Basic Classes of Linear Operators

427 Pages·2003·9.075 MB·English

by  Israel Gohberg, Seymour Goldberg, Marinus A. Kaashoek (auth.)

Basic
Classes of
Linear
Operators
Israel Gohberg
5eymour Goldberg
Marinus A. Kaashoek
Springer Basel AG
Israel Gohberg Marinus A. Kaashoek
School of Mathematical Sciences Department of Mathematics and Computer Science
Raymond and Beverly Sackler Vrije Universiteit Amsterdam
Faculty of Exact Sciences De Boelelaan 1081a
Tel Aviv University NL-1081 HV Amsterdam
Ramat Aviv 69978 The Netherlands
Israel
e-mail: [email protected] e-mail: [email protected]
Seymour Goldberg
Department of Mathematics
University of Maryland
College Park, MD 20742-4015
USA
e-mail: [email protected]
2000 Mathematics Subject Classification 47-01
A CIP catalogue record for this book is available from the
Ubrary of Congress, Washington D.C., USA
Bibliographic information published by Die Deutsche Bibliothek
Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie;
detailed bibliographic data is available in the Internet at <http://dnb.ddb.de>.
ISBN 978-3-7643-6930-9 ISBN 978-3-0348-7980-4 (eBook)
DOI 10.1007/978-3-0348-7980-4
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© 2003 Springer Basel AG
Originally published by Birkhăuser Verlag, Basel, Switzerland in 2003
Printed on acid-free paper produced from chlorine-free pulp. TCF ca
Cover design: Micha Lotrovsky, CH-4106 Therwil, Switzerland
ISBN 978-3-7643-6930-9
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Dedicatedtoour grandchildren
Table ofContents
Preface............................................................. xiii
Introduction....................................•.....•.............. xv
ChapterI HilbertSpaces....................•....................... 1
1.1 Complexn-Space 1
1.2 TheHilbertSpaceZj 3
1.3 DefinitionofHilbert SpaceanditsElementaryProperties 5
1.4 DistancefromaPointtoaFiniteDimensional Space 8
1.5 TheGramDeterminant 10
1.6 Incompatible SystemsofEquations ............................... 13
1.7 LeastSquareFit 15
1.8 DistancetoaConvexSetandProjectionsontoSubspaces 16
1.9 Orthonormal Systems 18
1.10 SzegoPolynomials 19
1.11 LegendrePolynomials 24
1.12 Orthonormal Bases 26
1.13 FourierSeries 29
1.14 Completeness oftheLegendrePolynomials 31
1.15 BasesfortheHilbert SpaceofFunctionsonaSquare 32
1.16 StabilityofOrthonormal Bases ................................... 34
1.17 SeparableSpaces 35
1.18 IsometryofHilbertSpaces 36
1.19 ExampleofaNonSeparableSpace 38
Exercises 38
ChapterII BoundedLinearOperatorsonHilbertSpaces............. 51
2.1 PropertiesofBoundedLinearOperators 51
2.2 ExamplesofBoundedLinearOperatorswithEstimatesofNonns 52
2.3 ContinuityofaLinearOperator 56
2.4 MatrixRepresentations ofBoundedLinearOperators 57
2.5 BoundedLinearFunctionals 60
2.6 OperatorsofFiniteRank 63
2.7 InvertibleOperators 64
2.8 InversionofOperatorsbytheIterativeMethod ..................... 69
2.9 InfiniteSystemsofLinearEquations 71
2.10 IntegralEquationsoftheSecondKind 73
viii TableofContents
2.11 AdjointOperators 76
2.12 SelfAdjointOperators 80
2.13 OrthogonalProjections ..................................... 81
2.14 TwoFundamentalTheorems 82
2.15 ProjectionsandOne-SidedInvertibilityofOperators 84
2.16 CompactOperators 91
2.17 TheProjectionMethodforInversionofLinearOperators 96
2.18 TheModifiedProjectionMethod 105
2.19 InvariantSubspaces 108
2.20 TheSpectrumofanOperator 109
Exercises 118
ChapterIII LaurentandToeplitzOperatorsonHilbertSpaces...... 135
3.1 LaurentOperators , 135
3.2 ToeplitzOperators .. .... ......... ... ....... .... ......... ....... 141
3.3 BandToep1itzoperators......................................... 143
3.4 ToeplitzOperatorswithContinuousSymbols 152
3.5 FiniteSectionMethod.......................................... 159
3.6 TheFiniteSectionMethodforLaurentOperators 163
Exercises 166
ChapterIV SpectralTheory ofCompactSelfAdjointOperators..... 171
4.1 ExampleofanInfiniteDimensionalGeneralization .. .............. 171
4.2 TheProblemofExistenceofEigenvaluesandEigenvectors......... 172
4.3 EigenvaluesandEigenvectorsofOperatorsofFiniteRank .......... 174
4.4 ExistenceofEigenvalues........................................ 175
4.5 SpectralTheorem 178
4.6 BasicSystemsofEigenvaluesandEigenvectors 180
4.7 SecondFormoftheSpectralTheorem 182
4.8 FormulafortheInverseOperator 183
4.9 Minimum-MaximumPropertiesofEigenvalues 185
Exercises 188
ChapterV SpectralTheoryofIntegralOperators................... 193
5.1 Hilbert-SchmidtTheorem ....................................... 193
5.2 PreliminariesforMercer'sTheorem .............................. 196
5.3 Mercer'sTheorem......... ... ... ................. ............ . 197
5.4 TraceFormulaforIntegralOperators 200
Exercises ...................................................... 200
ChapterVI UnboundedOperatorsonHilbertSpace................ 203
6.1 ClosedOperatorsandFirstExamples 203
6.2 TheSecondDerivativeasanOperator 204
TableofContents ix
6.3 TheGraphNorm 206
6.4 AdjointOperators 208
6.5 Sturm-LiouvilleOperators 211
6.6 SelfAdjoint OperatorswithCompactInverse ..................... 214
Exercises .............................................. 215
ChapterVII OscillationsofanElasticString....... 219
7.1 TheDisplacementFunction 219
7.2 BasicHarmonicOscillations 220
7.3 HarmonicOscillationswithanExternalForce " 222
ChapterVIII OperationalCalculuswithApplications............... 225
8.1 FunctionsofaCompactSelfAdjointOperator " 225
8.2 DifferentialEquationsinHilbertSpace 230
8.3 InfiniteSystemsofDifferentialEquations 232
8.4 Integro-DifferentialEquations 233
Exercises 234
ChapterIX SolvingLinearEquationsbyIterativeMethods 237
9.1 TheMainTheorem ............................................. 237
9.2 PreliminariesfortheProof " 238
9.3 ProofoftheMainTheorem 240
9.4 ApplicationtoIntegralEquations................................. 242
ChapterX FurtherDevelopmentsoftheSpectralTheorem.......... 243
10.1 SimultaneousDiagonalization 243
10.2 CompactNormalOperators 244
10.3 UnitaryOperators .................................... 246
10.4 SingularValues 248
10.5 TraceClassandHilbertSchmidtOperators 253
Exercises 254
ChapterXI BanachSpaces......................................... 259
11.1 DefinitionsandExamples 259
11.2 FiniteDimensionalNormedLinearSpaces 262
11.3 SeparableBanachSpacesandSchauderBases 264
11.4 Conjugate Spaces 265
11.5 Hahn-BanachTheorem 267
Exercises " 272
x TableofContents
ChapterXII LinearOperatorsonaBanachSpace .................. 277
12.1 DescriptionofBoundedOperators 277
12.2 ClosedLinearOperators 279
12.3 ClosedGraphTheorem 281
12.4 ApplicationsoftheClosedGraphTheorem 283
12.5 ComplementedSubspacesandProjections 286
12.6 One-SidedInvertibilityRevisited 288
12.7 TheProjectionMethodRevisited ............................... 289
12.8 TheSpectrumofanOperator ................................... 290
12.9 VolterraIntegralOperator 293
12.10 AnalyticOperatorValuedFunctions 295
Exercises ..................................................... 296
ChapterXIII CompactOperatorsonaBanachSpace............... 299
13.1 ExamplesofCompactOperators 299
13.2 DecompositionofOperatorsofFiniteRank 302
13.3 ApproximationbyOperatorsofFiniteRank..................... 303
13.4 FirstResultsinFredholmTheory 305
13.5 ConjugateOperatorsonaBanachSpace 306
13.6 SpectrumofaCompactOperator............................... 310
13.7 Applications 313
Exercises ..................................................... 314
ChapterXIV PoincareOperators: DeterminantandTrace 317
14.1 DeterminantandTrace 317
14.2 FiniteRankOperators,DeterminantsandTraces 321
14.3 TheoremsaboutthePoincareDeterminant 327
14.4 DeterminantsandInversionofOperators 330
14.5 TraceandDeterminantFormulasforPoincareOperators .......... 336
Exercises ..................................................... 340
ChapterXV FredholmOperators.................................. 347
15.1 DefinitionandExamples 347
15.2 FirstProperties 347
15.3 PerturbationsSmallinNorm................................... 352
15.4 CompactPerturbations 355
15.5 UnboundedFredholmOperators 356
Exercises 358
Tableof Contents xi
ChapterXVI Toeplitz andSingularIntegralOperators.............. 361
16.1 LaurentOperatorsonip(Z) 361
16.2 ToeplitzOperatorsonip •...•......•...•..................•.... 364
16.3 AnIllustrativeExample .... .... ................ ............... 372
16.4 ApplicationstoPairOperators .................................. 377
16.5 TheFiniteSectionMethodRevisited ............................ 384
16.6 SingularIntegralOperatorsontheUnitCircle 390
Exercises ..................................................... 395
ChapterXVII NonLinearOperators.. .... . 401
17.1 FixedPointTheorems 401
17.2 ApplicationsoftheContractionMappingTheorem 402
17.3 Generalizations ............................................... 405
Appendix 1: Countablesets andSeparableHilbertSpaces ............ 409
Appendix 2: The Lebesgueintegral and L Spaces 411
p
SuggestedReading 415
References 417
List ofSymbols 419
Index 421
Preface
The presentbook isanexpandedandenrichedversionofthetextBasic Operator
Theory, written by the first two authors more than twentyyears ago. Since then
the three ofus have used the basic operatortheory text in various courses. This
experiencemotivatedus toupdate and improve the old text by includinga wider
varietyofbasic classes ofoperatorsand their applications. Thepresentbook has
alsobeenwritten insuch awaythat itcanserveasanintroductiontoourprevious
books Classes ofLinearOperators, VolumesIand II.Weviewthethree books as
aunit.
We gratefully acknowledge the support ofthe mathematical departments of
Tel-Aviv University, the University ofMaryland at College Park, and the Vrije
UniversiteitatAmsterdam. ThegeneroussupportoftheSilverFamily Foundation
ishighly appreciated.
Amsterdam,November2002 The authors