Wn with certain compact operators Kij and Lk and with Wnas in the preceding example. It is the concrete form of this commutator which suggests
5.1.
221
REPRESENTATIONS
to introduce the following subsets of the quotient algebra ~’/G where, of course, ~ denotes the ideal of 5~ of all sequences tending to zero: 3-1 :=
0 {((
3o :=
0
WnL-1Wn
0
P~K~P~ {((PnKllP,~
))
,~>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_<~<~oof closed ideals of 92 which is labeled by the ordinal numbers between 0 and some fixed ordinal ao (called the length of the composition series) and which ownsthe following properties: (i) ~ is properly contained in ~+~for all a < a0. (ii) ~o={0}, ~o =92. (iii) If/9 is a limit ordinal, then J~ is the normclosure of U~<~J~.
226
CHAPTER 5.
REPRESENTATION
THEORY
In what follows, we will only meet finite composition series, for which condition (iii) is meaningless. With the notations as in the previous example, we thus have the following composition series for the algebras related to the finite section methodfor Toeplitz operators: { i}~=0 with
30={0),
(5.9)
31 =3, 32=P-1
is a composition series of length 2 of the algebra 92 = $(C)/G, and 3
{3i)i=o
with 30={0},
31 = ~, 32 = ~ + G, ~3 = S(C) (5.10)
is a composition series of length 3 for the algebra S(C). Theorem 5.18 (a) Every postliminal C*-algebra 9~ possesses exactly one composition series {3a}0
Dual
then all its ir-
algebras
The ideals K(H) and 3 in Examples 4 - 7 in Section 5.1.4 are not only liminal, but belong to the class of the dual algebras which we are going to introduce now. Westart with recalling some notations. Let (~t)teT be a family of C*-algebras. The product of this family is the C*-algebra of all bounded functions a from T into for every t. The restricted product of this family is the C*-subalgebra of the product which consists of all functions a such that, for each e > 0, there are only finitely manyt e T with
5.2.
POSTLIMINAL ALGEBRAS
227
Further, given a subset Mof a C*-algebra 91, we define its right annihilator R(M) to be the set of all a E 92 satisfying Ma= O. The left annihilator L(M) of Mis defined analogously. Theorem 5.20 The following assertions are equivalent ]or a C*-algebra 92: For every closed left ideal 3 o] 92, one has L(R(3) ) -= (i) For every closed right ideal 3 o]91, one has R( L( 3) ) -(i)’ (ii) The sumo] the minimal left ideals o] 91 is dense in 91. (ii)’ The sum o] the minimal right ideals o] 91 is dense in 91. (iii) If 3 is a primitive ideal of 91, then L(3) {0} and 92/3 is *-isomorphic to K(H) for some Hilbert space (iv) 91 is *-isomorphic to a restricted product of elementary C*algebras. 92 is *-isomorphic to a C*-subalgebra of K(H) for some Hilbert (v) space H. For a proof see [14], [93] and [49], 4.7.20.
¯
C*-algebras satisfying one of the conditions of Theorem5.20 are called dual. It is immediate from (iv) that the ideals K(H) and 3 in Examples 4 - 7 of Section 5.1.4 are dual algebras. Theorem 5.21 (a) Every dual C*-algebra is liminal. (b) The spectrum of a dual C*-algebrais a discrete topological space. Recall that a topological space is discrete if its one-elementic subsets (thus, all subsets) are open. A proof of Theorem5.21 can be found in [14] and [92], Theorem8.1, and comparealso [49], 4.7.20. Let ~3 be the restricted product of elementary C*-algebras J~t with t running through some index set T. Then every J~t is *-isomorphic to the closed ideal of ~3 consisting of all functions on T which vanish on T \ {t}. Hence, the elementary algebras in assertion (iv) of Theorem 5.20 can found amongthe closed ideals of 91, and the smallest closed subalgebra of 91 which contains all these ideals is just the whole algebra 91. Let now 91 be a C*-algebra and 3 be a dual ideal of 91. As we have just remarked, 3 is generated by its elementary ideals, 3t say. But every closed ideal of 3 is also a closed ideal of 2 by the second isomorphytheorem (Theorem 1.46). Thus, 3 can be identified with the smallest closed ideal of 91 which contains all elementary ideals ideals 3t. It is clear as well that 38 f3 3t = {0} if s ~ t. Thus, one can define the largest dual ideal of a C*algebra 92 as the smallest closed ideal of 92 which contains all elementary ideals of 91. Everydual ideal of 91 is a subideal of the largest dual ideal of 91.
228
CHAPTER 5.
REPRESENTATION
THEORY
We illustrate the new notions by a familiar example. Let H be an infinite-dimensional Hilbert space and (Ln) be a sequence of orthogonal projections from H onto its subspaces H,~ which converges strongly to the identity operator. By ~" we denote the C*-algebra of all bounded sequences (An) of operators An E L(Hn), and by 6 its closed ideal consisting of all sequences (An) with I[AnLnll ~ 0 as n -~ ~. One easily checks that ~" is just the product of the algebras L(Hn), whereas G is their restricted product. Further, if all spaces Hnare finitedimensional, then G is even a dual algebra by assertion (iv) of Theorem 5.20. On the other hand, if at least one of the spaces Hn has infinite dimension, then G is no longer dual. It makes sense in this case to introduce a smaller ideal of ~-, say ¢, the elements of which are the sequences (An) ~ such that IIAnLnll ~ 0 and An is compact on Hn for every n. Although this ideal cannot take the place of the ideal ~ in Kozak’s theorem (for instance, the sequence (0, L2, L3, ...) is stable, but the coset (0, L2, L3, ...) + not invertible in ~’/¢ if the dimension of H1 is i~nfinite), there are several interesting subalgebras A of ~" with A f~ ~ = G (see, e.g., [77], Sections 3.2.5 and 3.5.2) for which a restricted version of Kozak’s theorem remains valid. Theorem 5.22 (Restricted version of Kozak’s theorem~ I) Let A be a C*-subalgebra of a~ with A f~ 6 = ~. A sequence (An) ~ A is stable and only if the coset (An) + ¢ is invertible in the quotient algebra Ale. The ideal ¢ is dual by construction. Hence, there should be a Hilbert space /:/such that ¢ is *-isomorphic to a subalgebra of~ K(_f/). It is indeed not hard to identify this Hilbert space. For choose H as the orthogonal sum @n=l°°Hn of the Hilbert spaces Hn, i.e. let /~ be the set of all sequences ( n)n=x with Xn e Hn and ~n~__l [IXnll ~ < CxZ, and provide this set with the scalar product
:= n----I
Every sequence (An) ~ U gives rise to a linear operator on/~: (An): (xn) ~ (Anxn),
(5.11)
and this operator is bounded. Moreover, the mapping (5.11) is even isometry, [[(an)[[~- I](An)[[L([t) = sup I] anLn[I. Soonecan regard ~" a s a C*-subalgebra of L(/~), and a little thought reveals that A V~K(/~) is the ideal ¢. Thus, by the third isomorphy theorem, Ale ~- A/(A ~ K(I;I)) ~- (A + K(I:I))/K(I;I),
5.2.
POSTLIMINAL
229
ALGEBRAS
and since a coset (An)+K([-I) is invertible in (A+~K([-I))/K([-I) if and only if it is invertible in the Calkin algebra L([-I)/K(H) (inverse closedness), arrive at a further version of Theorem5.22. Theorem 5.23 (Restricted version of Kozak’s theorem, II) Let A be a C*-subalgebra of Y: with A N 6 = ¢. A sequence (An) E A is stable and only if the operator (An), defined by (5.11), is Fredholmon L([-I). If all spaces Hn are finite-dimensional, then G = ¢, and Theorem5.23 holds for every C*-subalgebra A of ~. It is evident that the application of Theorem5.23 is of particular interest when one is able to identify the sequence (An) with some familiar operator on/?/. This happens indeed sometimes, and one important example is provided by the finite sections of a Toeplitz operator. For this example, we rearrange the Hilbert space/~ -- @~=oImP,~ in a way which makes it isomorphic to the/2-space over the quarter plane Z+ × Z+ (see Figure 5.7). As Douglas and Howe[53] observed, the finite section sequence (P,~T(a)Pn)
!
if=
$
.
.
+xZ+) =/2(Z
ImP2 ImP1 Figure 5.7: The rearranged Hilbert space corresponds to the quarter plane Toeplitz operator T(2)(b) under this isomorphy,the generating functions a and b being related by -1). b(s, t) = a(st (Given a function b E L~°(T x T) with Fourier coefficients 2bran = (1/2~)
j~0 2w 2~ ~0
b(e~s, ei~)e-~mse-~n~ds dr,
the quarter plane Toeplitz operator T(2)(b) acts on/~(Z+ × Z+) via T(2)(b) : (x,~) (Ym,~)
where
Y,~n = E bm--k,,~-~Xk~.) k,l=O
230
CHAPTER 5.
REPRESENTATION
THEORY
This beautiful result stimulated several authors to look for extensions into various other directions, see [23], [75], [85], and also the monograph[26], Sections 8.29, 8.30.
5.2.3
Finite sections of Wiener-Hopf operators almost periodic generating function
with
In this section we are going to examinea class of operators the finite section method of which induces a non-postliminal C*-algebra in a natural way. The operators under consideration are convolution and Wiener-Hopf operators with almost periodic generating functions. The Fourier transform Ff of a rapidly decreasing function f on R is defined by
(F.f)(:~) - v~
e-~tI(t ) dr. ’
The operator F extends by continuity to a unitary operator on the Hilbert space L2(II~) (see Chapter IX.1 in [127]) 1
ei~t.f(t) dt.
Thus, if a ~ L~(It~), then the operator FaF-1 acts boundedly on L2(II~). This operator is the so-called operator of convolution b~ a, and its restriction to the subspace L2 (l~ +) of L~ (Ii~) is the Wiener-HopJoperator with the generating function a. Wedenote these operators by C(a) and respectively. Hence, if X+(x) is set 1 for x _> 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 x<0, if x_>0.
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.
REPRESENTATION
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]:
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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.
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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.
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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.
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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] <_1). Consequently, A ~ a(e)(a) in any case. If A ~ Ut~Ta(e)(Wt(a)), then Wt(a - Ae) is invertible and IIWt(a -~ Ae) II < 1/¢ for every t supteT ]lWt(a - Ae)-~[I < 1/¢, because the supremumis attained by Theorem 5.39 (b). Thus, A
246
CHAPTER 5.
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THEORY
Weturn back to the lifting theoremand consider sufficient families of ideal lifting homomorphisms.Our first goal is to show the converse of assertion (b) of the special lifting theorem. Then we employ this result characterize the ideals which can be lifted by a weakly sufficient family of homomorphisms. Theorem5.41 Let the notations and hypotheses be as in the special lifting theorem. (a) If {Wt} is a sufficient family, then the follow~n9 assertions are equivalent for a E 91: (i) The coset a + 3 is invertible. (ii) All operators Wt(a) are Iaredholm, and there are at most finitely many amongthem which are not invertible. (b) If {Wt} is a weakly sufficient family, then the following assertions are equivalent for a E 91: (i) The coset a + 3 is invertible. (ii) All operators Wt(a) are Fredholm, there are at most finitely many among them which are not invertible, and the norms II(Wt(a) + g(Ht)) -ill are uniformly bounded with respect to t. Proof. Weshall prove assertion (b); the proof for (a) is analogous. (i) =~ (ii): Let a + 3 be invertible. Then ab = e + j with certain elements b E 91 and j ~ 3. The first part of (ii) is a consequence of Theorem5.37 (b), and the second one follows from the identity (Wt(a) + K(Ht)) -1 -= Wt(b) + g(Ht) together with the estimate IIWt(b) + K(Ht)II <_ Ilb[I. (ii) =~ (i): Let all operators Wt(a) be Predholm, and assume that Wt(a) is moreoverinvertible for every t E T \ {tl, ..., tr}. Consider the selfadjoint element a*a. Since the Wt are *-homomorphisms, we conclude that all operators Wt(a*a) with t ~ T \ {tl, ..., tr} are invertible, whereas the selfadjoint operators Wt~ (a’a) are Fredholm with index zero. Hence, one can find compact operators Kt~ ~ K(Ht~) such that the operators Wt, (a’a) + Kt, are invertible for t ~ {tl, ..., tr}. Pick jr, ~ 3t, such that Wt, (jr,) = Kt, and set j := Jr1 +... + jt.. By the separation property (Theorem 5.37 (a)) we Wt,(a* a + j) = Wt,(a* a) + for ti ~ {tl ,. ...,t~} and Wt(a*a+j)=Wt(a*a)
for
t~T\{t~,...,tr}.
Thus, the operators Wt(a*a + j) are invertible for every t inverses are uniformly bounded. Since the family {Wt}teT ficient, we get the invertibility of a*a + j in 91. Finally, belong to the same coset modulo 3, whence the invertibility
~ T, and their is weakly sufa*a ÷ j and a*a of a + 3 from
5.3.
LIFTING
247
THEOREMS
the left hand side follows. Consideration of aa* in place of a*a yields the invertibility from the other side. ¯ Theorem5.42 Let the notations and hypotheses be as in the special lifting theorem. (a) If the family of homomorphisms{Wt} is weakly su~icient, then ~ is the maximaldual ideal of 91. (b) If 91 is postliminal and ~ is the maximalliminal ideal of 91, then the family {Wt} is weakly su~cient. Whenproving this theorem, we will several times make use of the following simple observation the proof of which is left to the reader as an exercise (apply Theorem 1.50). Proposition 5.43 Let 91, f~l, ~2 be unital C*-algebras and W1 : 91 --~ fS1, W2: 91 -’~ fB2 be unital *-homomorphisms.Then the following assertions are equivalent: (a) For every a E 91, the invertibility of W1(a) ~i implies the invertibility of W2(a)in fB2. (b) Ker W1C_ Ker W2. Proof of Theorem5.42. For the of proof (a), let ~max denote the maximal dual ideal of 91. Let a E 91, and suppose the coset a ÷ ~max to be invertible. Weclaim that the coset a ÷ 3 is invertible, too. Proposition 5.43 then yields the assertion. There are elements bl, b2 ~ 91 and jl, j2 ~ ~max such that abl = e + jl and b2a = e + j~, whence W~(a)W~(bl) = e~÷W~(ji), W~(b~)W~(a) = for every
t ~ T.
(5.21) The dual ideal 3maxis liminal due to Theorem5.21 (a), and the representations W~are irreducible by hypothesis. Thus, the operators W~(jl) dnd W~(j2) are compact, and the operators W~(a) are Fredholm for every t e T. Further, (5.21) entails that the normsof the inverse cosets of W~(a)+K(H~) are uniformly bounded(by IIblll e.g.). Let the ~ with s E S be the elementary subideals of 3max, which generate this ideal. Then one can find finitely many elements k~ ~ ~s~ such that IIJ~ - ks~ -... - k~ II < 1/2. (5.22) Further, for every s ~ S there is at most one t ~ T with Wt(~s) ~ (0}. Indeed, if both W~(~s) ~ (0} and W~(~s) ~ (0} for some s ~ tl ~ t2, then both W~and W~induce irreducible representations of ~ due to Theorem 5.7. Then, by the elementarity of~s and again by Theorem
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CHAPTER 5.
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THEORY
5.7, the representations Wt1 and Wt2 of 91 are unitarily equivalent, which contradicts the hypotheses. Thus, we conclude from (5.22) that I IW~(jl)ll 1/2 for all but finitely manyt E T, and an analogous assertion holds for j2. In combinationwith (5.21) this implies the invertibility of Wt(a) for all but finitely manyof the t. The weak sufficiency of the family {Wt} implies via Theorem5.41 (b) the invertibility of the coset a ÷3, whencevia Proposition 5.43 the inclusion 3maxC_ 3 follows. The reverse inclusion is a consequence of the duality of 3 by Theorem 5.38. Thus, 3 is maximal dual. For a proof of (b) suppose ~ to be the maximal liminal ideal of 91. Theorem5.38 we have Spec 3 ~ {W~}teT, and this space is dense in Spec 91 according to Theorem5.17 (b). The closure of the set {Ker Wt}teT in the hull-kernel topology is equal to (I E Prim 91 : ¢~eTKer W~C I}, and this set coincides with all of Spec A due to density. Thus, NKer Wt C Ker
71"
t~T
for every irreducible representation ~ of 91. But (~t~TKer W~= Ker W, where W refers to the homomorphism(5.17), and so Ker W C_ Ker ~ for every u ~ Spec 91. If now W(a) is invertible, then the preceding proposition provides the invertibility of u(a) for every ~, and since the irreducible representations form a sufficient family of homomorphismsfor 91, we conclude that a is invertible. Thus, the family {Wt} is weakly sufficient. ¯ Corollary 5.44 Let the notations and hypotheses be as in the special lifting theorem and suppose moreover that 91 is postliminal and that the largest liminal ideal o] 92 is dual. Then the ]ollowing conditions are equivalent ]or a closed ideal ~ o] 91: (a) ~ can be lifted by the special lifting theorem, and the corresponding ]amily o] homomorphisms{ Wt } is weakly su~cient. (b) ~ is the maximalliminal ideal o] 91. Corollary 5.45 Let the notations and hypotheses be as in’the special lifting theorem and suppose moreover that 91 is postliminal and that T is a finite set. Then the ]ollowing conditions are equivalent: (a) The family of homomorphisms(Wt} is weakly su~cieut. (b) The family of homomorphisms(Wt} is suj:ficient. (c) is themaximal dualidealo] 91. (d) ~ is the maximallimiual ideal o] 91.
5.3.
249
LIFTING THEOREMS
Proof. Assertions (a) and (b) are obviously equivalent, and so it remains to show that whenever the family {W,} is sufficient, i.e., whenever the mappingWin (5.17) is a *-isomorphism, then 3 is the largest liminal ideal of 91. Let k belong to the largest liminal ideal of 91. Then ~r(k) is compactfor every irreducible representation of 91 by Theorem5.17, which, in particular, implies the compactness of the operators Wt (k). Due to the finiteness T, one can find a j in ~[ such that W~(j) = Wt(k) for every t. Since Wis an isomorphism, we finally get j = k, i.e. k E 5. ¯ Weturn once more back to the special lifting theorern with arbitrary T and suppose that {W,} is a weakly sufficient family or, equivalently, that Wis a *-isomorphism from 91 onto a C*-subalgebra of the product of the C*-algebras L(Ht). This product can be viewed as a C*-subalgebra of the algebra L(H), the Hilbert space H being the orthogonal sum of the Hilbert spaces H, (the orthogonal sum of a countable set of Hilbert spaces is defined at the end of Section 5.2.2, for the general case one has to replace sequences by nets). A little thought shows that then the ideal 3 can be identified with W(91)g~ K(H) (recall Theorem5.20 (v) in this connection). As a consequence of the third isomorphy theorem, (W(91) + K(H))/K(H) ~ W(91)/(W(91) f~ K(H)) Wetherefore obtain the following. Corollary 5.46 Let the notations and hypotheses be as in the special lifting theorem and suppose that {Wt}is a weakly sufficient family. Then a coset a + 3 is invertible if and only if W(a) is a Fredholmoperator on L(H). Thus, to study invertibility in 9//3 is actually the same as to study Fredholm theory on L(H). These ideas will be developed for sequence algebras in Chapter 6 in detail. 5.3.4
Structure
of fractal
lifting
homomorphisms
Here we consider the special lifting theorem in its original context, viz. for algebras of approximation sequences, and we will examine how the assumptionof fractality of the algebra determines the structure of the lifting homomorphisms. Let H denote a Hilbert space, K(H) the ideal of the compact linear operators on H, ~- the C*-algebra of all bounded sequences of bounded linear operators on H, and G the ideal of ~" consisting of all sequences tending to zero in the operator norm. Our first goal is the structure of certain fractal and elementary subalgebras of ~’.
250
CHAPTER 5.
REPRESENTATION
THEORY
Theorem5.47 Let ~7 be a fractal C*-subalgebra of it r satisfying the following conditions: (i) ,7 is *-isomorphic to K(G) for some Hilbert space (ii) If (Jn) belongs to J, then the operators Jn are compact. Then there is a finite subset N1 o] N and ]or every n ~ N ~ N~ there are isomet~es Un ~om G into H such that that ~ consists o] all sequences ( Jn) with Jn = UnKU~ with K ~ K(G) if
n ~ N ~ N~.
Observe that the operator K is independent of n. For the proof we need some knowledge about C*-subMgebrasof the ideal of the compactoperators. The following result is essentiMly the equivalence (iv) ~ (v) in Theorem5.20; for a direct proof see [44], Theorem1.10.8. Theorem 5.48 Let H be a Hilbe~ space and E a *-subalgebra of K(H). Then there are mutually orthogonal closed subspaces Hi of H such that E is the restricted product of the ideals K(Hi). Proof of Theorem 5.47. Let W denote the *-isomorphism from ~ onto K(G), nnd consider the homomorphisms Hu : ~ + L(H), Then there
are natural
(Fk)k~
~
homomorphisms
H~W-~ : K(a)
~ ~(~)
~
(5.2a)
of K(G) onto the C*-subalgebras Hn(fl) of K(H). Because the algebra K(G) is simple, there are only two possibilities for the kernels of the homomorphisms (5.23), either
(case
-~ = K(G) or (case 1) Ker H~W
-1 = {0}. 2) Ker H~W
~rther, since the homomorphisms(5.23) are surjective, we get in case -~ = {0}, whereas H~W -1 is a *-isomorphism from that H~(fl) = Im H~W K(G) onto Hn(~) in c~e 2. Hence, either
(case 1) Hn(fl)
= {0} or.(case
2) Hn(~) K(G).
Weconsider c~e 2 and apply Theorem 5.48 to the algebra Hn(~) K(H). If H~(~) would be a restricted product of more than one ideal K(Hi), K(G) would possess non-triviM ideals, which is impossible. Thus, there exists a closed subspace Hn of H such that H = g~ $ Hn and Hn(~) = {0} K( Hn),
(5.24)
5.3.
251
LIFTING THEOREMS
{0} ~ K(H,~) referring to the subalgebra of L(H) of all operators having H,, and H~ as its invariant subspaces and acting on H. as a compact and on H.j- as the zero operator. Let En : H. -4 H, Rn : H -4 H,~ and P. : H ~ H. denote the operators of extension by zero, restriction and orthogonal projection. Then (5.24) can be rewritten
Since K(Hn) and K(G) are *-isomorphic, isometry V~ from G onto Hn such that Rn(HnW-~)(K)E.
= 2 for
there is, by Theorem 5.11, an all
K e K(G)
Multiplying this equality by E~ from the left and by R~ from the right, replacing E.R= by P=, and abbreviating the isometry E~V~from G into H to Un, we obtain P~(H.W-1)(If)P~
= U~I(U~ for
all
K ~ K(G),
whencefinally (HnW-1)(K)
= UnKU~ for
all
K G K(G),
because of PnH~(fl)Pn = H.(fl). So what results is that the natural numberssplit into two closes, N1 and ~2, such that ~ is just the set of all sequences (J.) with
~={ 0
U~KU~ ifn with ~ N~, someK~K(G)
ifn~N~.
The frac~ality of ~ finally implies that N~mus~be a finite set. In what follows, we suppose for simplicity that N~ = ~, i.e. that
¯ he isomorphism Wfrom ~ onto K(G) is necessarily
of the form
where k is arbitrary (but fixed). Indeed, since U~U~= I e L(G), one has
In the next step we consider the larger algebra ~ := ~ + ~.
252
CHAPTER 5.
REPRESENTATION
THEORY
Proposition 5.49 Let J be as in (5.25). Then the norm limit lim U~ JnUn exists ]or every sequence (Jn) E ~ = J + Proof. The assertion is evident for sequences (J,~) e ,~ (where the operators U~JnU,~are independent of n), and for sequences (G,~) ~ 6 one For sequences (J~) ~ 5, we abbreviate lim U~JnU,~to W(Jn). This notation is justified since the homomorphismWextends the mapping (5.26). Our next goal is subalgebras A of :~ which contain the set ~ = {(U, KU~) + (an) with K e K(G), (Gn) as their ideal. Theorem 5.50 Let ¢4 be a C*-subalgebra o] J: which contains 5. Then the ]ollowing assertions are equivalent. (a) ~ is a closed ideal of .4. (b) For any sequence (An) e A, there exist the strong limits s-lim and s-lim rr*A* Tr Proof. The implication (b) ~ (a) can be shown as in Theorem 1.19 Theorem4.22 (c). For the reverse implication, let (An) ~ A K e K(G) Since 3 is an ideal of A, there are an operator R ~ K(G) as well as a sequence (G,) e ~C such that
An. U.KU = U.RU + Multiplication by U,~ from the left and by U, from the right yields
V AnVn " t( = R + V G.V. (recall that U~Un= I e L(G)). The right hand side of the latter equality tends in the norm to R as n --> co. Hence, the sequence (U~A,Un. K)n>_l is norm convergent for every compact operator K ~ K(G). Let x ~ G, and write P~ for the (compact) orthogonal projection from G onto its one-dimensional subspace Cx. Then the sequence (U~ A,Un. P~) is norm convergent as we have just seen, and from U~A,U,~x = U~AnUnP~x, we conclude that the sequence (U~A~Unx)converges for every x, i.e. that (U,~A,~U,~)is a strongly convergent sequence. Employing the ideal property from the other side one obtains as above K. U,~A,~U, = R + U~G,~Un,
5.3.
LIFTING THEOREMS
253
whence, after taking adjoints, V~A~Vn. K* = R* + This implies the strong convergence of the adjoint sequence (U~A~Un). The mapping A --~ L(G), (An) ~-~ s-lim U~AnUnis evidently an extension of the homomorphismW from ~ to all of K(G). Wefinally ask for the specifications of the ideals ~ and the homomorphisms Wwhich are involved by the separation property. Theorem5.51 Let .4 be a C*-subalgebra of J: which contains the sets ~1 = {(vngv~) + (Gn) with g e K(G(1)), (Gn) 32 = {(VnKV~) + (Gn) with K ¯ K(G(2)), (Gn) as its ideals, where G(1) and G(2) are Hilbert spaces and the Un resp. Vn are isometries from G(1) resp. G(2) into H. Let further W1 denote the homomorphism(An) ~-~ s-lim U,~AnUn.Then the following assertions are equivalent. (a) The separation property W1(32)= {0} holds. (b) The sequence (V~Un) converges weakly to zero. Proof. The separation property (a) is equivalent to requiring that s-lim
U~VnKV~Un= 0 for all
g ¯ K(G(2)).
Thus, for every K ¯ K(G(~)) and x ¯ G(1), one has lim (U~VnK*KV~Unx,x) = lim (KV~*Unx, KV~Unx) = 0, and consequently KV~Unx-+ O. Given y ~ G(2), let Pu refer to the orthogonal projection from G(2) onto its subspace Cy. The operator Pu is compact, hence, PuV~*U~x-~ 0 and (V,~Unx, y) = (V~Unx, Puy) = (PuV;U,~x, y) for every x ¯ G(1) and y ¯ G(2), which implies the weak convergence of the sequence (V~Un) to zero. For the reverse implication note that the weak convergence of (V~Un) to zero involves the strong convergence of (KV~Un)to zero for every compact operator K, and thus the strong convergence to zero of the sequence (V~UnKV~U~).But this is the separation property. ¯ In applications, there usually appears a natural lifting homomorphism A -~ L(H) acting via (An) ~-~ s-lim An, i.e. with Un = I. In this case the separation condition implies that the V~ converge weakly to 0.
254
CHAPTER 5.
REPRESENTATION
THEORY
Notes and references Sections 5.1~ 5.2: The material of these sections is standard and can be found in (almost) every textbook on C*-algebras. The approach the finite section method for Wiener-Hopf operators presented in Section 5.2.3 is taken from [139]. One advantage of the approach to the classical Theorem 5.31 via describing a whole algebra of approximation sequences is that it renders possible the application of the results of Chapters 1 - 3 (determination of limiting sets of pseudospectra, for instance) to WienerHopf integro-difference equations. For a further extension of these results to LP-spaces see [139]. Section 5.3: This section is a polished and completed version of the paper [147] by two of the authors. Several of the results can be further generalized to the Banach algebra setting (with appropriate modifications). particular, there is a generalization of the Lifting Theorem5.35 to Banach algebras which works as nicely as its C*-version. For its formulation and proof see, for instance, [77], Chapter 1, Theorems1.2 and 1.8, while several applications can be found in [77].
Chapter 6
Fredholm sequences Thenotion of the index of a linear operator plays
an ever-increasing
role
in the study
of
differential andintegral operators. S. Goldberg The topic of this chapter is a special class of stably regularizable (but not necessarily stable) approximation sequences which we will call Fredholm sequences. These sequences are distinguished by several remarkable properties, two of which are: (A) The class of Fredholm sequences is (in contrast to general stably regularizable sequences) stable under small perturbations. (B) Fredholm sequences can be stably regularized without knowing explicitly a suitable cutting off parameter ~ (clearly a stably regularizable sequence can be regularized by using any sufficiently small cutting off parameter s, but the decision whether a given ~ is actually small enough maybe difficult in general). To get a first imagination, consider the C*-algebra S(C) related to the finite section method for Toeplitz operators. Since S(C)/~ is *-isomorphic to an algebra of ordered pairs of operators on 12 (Theorem1.55), there an evident way of introducing the notion of Fredholmness. Wesimply call a sequence (A,~) S(C) Fredholm if both ass ociated ope rators W(An) and I~V(An) are Fredholm operators in the commonsense. In this case, we can further introduce the nullity n(An) as well as the deficiency d(An) of the Fredholm sequence (An) n(An) := dimKerW(An) di mKerl~d(An), 255
256
CHAPTER 6.
FREDHOLM SEQUENCES
d(An) := dimCokerW(An) di mCokerl~(An), and we define the index ind (A,~) of a Fredholm sequence ind (An) := n(An) - d(A,~). It is a triviality to check that all standard properties of Fredholmoperators and their indices do have analogues for Fredholm sequences. In Section 6.1 we will extend this definition to a class of algebras of approximation sequences which is suggested and motivated by Corollary 5.46, and we will discuss a few quite simple consequences. In Section 6.2 we will analyse the condition of Predholmnessin detail, and the main result will be a relation between the nullity n(An) and an inner property of the Predholm sequence (An), namely the number of the singular values which tend to zero. This observation will be employed in the concluding Section 6.3 in order to define a weaker notion of Fredholmness, which also applies to sequences that are not supposed to belong to one of the special algebras considered in Section 6.1.
6.1 Fredholrn sequences in standard algebras Westart with introducing a class of subalgebras of the algebra 5~ of all bounded sequences which we call standard algebras or algebras satisfying the standard model. All concrete algebras considered in Chapter 4 will prove to be standard in this sense. Then we define the Fredholmness of sequences which belong to a certain standard algebra, and we examine the relations between ~-¥edholm sequences on the one hand and stably regularizable sequences and MoorePenrose stable sequences on the other hand. Wewill see that the Fredholmness of a sequence (An) involves a certain (asymptotic or exact) structure of the kernels of the operators An. 6.1.1
The
standard
model
Let H be an infinite dimensional Hilbert space and (Ha) be a sequence of closed subspaces of H such that the orthogonal projections Pn from H onto Hn converge strongly to the identity operator I on H. Wedenote by ~" the set of all bounded sequences (An) of operators An E L(Hn), and make it to a C*-algebra with identity in the usual way. Welet ~ stand for the closed ideal of 9v consisting of all sequences (G~) with [IGnPnl[ -~ 0 as Let further T be a (possibly infinite) index set and suppose that, for every t E T, we are given an infinite dimensional Hilbert space Ht with
6.1.
FREDHOLMSEQUENCES IN STANDARD ALGEBRAS
257
identity operator I t as well as a sequence (Etn) of partial isometries Etn Ht -~ H such that - the initial projections p~t of Etn converge strongly to I - the range projection of E~ is Pn, - the separation condition (E~)*Etn -, weakly as n ~ c~
(6.1)
holds for every s, t ¯ T with s 7~ t. (Recall that an operator E : ~ - ~ H" i s a partial is ometry if EE*E = E and that E*E and EE* are orthogonal projections in this case, which are called the initial and the range projections of E, respectively. The restriction of E to Im (E’E) is an isometry from Im (E’E) onto Im (EE*) Im E.) For brevity, write Et_n instead of (Etn) *, and set Hn := Im Pn and
:= Im
Let o 5t-T stand for the set of all sequences (An) ¯ ~: for which the strong limits s-lim E_t nAnEtn and s-lim , (E t A exist for every t ¯ T, and define mappingsWt : .TT --~ L(Ht) by Wt(An) := s-lim Et__nAnEtn. As in Theorem4.22, one easily checks that ~-T is a C*-subalgebra of 5c which contains the identity, and that the Wt are * homomorphisms. The separation condition (6.1) ensures that, for every t ¯ T and every compact operator Kt ¯ K(Ht), the sequence (EtnKtEt__n) belongs to the algebra ~-T, and that for all s ¯ T WS(Et~KtEt-n)
Kt if s = t = 0 if s ¢ t.
(6.2)
Conversely, (6.2) implies the separation condition (6.1) (Theorem 5.51). Moreover, the ideal ~ belongs to ~T. So we can introduce the smallest closed ideal jT of ~c-T which contains all sequences (EtnKtEt_n) with t ¯ T and Kt ¯ K(Ht) as well as all sequences (Gn) Specifying the special lifting theorem to this context yields the following. Theorem 6.1 (a) A sequence (An) :Tis st abl e if an d only if th e o perators Wt(An) are invertible in L(Ht) for every t ¯ T and if the coset (An) -b (~T invertible in thequotient alge bra
258
CHAPTER 6.
FREDHOLM SEQUENCES
(b) If (A,~) E j:T is a sequence with invertible coset (An) T, then all operators Wt(An) are Fredholm on t, and t he n umber of t he n on-invertible operators amongthe Wt(An) is finite. Proof. Apply Theorem 5.37 with 2, := ~T/6, ~-t := {(E~K~Et_~) + with Kt ~ K(Ht)} and with Wt : (A,~) + 6 ~ Wt(A,~). " Wesay that a C*-subalgebra ,4 of ~" which contains the identity is standard or satitisfies the standard modelif (A) there is an algebra .TT together with an associated ideal ~7T as above such that ~7T C_ A C_ z-T, and if (B) the family {wt}tET iS sufficient for .4 in the sense that, if (An) ~ and all operators Wt(An) are invertible, then the sequence (A~) stable. Observethe tiny difference to the notion of sufficiency introduced in Section 5.3.3: If the family {Wt} is sufficient for .4 in the sense of (B), then the homomorphisms (A~)+G~-~ Wt(A,~) form a sufficient family for the algebra ,4/~ in the sense of Section 5.3.3, and conversely. In the present context it will be more convenient to think of the Wt as acting on the sequences itself, not on cosets of sequences modulo~. 6.1.2
Fredholm
sequences
Let ~, ~T and ~7T be as in Section 6.1.1, and let .4 _C ~T be a standard C*-subalgebra of Definition 6.2 (a) A sequence (An) ~ A is Fredholm se quence, if its T is invertible in A/f i coset (An) + T. (b) If the sequence (A~) ~ A is Fredholmthen null ity n(A,~), defi ciency d(An) and index ind (An) are defined n(A,~)
:= EdimKerWt(A,,), tET
d(An)
:= EdimCokerWt(An)
(6.3)
t~T
and ind (An) := n(An) - d(An). Observe that the numbers n(A,~) and d(A,~) and, hence, ind (An) are finite due to Theorem 6.1(b). It is a triviality to carry over the well-knownproperties of Fredholmoperators to Fredholm sequences. Here is what results:
6.1.-
FREDHOLM SEQUENCES IN STANDARD ALGEBRAS
259
Theorem 6.3 Let .4 be a standard algebra. (a) Every stable sequence (An) E .4 is Fredholm, and n(An) = d(An) ind (An) = 0 in this case. (b) If (A,~) ~ .4 is a Fredholm sequence and (En) ~ A is a sufficiently small sequence (in the sense of -4/~) sequence then the sum (An + is Fredholm, too, and n(An ÷ En) <_ n(An), d(An ÷ En) <_ d(An), ind (An + En) in d (An). (c) If (An) ~ -4 is a Fredholm sequence and if the sequence (Kn) belongs to fit then (An ÷ Kn ) is a Fredholm sequence, too, and ind (An + Kn) ind (An). (d) The adjoint sequence (An)* = (A~) of a Fredholm sequence (An) Fredholm, and n(d~) = d(An), d(A~) -- n(An), ind (A~)= -in (An). (e) If (A,~), (Bn) ~ A are Fredholmsequences, then their product (An)(Bn) is also Fredholm, and ind (An)(Bn) = ind (A,~) + ind (Bn). The corresponding results for Fredholm operators can be found in textbooks on functional analysis and operator theory, such as [50], [72], [107] or [157]. In Section 6.2 we will point out that and howthe quite formally introduced numbers n(An) and d(An) are related with the asymptotic behavior of the singular values of An. 6.1.3
Fredholm
sequences
and stable
regularizability ~ denote the orGiven a closed subspace Mof a Hilbert space K, let PM thogonal projection from K onto M. Theorem 6.4 Let A C_ ff~T be a standard algebra and (An) ~ A a Fredholm sequence. Then (An) is stably regularizable, and the Moore-Penroseinverse of (An) + ~ in -4/~ is the coset (Bn) + ~ where B, = (A~An + E Et~ P~rW’(A.)Et--n)-XA~
(6.4)
tET
for all sufficiently large n. Thus, Fredholm sequences are stably regularizable, and one can describe their Moore-Penroseinverse modulo ~ ’explicitly’. Observe that the sum in (6.4) is actually finite due to Theorem6.1(b). The existence of the inverse in (6.4) for large n is part of the assertion. t¯p~H Proof. Consider the operators C,~ := A~An + ~teT ~ KerW~(An)~-,~ ~.:~t The sequence (Ca) belongs to -4 because fit C_ -4, and taking into account the separation property (6.2), one gets, for every s E Ws(Cn) = WS(An)*W~(An)
+ PKH:rw.(An).
(6.5)
260
CHAPTER 6.
FREDHOLM SEQUENCES
The operators WS(An) are Fredholm by assumption, hence they are MoorePenrose invertible (Theorem 2.4). Theorem 2.15 in connection with the discussion in Example2.16 yields the invertibility of the operators Ws (Cn). Since {WS}s~T is a sufficient family, we conclude that (Cn) is a stable sequence and, hence, the operators Cn are invertible for all sufficiently large n. For these n, set Bn := Cn An. From (6.5) and Theorem 2.15 conclude WS(Bn) ~- (W~(An)*W~(An)
~ ~ Ke I:)H rW ’(A.)] rr
h-- I T])’S A "~ k~nj[ = WS(An+,
whence W~(AnBnA,~ - An) = W~(BnAnBn = W~((AnBn) * - AnBn) = W~((BnAn) + - BnAn) -= for every s E T. Since
II(Dn)+ ~[[ = supIIWS(D,,)ll for every sequence (Dn) E A by Theorem 5.39, one has (AnBnAn - An) ~ G, (BnAnB,~ - Bn) ((A,~B,~)* - A,~Bn) ~ ~, ((B,~A,~)* - B,~An) i.e. (Bn) + ~ is indeed the Moore-Penrose inverse of (An)
6.1.4
Fredholm sequences ity
and Moore-Penrose
stabil-
The general criterion for Moore-Penroseinvertible elements in C*-algebras (Theorem 2.15) immediately yields the following characterization of MoorePenrose stable sequences. Proposition 6.5 The sequence (A~) ~ jz is Moore-Penrose stable i] and H,~ is stable. If this condition is satisfied only the sequence (A~An+ P~erA~) . H,~ --1 * then the sequence (Bn) with Bn : (AnAn + P~erA~) An for all sufficiently ~. large n is the Moore-Penroseinverse of (An) in
A comparison of the expressions for Bn in the preceding proposition and in (6.4) suggests that there might be a connection between the kernels of An and the kernels of the limit operators Wt(An). This connection is established in the following result:
6.1¯
FREDHOLM SEQUENCES IN STANDARD ALGEBRAS
261
Theorem 6.6 Let A C_ ~T be a standard algebra and (An) E .4 a Fredholm sequence¯ Then (A,) is Moore-Penrose stable Hn ~-~ lira ][Pi~erA, -- Z~ ",*"
KerWt(A.)
~:~-n
]1
: 0.
(6.6)
tET
Proof. Let the sequence (An) be Moore-Penrose stable, and abbreviate the H~ + ~ and ~[-~tET ~n KerW’(A.)~-,~ + in ~-/ ~ cosets (An) + G, (P{~erA.) Hn to a, pl and pz, respectively. Clearly, A,~ Pl~erAn ---- 0, and the sequence (A~An+ P(~ora.) is stable by Proposition 6.5. Hence, P~ = Pl, apl = 0, and a*a +pl is invertible
in ~-/~.
(6.7)
On the other hand, as we have seen in the proof of Theorem 6.4, the tt n * t sequence (AnAn + ~t~T E~nl:)~erW,(A.)E-n) si st& ,blei.e. a*a + p~ is invertible. Moreover, ap~ = 0, which can be checked as .follows: For every sET, t n’
W’(A,~
¯ P{~rw’(a.) =
t~_T
whence, via Theorem 5.39, (An ~-~ 15~t t~T
i.e. ap2 = 0. Finally, one has p22 = p~: Indeed, the separation property ensures that
$~T
for all s E T. Via Theorem5.39, this shows the idempotency of P2. Thus, as for pl, we find that P22 =p2, ap2 = 0 and a*a+p2 is invertible
ingV/G.
(6.8)
Since both Pl and P2 are self-adjoint, we conclude from (6.7), (6.8) Theorem2.15 that pl = p2. Let, conversely, (An) E ,4 be a Fredholmsequence which satisfies (6.6). Then a + p2 is invertible by Theorem6.4, and since pl = P2 due to (6.6), one also has invertibility of a + p~ in ~-/G. The latter is equivalent to the Moore-Penrose stability of (A~) by Proposition 6.5¯ Thus, Moore-Penrose stability
of a Fredholm sequence (An) is equivalent
262
CHAPTER 6.
FREDHOLM SEQUENCES
to a certain asymptotic kernel structure of the operators An. It is natural to ask whether this kernel structure is exact under certain circumstances, i.e. whether e~’~ l~pt lt)H H. l~Pt (6.9) P~erAn : /__~t-~n* KerWt(A,.)~-~-n for all sufficiently large n. It is indeed quite easy to establish somesufficient conditions for the equality (6.9). In fact, it is enoughto guarantee that the l~t are exactly equal to 0 (and not only operators An ~teT l~,t --~"pHt XerW’(A.)~-n convergent to 0), and that the sequence (~t~T--n*~t PH’KerW,(An)~_njm ~ is an exact projection (and not only a projection modulo 6). If these conditions are satisfied, one can repeat the arguments of the preceding proof for the algebra ~" instead of ~:/G, and what results is Theorem6.7 Let A c_ ~=’T be a standard algebra and (An) ~ A a Fredholm sequence, and assume that ~,n ~n* KerW~(A~)~-n t tt U (ii) (En~erW,(A=)E_n) (
= O~ and ~£at Es H" s nP~erW~(A=)E_n)
tt H equas l (E~P~erWt(A=)E_n)
t
s = t and is equal to 0 i] s # t for all s, t ~ T and all su~ciently large n. Then, if only n is large enough, H. P~erA,,
t~’~ 12t p~H K’t = ~ ~:% KerW’(A.)*~-n" t6T
In particular, condition (i) of this theoremis satisfied if, for every t ~ (iii) Im p~t C_ Imp~t+~for all (iv) there is an no such that KerWt(An) C_ ImPnto, and (v) Et_nAnEt~ = pt~wt(An)Ptn. Indeed, under the assumption of (iii)-(v)
we
tl~t p~H An ~n KerWt(A,~)~-~-n
for n > 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 R,~. Proof. For contrary, assume there is an infinite subset N1 of N such that dim Im Q,~ is strictly larger than dim Im Rn for all n ~ 1~1. For n consider the restriction of Rn onto Im Q,~. Since the dimension of the range Im Rn is less than that of the initial space Im Qn by assumption, the operator Rnllm Q. must have a non-trivial kernel, i.e. there are vectors v~ ~ Im Qn with IIv,~ll = 1 and Rnvn = O. Let further (Bn) ~ iT be sequence such that Bn(A~An+ Rn) = Pn for all sufficiently large n. (The existence of (B~) is a consequence of the stability hypothesis.) Thus, all sufficiently large n E N1, we obtain Bn(A~An+Rn)vn = v~ and, hence, BnA~Anv,~ = vn for large
n ~ N1.
(6¯21)
Because vn ~ Im Qn and Qn is a projection, this identity can be rewritten B,~A~A,~Qnv~ = vn for large n ~ N1.
(6.22)
The sequences (Bn) and (vn) are bounded, and (A~AnQ~) is a zero sequence by hypothesis. Thus, the norm of the left hand side of (6.22) tends to zero asn -~ oo, whereas the right hand side of (6¯22) has norm 1 for all n E N~. This contradiction proves the assertion¯ ¯ The desired estimate dim Im II n _~ dim Im f~ follows from applying Lemma 6¯17 with H,~ and fin in place of Q,~ and R,~, respectively¯
270 6.2.4
CHAPTER 6. Lower
estimate
of
FREDHOLM SEQUENCES
dimImHn
It is obvious to try to prove the lower estimate dim Im Hn > 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 <- (1 - -1/2. []P~PI~]I) Proof. (a) Every element of M + N can be uniquely written as m + where m ~ Mand n ~ N, and the projection P maps m + n onto the first component m of this sum. Hence, IlPll
= sup Ilm[~ - sup Ilmll ~,=~ Ilm÷nll -~,.=e~, lira÷nil" m+n~O
(6.23)
For the latter equality in (6.23) observe that the second supremumis taken over a smaller set than the first one and is, hence, not greater than the first supremum.If the first supremumwould be strictly larger than the second one, then one could find elements mo ~ Mand no ~ N with mo+ no # 0 such that sup . (6.24) Ilmo + no]l > m~0
6.2.
271
FREDHOLM SEQUENCES: SINGULAR VALUES
Evidently, this cannot hold for m0~ 0. Hence, m0= 0, which contradicts (6.24). Thus, (6.23) holds, and from this equality we conclude lIPIl=
1 1 1 sup --= m~Sa(M)lira + nil inf,~S,(M),,~ N Ilrn + nil dist ($1 (M),
(b) Let (., .) refer to the inner product in H. For all m E M= H and n E N = IMPS, ~ ~Im - ~I
=
(P~m
=
- gn, P~m - P~n> n~ ~ H ~]m~[2+~]n[~2-
H H
whence, via the Cauchy-Schwarzinequality, lira - nil 2 _> Ilmll2 + ][nl]2 - 2 ]IpHMp~II IIrn]l IInll. If, in particular,
m ~ SI(M), then k 1 + Iln]] ~ - 2 I]P~P~ll = (]]P~P~I[- ]ln]l) 2 ~ + 1 -IIP~PNH]I >~ 1 -]lP~P~,l]
for all n ~ N. Consequently, dist ($1 (M), 2 _>1- IIP~IP~H Hll 2, and part (a) yields the assertion.
¯
Assertion (a) of the preceding lemma remains valid in the Banach space setting. Corollary 6.20 Let (Mn) and (Nn) be sequences of finite-dimensional subspaces of a Hilbert space H with Mn N N,~ = {0} for all sufficiently large n and [[PMH.PNH~[[ --> 0 as n --~ oo. Thenthe normof the (in general, non-orthogonal) projection operator firom Mn+ Nn onto Mnparallel to N,~ is less than 2 if only n is sufficiently large. The following result completes Lemma6.16. Lemma6.21 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 [[p - P~mO. H pQ.[] -~t 0 a8
272
CHAPTER 6.
FREDHOLM SEQUENCES
Proof. Because of I[Q,~PQn - Pll -4 o, the operators QnPQ,~are almost projections, and applying Proposition 2.20 we see that Q,~PQn = Rn + Gn
(6.25)
with certain orthogonal projections R,~ belonging to alg (QnPQ,~, I) and with a certain sequence (G,,) tending to zero in the operator norm. check of the proof of Proposition 2.20 further shows that the operators Q,~PQ,~+ (I - R,~) are invertible for n large. Starting with the trivial identity (Q,~PQ,~ + (I --R,~))(I -Pin, we thus obtain H (I-PimQ,,pQ,)
Q,,PQ,) =
(I --
Rn)(I--
= (QnPQn+(I-R,,))-I(I-R,,)(I-P~Q,,pQ.,),
PimQ,,PQ,,)
(6.26)
where all factors on the right hand side commutewith each other since PiH~ commutes with Q,~PQ,~ and since R~ belongs to the algebra alg (QnPQn,I). Thus, multiplying (6.26) by I - Rn yields I -PimQ,PO,,,H~_ (I
_ pimQ,,p~,,~)(iH _ Rn) =(I - n,~)(I - pltfmQ,,pc~.,)
or, equivalently,
=
QpQo = P ’mQ.p .
whence, finally, Im Rn C_ ImQ,~PQn. (6.27) Weclaim that in (6.27) equality holds. Since dim Im QnPQ,~= dim Im P by Lemma6.16, it remains to verify that dim Im Rn = dim Im P, too. Then the finite dimensionality of Im P will provide the desired equality. From (6.25) we derive that [IR~ - P[] -4 0. Set C,~ := P+R~ - I = (2P- I) +R~ -P. The operator 2P - I is invertible (it is its owninverse)i, and since ]]Rn P]] -~ 0, a Neumannseries argument shows the invertibility of C~ for all large n. Further one has C,~P = (P + Rn - I)P = R,~P = Rn(P + Rn - I) = R,~Cn, i.e. Rr~ = CnPCg~, which gives dimImR~ = dimImP. Thus, instead of (6.27) one actually has Im R,~ =Im Q,~PQ,~ or, what is the same, R,~ = P~mQ,PC~," Nowit is clear that ][P- P~mc~,,PCJ,,]] =]]P - Rn]] -> 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<~
and
sup
dimImKn
Proof. If (An) is of central rank one in 9v, then every matrix A,~ is of rank one in Cn×n, hence dim Im An <_ 1. Conversely, let (An) ¯ J= be a sequence of matrices with dim Im An <_ 1 for every n, and let (Bn) ¯ ~: arbitrarily. Then there are numbers #n such that AnBnA,~ = #nAn. (6.36) The numbers #n are uniquely determined if An ~ 0; in case An = 0 we choose #n = 0. The so-defined sequence (#,~) is bounded. Indeed, (6.36) implies I#nl IId,~ll <- IIAnll211Bnll for every n whence I#nl <- IIAnll IIBull for every n with An ¢ 0. This observation identifies the elements of Central rank one. It is clear now that the elements of finite central rank are just the sequences (An) ¯ with sup dim Im An < o~ which yields the assertion.
.
Observe that J(5 r) is a proper ideal of ~’. Indeed, suppose for contrary, that there is a sequence (Kn) ¯ v such t hat supllI~-K~ll
< 1/2 and
supdim ImKn < oo.
Then every matrix Kn is invertible, hence, dim Im Kn = n, which contradicts the second condition of the choice of (K,~).
6.3.
285
A GENERAL FREDHOLM THEORY
Fredholm sequences. Based on the ideal of the centrally compact sequences in jr one can introduce an appropriate class of Fredholmsequences. Definition 6.34 A sequence (An) E = i s a l~redholm se quence if it is invertible modulothe ideal J(jr) of the centrally compactsequences. The following properties of Fredholm sequences are obvious. - Every stable sequence is Fredholm. - The adjoint of a Fredholm sequence is Fredholm. - The product of Fredholm sequences is Fredholm. - If (A,~) is Fredholm and (Kn) E J(jr), then (A,~ + Kn) is Fredholm. - The set of the Fredholmsequences is open in jr. For another characterization
of Fredholm sequences, let again 0 ... _<~(~n) denote theeigenvalues ofA~An andwrite a~n):= ,~i( (n)~1/2, _> for the singular values of An. Theorem6.35 Each of the following conditions is equivalent to the Fredholmness of a sequence (An) (a) There is a sequence (Bn) ~ jr and a sequence (Jn) ~ J(jr) of finite rank such that B~A~A~ = In + Jn. (6.37) (b) There is a k such that li, m~i~nf a(k~,> 0.
(6.38)
Proof. Let (An) be a Predholm sequence. Then (A~An) is a Fredholm sequence and, by definition, there are sequences (B,~) ~ jr and (J,~) such that (Bn)(A~A,~) = (In) + (Jn). (6.39) One can assume that sup dim Im Jn < ~. Indeed, by Proposition 6.33, there exists a sequence (Kn) ~ jr such that II(Jn) - (Kn)ll < 1/2 and that sup dim Im K~ < ~. Writing (6.39) (B,~)(A~A,~) = (I,~) + (Jn and taking into account the invertibility
of (In) + (Jn -- Kn) in jr, one
(In + Jn - Kn)-I(B,~)(A~An) = (In) + (In + J,~ - K,~)-I(Kn) with dim Im (I,~ + Jn - K~)-I K,~ ~_ dim Im Kn.
286
CHAPTER 6.
FREDHOLM SEQUENCES
Denoting (IN + J, - Kn)-1 (Bn) and (In + Jn - Kn)-1 (Kn) by (Bn) and (J,~) again, we arrive at (6.37). Let now the sequences (Bn) and (Jn) be as in (6.37) and A*~A,~ = U~A,~U,~with A := diag (A~n),..., and with unitary matrices U~ refer to the diagonalization of A~An. After multiplication by U, and U~, the identity (6.37) becomes (U,B,U~)(A,) Abbreviating
= (In)
C, := UnB, U~ and F, := U,J,U~ we get.
CnAn= Cndiag (A~~), ..., where still
+ (U,J,U~).
~)) = In + Fn for all
(6.40)
sup dim Im F, < ~. Set k := lim supdim Im Fn.
Weclaim that lim inf £~ > 0. Assume this is wrong.. Then there is an infinite subsequence (n~)~kl of N such that hm~Ak+i -- 0. Multiplying (6.40) from both sides by Pk+~= diag (1, ..., 1, 0, ..., 0) (with the occurring k + 1 times) we get Pk+~C,,An, P~+I = Pk+l + Pk+IFn, Pk+I, and, since
[IA~,P~+II[= [[diag (A~~’) x(~’) 0, 0)11 x("’) ~ lim
IIPk+~ +Pk+~Fn, P~+~ll =O.
Thus, the matrices P~+IF,~P~+l ~ C(~+~) x (k+~) are invertible for all sufficiently large n~. This is impossible because of dim Im Pk+~F,,Pk+~ ~ dim Im F,, ~ k < k + 1 which proves the claim (b). Finally, for the proof that (b) implies the ~edholmness of (An), let k be a number such that inf and let A, = UnEnVgwith En = diag (a~n), ..., a~")) and with unitary matrices U, and I~ refer to the singular value decomposition of A,. The
6.3.
287
A GENERAL FREDHOLM THEORY
choice of k guarantees that the sequence (En + Pk)n>_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 <_ ~rt =r=rankk which verifies the rank identity.
¯
Weproceed with relations between the rank of an element and the range dimension of its image under irreducible representations. Proposition 6.48 I] k ~ It is o] finite
rank, then rankk = dim Im~rt(k).
Proof. Recall from the proof of Theorem 6.37 that the irreducible representation r, of .4 maps elements of rank one onto operators with range dimension at most one. Hence, if k is the sum of r rank one elements, then rt (k) is the sum of r operators of rank one, whence dim Imut(k) _< r = rankk. Conversely, suppose nt(k) is a compact operator with range dimension Choose an orthonormal basis e~, ..., e, in Im~rt(k), let Pi stand for the orthogonal projection from Ht onto Cei, and let Pi denote the (uniquely determined) element in It such that r(pi) = P~. Since ~ Pi is the orthogonal projection from Ht onto the range of ~r,(k), one has ~t(k)
= Pirt(k)
= rt(pik) i~l
= rt(~pik).
6.3.
295
A GENERAL FREDHOLM THEORY
Due to Proposition 6.42, this implies k = ~p~k. One easily checks that every p~k is a rank one element. Thus, k is the sum of r rank one elements whence the estimate rank k _< dim Im~t(k). Fredholmness modulo G(‘4). Now we will have a closer look at the Fredholm theory associated with the ideal G(‘4) of the compact elements. The remainder of this subsection is not needed in what follows. Recall that a E A is G(A)-Fredholm if the coset a + G(A) is invertible in the quotient algebra A/G(A). If a is G(A)-Fredholm then a*a and an* are G(‘4)-Fredholm, too, and there exist elements b, c E .4 as well as elements kl, k2 of finite rank such that ba*a=e+kl
and
aa*c=e+k2.
Let a(a) stand for the smallest non-negative integer which owns the following property: there are a finite rank element kl with rank kl = and an element b ~ .4 such that ba*a = e + kl. Analogously, ~(a) is defined as the smallest possible rank of k2. Finally, define the index of a by inda := c~(a) - ~(a). In case .4 = L(H), one has G(L(H)) = K(H) and c~(A) = dim KerA and ]~(A) = dim CokerA for every Fredholm operator. Here is a generalization of these results to arbitrary G(A)-Fredholm elements. Theorem 6.49 Let a ~ A be a Fredholm element modulo G(A). Then a(a) = ~’~.dim gerrt(a)
and f~(a)
= ~dim Coker~rt(a).
tET
tET
Observe that the occurring sums are actually 6.46.
finite
thanks to Theorem
Proof. Wewill verify the first assertion only. Let a ~ .4 be a Fredholm element, and let b ~ .4 and k ~ G(.4) be elements such that ba*a=e+k
and
rankk=a(a).
Write k as ~ kt with kt ~ It (which can be done uniquely). By Proposition 6.47, ba*a = e + ~ k~ and a(a) = ~ rank k~. The separation property Proposition 6.45 (b) implies that ut(b)~rt(a*)~rt(a) =et + ~rt(kt)
296
CHAPTER 6.
FREDHOLM SEQUENCES
with et referring to the identity operator on Hr. Hence, dimKer~rt (a) a(Tct (a)) .
_< rank~rt(kt)
= E ra nk kt = r ankk = a (a ).
For the reverse inequality, let Pt stand for the orthogonal projection from Ht onto Ker~rt(a), and write Pt for the (uniquely determined) element It such that 7ct(pt) = Pt. Consider the element 5 := a*a + ~’~Pt (again this sum contains only a finite number of non-zero terms due to Theorem6.46). Since a is a Fredholm element modulo G(A), also & is a G(A)-Fredholm element. Moreover, the operators ~rt(a) = ~rt(a*)rt(a) are invertible for every t E T. The lifting theorem implies that ~ is an invertible element of A. In particular, there exists a b E A such that b~ = e resp. ba*a =e-bEpt = e- Ebpt. Hence, o~(a) _< rank Ebpt ~ E rankbpt <_ Erankpt = E rank Pt = E dim ImPt = E dim Ker~-t(a), yielding finally the desired kernel dimension identity.
¯
Let us mention a consequence of the kernel dimension identity. Clearly, the Fredholm element a determines its a-number uniquely, but the finite rank element k in ba*a = e + k is not determined uniquely by a (it depends on the choice of b ~lso). Thus, it is a priori not evident whether a uniquely determines the ranks of the elements k, (only their sum rank k is determined by a). The preceding theorem states that a also determines the ’local’ ranks rank kt uniquely since rank kt = dim Ker ~rt(a). Corollary 6.50 Let a ~ .4 be a Fredholm element modulo G(JI). Then ind a = E ind rt(a). tET
With this corollary it beco~nes obvious that the introduced functionals a, ~ and ind for G(A)-Fredholmelements satisfy all the commonproperties one knows for the functionals dim Ker, dim Coker and ind in case of l~redholm operators.
6.3. 6.3.3
A GENERAL FREDHOLM THEORY Fredholm
sequences
in
standard
297 algebras
Nowwe return to Fredholm sequences, i.e. to sequences which are invertible modulo the ideal J(~’) of the centrally compact sequences in ~-. The main goal of this subsection is to single out a class of subalgebras .4 of ~" such that, for sequences belonging to A, their Fredholmness coincides with the invertibility modulo an ideal of compact elements as considered in the previous subsection. This makes the lifting theorem and its consequences (the dim Ker identity) available to the determination of the a-number Fredholm sequences. Let us agree upon reserving the notation a(An) for the a-number of a Fredholm sequence (An) (i.e. modulo J(gV)). If we to consider a-numberswith respect to other ideals of compactor essentially compact sequences, we will mention this explicitely. Algebras with center c. Let .4 be a unital C*-subalgebra of the algebra 9~, and throughout what follows suppose that the ideal G = G(~) belongs to .4. Then G is a closed ideal of A, and it is easy to check that coincides with G. Lemma6.51 I] ~ C_ A C_ Y= and A is unital, then the center o].4 is isomorphic to a subalgebra o] l ~ which contains c. Proof. Let In denote the n × n identity sequence in c with limit a. Then
matrix, and let (an) C bea
The sequence a(In) belongs to A since this algebra is unital, and the sequence ((an -a)In) tends to zero in the norm whence ((an -a)In) Clearly, (anIn) belongs to the center of A. Conversely, let the sequence (Cn) be in the center of A. Since G _c ~4, every matrix C,~ commuteswith every other matrix in Cn×n, hence, anI,~ with a sequence (an) E ~. ¯ Wesay that the center of the algebra A C_ Y: is c if this center consists exactly of the sequences (cnIa) with (c,~) ~ c. Here are two instances algebras ,4 with center c. Suppose H is a separable infinite-dimensional Hilbert space and (Pn) is a sequence of orthogonal projections Pn from H onto an n-dimensional subspace of H such that (Pn) converges strongly to the identity operator on H. Assumefurther that all sequences (An) in ~l possess the following property: If the matrix An is identified with an operator on Im-Pn, then the strong limit s-lim AnPnexists. If, in particular, An = Chin with complex numbers Cn, then this strong convergence implies that (ca) ~
298
CHAPTER 6.
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Another class of algebras with center c is constituted by the fractal algebras (see Section 1.6.1). Theorem1.66 immediately implies that the center of a fractal unital C*-subalgebra of ~- which contains the ideal ~ is C.
Fredholm inverse closed subalgebras. If ,4 is a unital subalgebra of ~" which contains 6 and has center c, then every central rank one sequence in A is also a central rank one sequence in ~’, hence, J(A) C_ J(J:). Definition
6.52 The subalgebra A o] ~ is called Fredholm inverse closed
if J(A) = A n J(~).
(6.46)
Thus, if .4 is Fredholm inverse closed, and if (An) E A is a Fredholm sequence (i.e. invertible modulo J(~’)), then (An) is invertible .4 A J(9 r) (due to the commoninverse closedness of (..4 + J(.~))/J(.~) A/(A ~ J(Sr))) and, hence, (An) is invertible modulo JtA). It will be pointed out that, for Fredholm sequences (An) which belong to a fractal and Fredholminverse closed subalgebra A of 3r, their a-number can be determined by an analogue of the kernel dimension identity in Theorem 6.49. Sequences of essential rank one. Let .4 be a unital C*-subalgebra of Y which contains the ideal ~ and has center c. A central rank one sequence of .4 is said to be of essential rank one if it does not belong to the ideal ~. For every essential rank one sequence (K~), let J(Kn) refer to the smallest closed ideal of A which contains the sequence (Kn) and the ideal ~. Since .4 has center c, i t is obvious that, for every essential rank one sequence (Kn) of ,4, the coset (Kn) + is a r ank one element of t he quotient algebra A/G and that J(K~)/~ = I((Kn) (with the notation I as in the previous subsection). This shows that Fredholmness of a sequence (An) E A is closely related to the (common)Fredholm theory in the algebra A/6. In particular, if T refers to the set of all equivalence classes of rank one elements in A/G, then there exists a t ~ T such that J(K~)/~ = h. (6.47) Let Tess stand for the set of all t E T for which there is an essential rank one sequence (Kn) ~ .4 such that (6.47) holds. Further, write Gess(.A/~) for the smallest closed ideal of A/~ which contains all ideals It with T ~ Tess.
6.3.
A GENERAL FREDHOLM THEORY
299
It is easy to check that the lifting theorems (Theorems6.44 and 6.46) and their consequences (in particular, Theorem6.49) remain valid if the set and the invertibility modulothe ideal G(A/G) are replaced by the set Tess and by invertibility modulo Gess(A/~). In particular, the coset (An) + is invertible in A/G if and only if the operators ~rt((An) + G) are invertible for all t E Tess and if (An) ÷ is invertible mod ulo Gess(‘4/~). Moreover, if the coset (A,~) + 6 is invertible moduloGess(A/G), then ~e..(A/~)((Au)
= ~ dimger~rt ((g,~) + G).
For t E Tess, abbreviate the homomorphism A ~ L(H~), (An) ~ ~rt((An) by Wt. The homomorphismWt is an irreducible representation of the algebra ‘4 (Theorem 5.7) which maps the ideal G to zero. With these notations and identifications we have J(A)/O = Gess(A/O), and the following version of the lifting theorem holds. Theorem6.53 Let .4 be a unital and l~redholrn inverse closed C* - subalgebra o] Y: which contains the ideal ~ and has center c. A sequence (An) ~ A is stable i] and only i] (An) is a ~redholm sequence and i] operators Wt(An)are invertible ]or all t Tess. Fredholm sequences in fractal and Fredholm inverse closed algebras. Nowwe are going to formulate and prove the main result of this section: a relation between the c~-number of a Fredholm sequence (An) and the kernel dimensions of the operators Wt(An). This result is a generalization of the identity (6.3) which served above as the definition of the a-number of a Fredholm sequence. In particular, it is not needed for this generalization that the underlying algebra is standard (Section 6.1.1). The point is that condition (B) in the definition of a standard algebra is not required here, and also the explicit form of the homomorphisms Wt as strong limits is not assumed in what follows. Theorem6.54 Let ,4 be a unital, ]ractal and Fredholm inverse closed C*subalgebra o] Y: which contains the ideal ~, and let (A,~) ~ A be a Fredholm sequence. Then (6.48) ~(An) = ~ dim KerW~(A,~). tET~
300
CHAPTER 6.
FREDHOLM SEQUENCES
The remainder of this section is devoted to the proof of this result. Let (An) E A be a Fredholm sequence. Since A is Fredholminverse closed, (An) is invertible modulo J(A). Then, by Theorem 6.37, the operators Wt(An) are Fredholm for every t E Tess and, by Theorem 6.46, there are only finitely many operators Wt(An) which are not invertible. Let PKerW~(A~) denote the orthogonal projection from Ht onto the kernel of Wt (An) (only a finite number of these projections are not zero). Decomposeeach of these projections into a sum of dim Ker Wt(An) orthogonal projections of range dimension one: dim Ker
PKerWt(A,,)
=
E
Pi,t
such that Pi,tPj,t = Pj,tPi,t = 0 whenever i ~ j (for example, by choosing an orthonormal basis in Ker Wt(An) and by defining Pi,t as the orthogonal projection onto the ith element of this basis). Since ~rt is an isomorphism between It and K(Ht), every projection Pi,t corresponds uniquely to a coset Pi,t ~ It. Clearly, Pi,tPj,t = Pj,tPi,t = 0 if i 7t j. Since Is fq It = {0} for all s, t ~ T with s 7t t, one has moreover Pi,tPj,s
= Pj,sPi,t = 0 whenever (i,t)
~ (j,s).
(6.49)
Further, Proposition 2.21 guarantees the existence of sequences (Pin’t) of projections such that (p~,t) belongs to the coset P~,t. Observe that (P~n’t) ~ .4 since .4 contains 6. Moreover,since (pie, t) + belongs toIt and t is in Tess, every sequence (P~n’t) belongs to an ideal of the form J(Kn) with an essential rank one sequence (Kn). In particular, (P~n’t) e J(A). Weclaim that the lifting of the projections Pi,t to the sequences (Pin’t) can be done in such a way that all projections p~,t have one-dimensional range and that p~,t p~,~ =pjn,S p~,t = 0 whenever(i, t) ~ (j,
(6.50)
for all sufficiently large n. Proposition 6.55 Let ‘4 be as in Theorem6.54 and (p~,t) the lifting of rank one projection Pi,t ~ A/6. Then the projections P~n’t can be chosen in such a way that their range dimension is one. Proof. Let (Kn) e A be an essential rank one sequence such that (p~,t) J(Kn). Then the operator K := Wt(Kn) has rank one (see the proof of Theorem 6.37), and Pi,t = Wt(P~’t) is a projection operator with onedimensional range by construction of (p~,t). Choose compact operators E, F ~ L(Ht) such that EKF= Pi,t, and let (En) and (Fn) be pre-images of E and F under the mapping Wt in J(Kn), respectively. Then EnK~Fn= P~n’t + Gn
6.3. A GENERAL
FREDHOLM THEORY
301
with a sequence (Gn) tending to zero in the norm (recall that the mapping Wt is a homomorphismfrom J(Kn) onto K(Ht) with kernel 6 due to its definition and Proposition 6.42). Multiplying this equality from both sides by p~/,t yields P~’tEnKnFnP~’t = P~’* + P~’tGnP~’~. (6.51) Consider the operators occuring in (6.51) as acting on Im P~,~. Then the right hand side of (6.51) is invertible for n large enough (say, for n such that [[Gn[[ < 1/2), whereas the left hand side of (6.51) has range dimension at most one for every n. In case dim Im P~,~ >_ 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) <... < a(~~). Let k = a(An), i.e. liminfa-,oo a(~n) = 0 and lim infn-,oo a~1 > 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)
<_ n(A,~)
~dim KerW~(A,~) * <_ d(A, ~).
l=l
/=1
(6.59) Weprepare the proof by a few lemmas.
310
CHAPTER 6.
FREDHOLM SEQUENCES
Lemma6.62 Let H be a Hilbert space and A, An, W, Wu E L(H). (a) I] An -+ A strongly and W,~ --, W weakly, then W,~An --~ WAweakly as n --~ cx). (b) I] A~ --~ A* strongly and Wn --, W weakly, then AnWn-~ AWweakly as ~ --~ (:x:).
Weomit the elementary proof.
¯
Observe that, in both instances considered in Lemma6.62, the product sequence does not converge strongly in general, and that, on the other hand, the product of two weakly convergent sequences need not be weakly convergent again. Moreover, even in case the product of two weakly convergent sequences is strongly convergent, then the limit of the product will in general be different from the products of the limits (whereas the product of two strongly convergent sequences converges strongly to the product of the limits of the sequences in any case). The latter fact also implies that the strong limit of a sequence of orthogonal projections is an orthogonal projection again, whereas the analogue assertion
for weakly convergent
sequences
of projections
is false.
Example. Choose H = l 2 @12. As in Section 1.4.2, let Pn and/~n denote the orthogonal projection from l ~ onto its subspace of all sequences (al,.. ¯, an, 0, 0,...) and the associated reflection operator, respectively. The operators Qn :=
1( P,~
are orthogonal projections But Q is not idempotent.
which converge weakly to Q := ½diag (I, I).
Lemma6.63 Let H be a Hilbert space, and let (P,~) be a sequence orthogonal projections on H which converge weakly to an operator P L(H). Then P is sel]-adjoint and non-negative, and dim Im P _< lim sup dim Im P,~. Proof. The non-negativity of P is clear. Set k := limsup dimImPn. There is nothing to prove in case k -- ~. So let k < ~c, denote by Q _> 0 the non-negative square root of P (which can be defined via a functional calculus based on Theorem 1.48), and assume that dim Im Q > 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)) j<0
-~ + ~ajY j and ~b(p):=p(~), j>0
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 j<0 j>0 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 <_ k. It is furthermore obvious that the operators Bn := (An -
A[IH.)([I.o-
Qn) + Q.
are invertible for all n ~ ~/and that their inverses are uniformly bounded: sup JIB, ill < max{l/e, 1}.
326 CHAPTER 7.
SELF-ADJOINT
APPROXIMATION
SEQUENCES
In contrast to the first part of this proof, now we cannot conclude that (Bn)ne, is a sequence in 5c~c, so our previous arguments do not apply. Instead, we will make use of the fact that the algebra L(H) is, when considered as a Banach space, the dual of some Banach space (viz, the space of the trace (:lass operators). So the Banach-Alaoglutheorem tells that the unit ball of L(H) is compact with respect to the *-weak topology inherited by this duality. Since this topology is stronger than that one of the weak convergence of operators, the unit ball (and hence every bounded ball) in L(H) are also compact in the weak operator topology (for details see, e.g., [114], Section 3.5). Consequently, one can find an infinite subsequence # of y such that the (uniformly bounded) sequences (B~l),~e~ (Qn)net, are both weakly convergent to some operators B and Q in L(H), respectively. Thus, letting n E # go to infinity in (An - .~IIH,~)B~I(pn - Qn) = (Pn -and taking into account Lemma6.62 we obtain (A-M) This identity
B(I-Q)
implies the assertion
=
since Q is compact by Lemma6.63. ¯
We conclude this subsection by an example (due to W. Arveson) which illustrates that the inclusions in Theorem7.2 can be proper. Specifically, we will construct a self-adjoint and unitary operator A ~ L(l 2) with a(A) = aess(A) = {-1, 1}
(7.2)
but 0 e lim infess
a(PnAPn)
(7.3)
where Pn again refers to the orthogonal projection from 12 onto the space of all sequences(al, a:,..., a,~, 0, 0...). Let E1 := 4N, E2 := 2N \ 4N, and define a function f on E1 by f(k)
= k 2+1 (~ 2N-l).
Let O1 := f(E1), and set 02 := (2N - 1) \ O1. Clearly,’ both E2 and are infinite subsets of N, and we define f on E2 to be any bijection from E2 onto 02. This construction involves a permutation ~ of N with r~ = Id via if k even f(k) ~(k) := f-l(k) if k odd,
7.1. SPECTRUM OF SELF-ADJOINT
SEQUENCE
327
and we claim that the operator A: 12 ~ l ~ , (an)heN ~ (a~(n))neN has the properties announced. Evidently, A = A*= A-1 # =t=I, whence(7.2) easily follows. For a proof of (7.3), let In denote the set {1,2,... ,n}, and write ~S for the numberof the elements of a set S. It is elementary to check that ~(](EI ¢3 In) \ tends to infinity as n ~t ~x~, which, obviously, implies that lim ~(f(E¢3In) \ It) and, consequently, lim ~(~r(In) \ In)
(7.4)
Since A maps the basis element ek := (0,... 0, 1,0,...) of 12 (with the standing at the kth place) to e.(~), it follows that for every k belonging set Sn := {k E In : ~r(k) ~. In}, we have PnAPnek= 0. But the cardinal.ity of Sn tends to infinity as n -> 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.
SELF-ADJOINT
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( , (~) .(.), (~
.<~))11~. (7.11)
(recall that the sequence (C~k) - D~k)) is fractal by construction). Since ~ ~ both sequences tt~c(k) .(n))n~1}k21 and tt~D(k) .(n))n21)k21 have the same limit
7.1.
SPECTRUM OF SELF-ADJOINT
SEQUENCE
333
as k -~ c~, the right hand side of (7.11) tends to zero, which gives the claim. Thus, every self-adjoint sequence (A,(n)) E corresponds uni quely to a coset (C,~) ÷ 6 (depending only on the sequence (A~(n)) itself) denote by ~r~(A~,(n)). Claim 5. For every self-adjoint
sequence (An) ~ .A and every sequence
r,(A,(n))n>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
<
(7.13)
kEZ
Observe that this set is strictly larger than the corresponding set in [8], where ((1 Ik l)~/2dAk)keZ ~ /I (Z) is required. Clearly, the class B encloses all band operators, i.e. all operators A with d~A = 0 for all sufficiently large Ikl. Furthermore, if a ~ L°~(~), then dL(a)
k is the absolute value of the kth Fourier coefficient ak of a; hence, the Laurent operator L(a) (a ~-j)i,jez be longs to /3 if andonly
I]L(a)]l"
(1 ~ + Ikl)lak]
= E]akl + k~Z
< oc.
(7.14)
\k~Z
The condition ~ lakl < oC means that a belongs to the Wiener algebra W, whereas ~kez(1 + Ikl)levi 2 < c~ is a Besov-space condition (which
7.1. SPECTRUM OF SELF-ADJOINT
SEQUENCE
335
precisely means that a belongs to the Besov space B~/2). Thus, L(a)¯13
if and only if a¯WrhB~2/2.
A well-knowntheorem by M. G. Krein states that WrhB~/2is a (non-closed) subalgebra of L°° (~’), and that
Ilallw,~W~ := II/(a)ll~ RI/~ is a norm on W n ~2 which makes this set to a Banach algebra (see [97] or [27]). Krein’s result easily carries over to the set B. Theorem7.9 The class ]~ is a (non-closed) unital subalgebra of L(12(Z)), and (7.13) is a norm on B which makes B to a Banach algebra. Proof. With every operator A E B, we associate a function aA ~ WNB12/2 via aA(z) := Zd~zk’ Izl = 1. kEZ
Clearly,
IIAII~= IIL(a.~)ll~ forall A ¯
B.
(7.15)
Further, given A = (aij) and B : (bij) in B, note that dA+B < d~ + d~,
(7.16)
d~"_< ~ dA~_r d’r.
(7.17)
k
--
Indeed, (7.16) is evident, and (7.17) can be seen as follows: d~" = sup l~a,+~,~b~, i
r~Z
I = I~ r~Z
~ ~s~p~ai+k,i+r~’sup]bi+r,i] r~Z
~
A B =~ ~ dk-r dr"
i
From (7.16) and (7.17) we conclude
I~n(~+.)~l, g ~]L(~)+L(~.)~., ~L(~.)~. g ]~L(~)-L(~.)~., and (7.18) and (7.19) imply that B is an algebra: If A, B e B, ~]AB~]~ (7.~5)~L(aAB)]~ (*)
(~9) _ ]]L(aA).L(aB)~]~
~ ~L(~)~],.~L(~.)I~
(7.~S) (~.19)
336 CHAPTER 7.
SELF-ADJOINT
APPROXIMATION
SEQUENCES
where (.) is a consequenceof Krein’s theorem 7.9. The proof of the linearity of B is analogous, and that B is a Banach algebra under the norm [1" can be checked straightforwardly. ¯ The main result of this section is as follows. Theorem7.10 Let A be a sel]-adjoint
operator in the algebra l~. Then
(a) lim infess a(PnAPn)aess(A). (b) The sequence (PnAPn) has the Arveson dichotomy. For the proof, we need a theorem by H. Weyl which describes the difference between the spectrum and the essential spectrum of a self-adjoint bounded operator. Theorem 7.11 Let A e L(H) be self-adjoint, and A E a(A) aess(A). Then A is an isolated point in a(A) whoseeigenspace has a finite dimension. A proof can be found in [128], Section XIII.4, Example 3. Further we will need the notion of the trace of a non-negative operator. If H is a Hilbert space with orthonormal basis (eo, el, e2, ...) and A ~ L(H) is non-negative, then the trace of A is the (finite non-negative or infinite) number trA := ~ (Aek, The trace of an operator is independent of the choice of the basis. For details on traces and trace class operators we refer to the monographs[68] and [127], Section VI.6, and to the article [66]. Proof of Theorem 7.10. Weclaim that every point in l~ \ aess(A) transient, which implies both assertions (a) and (b). Indeed, aess(A) lim infess a(P~APn)by Theorem7.2. If there were a point in liminfa(PnAP~) aess(A) _C[¢\ aess (A), then this point were transient, which is impossible. This gives (a), and (b) follows easily since all points in ll~ \ lim infess a(P~AP~)= ll~ \ aess(A) transient. To prove the claim, choose a point A in ll~\aess (A). FromWeyl’stheorem we infer that A is either an isolated point in a(A) with finite-dimensional eigenspace or that A - AI is invertible. In both cases, Ker (A - AI) finite-dimensional, and if we let Q denote the orthogonal projection from H onto this kernel, then Q commutes with A, and the operator (I-Q)(A-AI)(I-Q)+Q
= A+Q-AI
7.1. SPECTRUM OF SELF-ADJOINT
SEQUENCE
337
is invertible. Because the invertible operators form an open subset of L(H), there is an ¢ > 0 such that all operators A + Q - #I with/~ E [A - ~, A + ~] are invertible, and the continuity of the inversion further gives M := sup II(A
+ Q - #i)-1[[
< oo.
(7.20)
Set U := (A - ~, A + ¢). Wewill show that the sequence (Nn(U)), where N,,(U) refers to the numberof eigenvalues of P,~APnin U, is boundedand, hence, that A is transient. Fix n, let let A1, ..., Ap be the eigenvalues of PnAPnwhich lie in U (with repetitions according to the multiplicity, i.e. p = Nn(U)), and choose an orthonormal set {xl, x2, ..., xp} C_ ImPn of eigenvectors of P,~AP,~ such that P,~APnXk = )~kXk. In the next step we will prove the inequality 1 _< M2 [[(I-Pn)AP,~xk[[ 2 + (Qxk,xk), which holds for every k = 1, ...,
(7.21)
p. To get this, observe that
APnxk - A~xk = APnxk - PnAPnxk = (I-
Pn)APnxk
(7.22)
and write 1 - (Qxk,x~) by (7.20)
by (7.22)
which gives (7.21). Summarizingthe inequalities (7.21) over k yields P
P
<- M~ Z I1(I-
P
P~)APnxk[12
+ Z (Qxk’ Xk)
k----1
k:l
P
= M~ Z(P,~A(I-
P
P,~)AP,~xk,
xk) + Z(Qxk,
k----1
(7.2 3)
k=l
_< M2 tr (PnA(I - P,~)AP,~) tr Q. Since Q is a projection, we have tr Q = dim Im Q, and for the trace of the operator P,~A(I - P,~)APn we find tr (P,~A(I - P,~)AP,~)
338 CHAPTER 7.
SELF-ADJOINT
APPROXIMATION SEQUENCF, S
n--i
n--I oo
= ~
Pn)APnek,
k=0
k=0
< 1.(dA~) 2+2.(d2A)2+...+n(dA~)2+n(dn+~)
A
2
+..-
Hence, (7.23) reveals that Nn(U) = p ~ 2 ] [A][~ + di mImQ, and since the right hand side of this inequality is independent of n, we conclude that A is transient. 7.1.4
Arveson
dichotomy:
standard
algebras
A further instance where Arveson dichotomy holds is self-adjoint in standard algebras. Let the notations be as in Section 6.1.1.
sequences
Theorem 7.12 Let .A C .~T be a standard C*-subalgebra o] J:. Then every sel]-adjoint sequence (An) in A has the Arveson dichotomy. More precisely, a numberA belongs to lim infess a(An) i] and only if (An - APn) is not a Fredholm sequence. I] the sequence (An - APn) is Fredholm, then there is an open neighbourhood U of A such that sup N,~(U) = n(Ak APk) n>l
]or all sufficiently large k. Proof. Since the An are self-adjoint, we only have to consider real A. Let tt~ n), #~n), --., tt(~n) denote the eigenvalues of the self-adjoint operator An - APn and label them so that
Then (tt~n)) 2, (#(~n))2, ..., (#(nn))2 are the eigenvalues (= the values)of the non-negative operator (An - APn)2. If now (An - APn) is Fredholm sequence, then its square ((An -APn)2) is Fredholm, too, and by Theorem6.12, this sequence exhibits the splitting property of its singular values. That is, there are numbers d > 0 and ~n > 0 such that lim ~,~ = 0 and a2(dn) C_ [0, ~2~] [3 [d2,oo) and, moreover, the numberof the singular values of (An - APn)2 in [0, ~2~], counted with respect to their multiplicity, is independent of n and coincides
7.2.
SZEGO-TYPE
THEOREMS
339
with the nullity jo := n((An APn) 2) ofthe sequence ((An APn)2). Thus,[~Jo(n)~2-< ~n~and #j0+l(n) [~ ~, which implies for the eigenvalues of . (n) An - APn that ~n), ¯ ", UJo are located in the interval [-~n, en], whereas n) . (n) "’" , #(n the eigenvalues ~0+1, belong to (-oo, -4 U [d, oo). Consider the open interval (-d/2, d/2). The above discussion shows that the number of the eigenvalues of Amwhich lie in U := (A-d/2, A+d/2) is independent of n and equal to j0. Hence, A is a transient point. If (An - APn) is not a Fredholm sequence, then similar arguments combination with Theorem 6.67 yield that 0 e liminfess 2) a((An -APn) and that, consequently, A is an essential point. ¯ Westill mention an analogue of Weyl’s Theorem7.11 which holds for selfadjoint approximation sequences. Theorem 7.13 Let .4 C_ .~T be a standard C*-subalgebra o] ~: and (An) a sel]-adjoint sequence in .4. Then every point in limsupa(An) which not in lim infess a( An ) is isolated in lim sup a( An ). Proof. From Corollary 3.18 and Theorem 7.12 we infer that lim sup a(An) = a~=/g((An)
(7.24)
and lim infessa(An) = a.rr/:Tr
((An)
+ jT),
(7.25)
respectively. As we have seen at the end of Section 5.3.3, there is a Hilbert space H such that one can identify the cosets (An) + ~ .TT/~ with li near bounded operators on H, and the cosets (An) + which be long to theidea l jT + ~ with compact operators on H (Corollary 5.46). Having this identification in mind we see that the equalities (7.24) and (7.25) actually state that lim sup a(An) coincides with the spectrum of a certain self-adjoint operator, and lim infess a(An) with the essential spectrum of that operator. Thus, the assertion in an immediate consequence of Weyl’s theorem for operators. ¯ Moreover we conclude from Theorem 7.11 that the multiplicity point in lim sup a(An) li m infess a(An) is fin ite.
7.2
Szegii-type
of every
theorems
A classical theorem by Szeg5 states that if a E L~(~) is a real-valued function, if ~n),..., -,n~(n) are the eigenvaluesPnT(a)Pn, andif f : ~-~Ii~
340
CHAPTER 7.
SELF-ADJOINT
APPROXIMATION
SEQUENCES
is a compactly supported continuous function, then ~)) 1 f(A~n))+...-I-f(A(n n~olim = 2--’~ f(a(e~t)) n
(7.26)
In this section, we are going to present some recent investigations, which were initiated by William Arveson [8, 9] and further developed by Erik B6dos [12, 13] and Dylan SeLegue [158] and which provide a general setup for SzegS-type theorems in the context of C*-algebras. 7.2.1
F01ner
and
Szeg8
algebras
Westart with describing the algebraic frameworkof generalized Szeg5 theorems. Let, as in Section 7.1, H be an infinite-dimensional Hilbert space and 7-/= (Ha) be a sequence of finite-dimensional subspaces of H such that the orthogonal projections P,~ from H onto Hn converge strongly to the identity operator I on H. Given an operator A E L(H) let ]AI denote its absolute value, i.e. the non-negative square root of A’A, and let further tr denote the canonical trace on H. If A has the matrix representation (a~j) with respect to some orthonormal basis of H, then tr(A*A) =
2. la ijl i,j
In particular, tr Pn = dim Im Pn = dim Ha. Let 5(7-/) stand for the collection of all operators A ~ L(H) having the property that lim tr (]P,~A - APnl) = 0. ~-~o~ tr Pn
(7.27)
Lemma7.14 ~(7/) is a unital C*-subalgebra o] L(H). Proof. Recall that the set Af(H) := {A L(H) : tr (IAD < oo} of the nuclear or trace class operators is a (non-closed with respect to the common operator norm) two-sided ideal of L(H), that the mapping A ~t tr (IAI) defines a norm on Af(H) which makes this set to a Banach space, and that tr (IA + BI) _< tr (IAI) + tr (IBI), tr(IACI) _< tr(IAI) llCll , tr(ICAI) _< IlClltr(IAI), tr (IAI) = tr (IA*])
(7.28) (7.29) (7.30)
for arbitrary operators A, B E Af(H) and C ~ L(H). For details see [127], Section VI.6.
7.2. SZEGO-TYPE
THEOREMS
341
If nowA, B E ~(74), then tr (IP,(A + B) - (A + B)P,I) tr (IP, A - AP, I) + tr (I P, B - BP,I) and tr ([P~(AB) - (AB)P~I) = tr ([(P~A APn)B + A(PnB - BP~)[) <_ IIBIItr(IP,~A-AP,~I) + IIAIItr([PnB due to (7.28) and (7.29), which clearly implies that A + B AB are in ~(74) again. Further, if Am e ~(74) and Am -~ A ~ L(H) in the norm of L(H) then tr OPnA- APn]) _< tr (IPn(A-
Am) -
<_ 2trP~l]A-Amll
+ tr(IPnA.~-A.~P~I),
which gives the closedness of ~(74) in L(H). Finally, the symmetryof ~(74) is a consequenceof (7.30), and that ~(74) contains the identity operator trivial. . Werefer to ~(7/) as the F¢Iner algebra associated with 74. Here is an equivalent characterization of operators in ~(74), which is often more convenient to check than (7.27). Lemma7.15 An operator A E L(H) belongs to the F¢Iner algebra if and only if lim ~-~
tr(I(I-
P~)AP~I2)=lim tr(IP’~A(I- P~)[2) = 0. (7.31)
tr P,~
Proof. Let A ~ L(H) satisfy tr (l(I
2) P~)AP~I
tr P~
(7.27).
Then
tr (P~A* (I - Pn)APn) tr (P~(A*Pn - PnA*)(PnA - APn)P~) tr ((P~A - APn)*(PnA - APn)) IIPnA - APril tr (IPnA - APnl) 2 IIAII tr (IP,~A
whencethe first identity in (7.31) follows. The second one follows similarly by taking adjoints.
342 CHAPTER 7.
SELF-ADJOINT
APPROXIMATION
SEQUENCES
Let now, conversely, A be subject to condition (7.31). Then the assertion will follow once we have established the inequality tr (]PnA~r~nAP~I)
_< Itr (](I -trPnPn)APnl2)
+
I
tr (IP,~A(I ~) P~)I
" (7.32)
To this end note that tr
(IP~A - AP,~I) = tr (IP~A(I - P~) - (I - P~)AP~I) (7.28) < tr([(I-P~)AP~]) ).
+ tr(IP~A(I-Pn)l
The operator [(I- P~)APnl 2 = PnA*(I- Pn)APn acts on H,~; hence, its non-negative square root I(I - P~)APn[acts on this finite-dimensional space, too, and if we let A1,..., Adim H~ denote the eigenvalues of this square root, then the Cauchy-Schwarzinequality yields dim Hn
tr(l(I-P~)AP~l)
dim H,~
= ~ Ar < (dimH~)’/2( r=l
= ~. Taking into account that similarly obtain
v/tr
tr([P~A(I-
r----1
([(IP,~)[)
Pn)Ap~I2). = tr(l(I-
tr (IPnd(I Pn)l) < tr ~/~"~n - V/t r ([PnA(I - Pn)12).
(7.33) P~)A*P~D, we (7.34)
Adding(7.33) and (7.34) and dividing the sum by Pn,we arri ve at ( 7.32). ¯
Let us mention two important examples of operators F¢lner algebra. Example 7.16
belonging to the
: Band operators
Wecall A ~_ L(H) a band operator with respect to 7/ if sup~ tr (IPnA APn]) < c~. Clearly, every band operator belongs to Example 7.17 : Toeplitz
operators
Let H be the Hilbert space l 2 =/2(Z+) with its standard basis ( i)i=0, let Hn be the span of (e0,..., e~_l}. Then every Toeplitz operator T(a) with a ~ L°°(~) belongs to ~(7/). Indeed, given n ~ N, let b,~ stand for
7.2.
SzEGO-TYPE
343
THEOREMS
smallest integer which is greater that or equal to v~. A straightforward computation shows that tr (l(I - Pn)T(a)Pnl 2) 1 = -tr (PnT(-6)(I - P~)T(a)P~) n tr Pn
i~l~ + ~l k=l
~ k=b~+l
I~12+n~
I~l~ 2+... + ~ la~l ~k:l
k=b~
~) 2 +,,, + b~l~b~l = ~ I~kl~ + ~(la, 2 +2la~l
k=b~+l
Because ~=0 I~1~ < ~, lira ~tr
~) =0. (1(I- P~)r(~)P~l
Analogously we find that lim ~tr 7.15, we arrive at the assertion.
(IP~T(~)(I- ~) = 0.Via L emma
Wereturn to the general setting and proceed with a basic technical lemma. Lemma ~.18 ~et A~, ..., tr (IP~A~A~...
A~ ~ L(H). Then A~P~ - P~A~P~A~P~... P~A~P~I)
Proof. Without loss of generality, we can assume ~hat r. ~hen we have tr (IP~A~A~...
IIA~II= I for every
A~P~ - P~A~P~A~P~...
= tr (IP~A~A~...
A~-IP~A~P~ - P~A~P~A~P~... P~A~P~
+ P~I~... ~-~(I - P~)A~P~) <
tr ([(PnA1... Ak-lPn - PnAIPn... PnA~-~Pn)P~AkPnI) + tr ([PnA1Az... Ak-l(I - Pn)AkPn[) tr (IP,~AI ... Ak-~P~-- PnAIPn... P~Aa-~P,~I) . IIP~A~P~II + IIPnAIA2... Ak_~ll. tr (l(I Pn , )AkPnI)
344 CHAPTER 7.
SELF-ADJOINT
APPROXIMATION
SEQUENCES
whence, via induction, k
tr (IPnAI... AkP,~ -- PnA1P,~... P,~A~P~I)<_ tr (l( I - P~)AtPnI). The assertion is a consequence of this estimate together with the inequalities tr (l(I-
P~)AIP,~I)
tr (](I Pn)(AtP,~ - PnAI)]) ]]I - Pnl] tr (IPnAt - AtPn]),
which hold for every At by (7.29). Weare now in a position to formulate and prove the main result of this section. In order to realize both the essence of this theorem and of the forthcoming definition, let us reformulate SzegS’s classical theorem in terms of *-weak convergence of linear functionals. Given a real-valued function a ¯ L°° (~P), set a := ess inf~eva(x),
/~ := ess sup~va(x),
and let (fx~,) C[a, fl ] -- ~ C bethe Dirac func tional at A I ’~) where the ,kl ’~) refer to the eigenvalues of PnT(a)Pn. Further introduce functionals "r,~, "r ¯ C[a,/~]*by
r,~ :=
+~x~n~+...+~), r(I) := ~
f(a(eit))dt.
Being a convex linear combination of multiplicative functionals, the rn are linear functionals satisfying r,~(1) = 1 IIr~ll, i. e. th ey are st ates in sense of Section g.4.1. Clearly, these functionals are no longer multiplicative, but one still has T~(fg) = T~(gI) for all f, g ¯ C[c~,~]. This equality can be viewed both as a consequence of the commutativity of the algebra C[~, fl] and of the fact that T,~ is a linear combination of multiplicative functionals. Generally, a state # on a unital C*-algebra 9/is called a tracial state if #(AB) = #(BA) for
all
A,B¯9/.
In this sense, the ~-,~ as well as T are tracial states. Nowthe Szeg5 theorem(7.26) can be reformulated as the statement that the states Tn convergence to the tracial state ~- in the *-weaktopology
of c[~,~]*.
7.2.
SZEGO-TYPE THEOREMS
345
Theorem 7.19 Let ‘4 be a unital C*-subalgebra of the F¢lner algebra ~(7-l). For every n >_1, let Pn be the state of ,4 defined tr (PnAPn) tr Pn ’
p,~(A)
and let R,~ be the *-weak-closedconvexhull of the set {Pn, Pn+l, Pn+z, ¯ ¯ -}. Then Roo(A) := V~n>_lRn is a non-empty set of tracial states Proof. R1, Rz, ... is a decreasing sequence of non-empty *-weakly closed convexsubsets of the unit ball of the dual of .4, which is *-weakly compact by the Banach-Alaoglu theorem. Hence (see [127], Section IV.3), Roo(.4) is a non-empty *-weakly closed (and thus *-weakly compact) convex set states. It remains to showthat every element of Roo(.4) is a tracial state, i.e. that p(AB) = p(BA) for all
A, BE.4 and pERoo(.4).
Because [p(AB) - p(BA)I limsup Ip, ~(AB) - p ,~ (BA)[, it suffices to showthat Ip,~(AB) - p,~(BA)I -~ as n ~ (x). Since tr (PnAPnBPn)tr (P, ~BP,~APn) for ever y n, w e c an util ize the case k = 2 of Lemma7.18 as follows: 1 [pn(AB) - p~(BA)[ = tr--~ [tr ~_ t-~p. ([tr
(PnABPn) tr (P~BAP~)[
(P,~ABPn - P,~APnBP,~)[ [t r (P,~BAPn - P,~BP,~APn)[)
2 _<~tr---- ~- [[A[[ [[B[[ ([tr (P~AAP,~)[ + [t r (P,~B - BP~)[). The right hand side of this estimate tends to zero due to the assumption A, B ~ ~(7/). Wewill call a unital C*-subalgebra A of the F¢lner algebra ~(7/) Szeg5 algebraif Roo(.4) is a singleton. Observe that Theorem7.19 also provides a necessary condition for an algebra to be a subalgebra of the FOlner algebra. Indeed, there exist unital C*-algebras without any tracial state (the Cuntz algebras On with n _> 2, or the algebra L(H) for infinite-dimensional Hilbert spaces H can serve as examples, see [44]). For these algebras, Theorem7.19 asserts that there is
346 CHAPTER 7.
SELF-ADJOINT
APPROXIMATION
SEQUENCES
no sequence 74 of finite-dimensional subspaces of H such that the F01ner algebra ~(74) contains the given algebra. In the next sections we are going to specify Theorem 7.19 to two instances: first to the context of the classical Szeg5 theorem, and then to algebras .4 C_ 5(7/) which possess unique tr acial st ate 7 (t he ir rational rotation algebras can serve as examples), in which case one can identify the set Roo(A) with the singleton {T}. 7.2.2
SzegS’s
theorem
revisited
Wereturn to the context of the classical Szeg5 theorem, i.e. we let H = /2(Z+), denote by Pn the orthogonal projection from H onto the span e0, el, ..., en-1, set 74 := {Im P,~, n >_ 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.
SELF-ADJOINT
APPROXIMATION
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.
SELF-ADJOINT
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
