Table Of ContentLecture Notes in Mathematics 2250
Anna Skripka
Anna Tomskova
Multilinear
Operator
Integrals
Theory and Applications
Lecture Notes in Mathematics 2250
Editors-in-Chief:
Jean-MichelMorel,Cachan
BernardTeissier,Paris
AdvisoryEditors:
KarinBaur,Leeds
MichelBrion,Grenoble
CamilloDeLellis,Princeton
AlessioFigalli,Zurich
AnnetteHuber,Freiburg
DavarKhoshnevisan,SaltLakeCity
IoannisKontoyiannis,Cambridge
AngelaKunoth,Cologne
ArianeMézard,Paris
MarkPodolskij,Aarhus
SylviaSerfaty,NewYork
GabrieleVezzosi,Florence
AnnaWienhard,Heidelberg
Moreinformationaboutthisseriesathttp://www.springer.com/series/304
Anna Skripka (cid:129) Anna Tomskova
Multilinear Operator
Integrals
Theory and Applications
123
AnnaSkripka AnnaTomskova
DepartmentofMathematicsandStatistics SchoolofComputerScienceand
UniversityofNewMexico Engineering
Albuquerque,NM,USA InhaUniversityinTashkent
Tashkent,Uzbekistan
ISSN0075-8434 ISSN1617-9692 (electronic)
LectureNotesinMathematics
ISBN978-3-030-32405-6 ISBN978-3-030-32406-3 (eBook)
https://doi.org/10.1007/978-3-030-32406-3
Mathematics Subject Classification (2010): Primary: 46L51, 47B49, 47A60, 47A63, 47B10, 47C15,
47A55,15A60;Secondary:46N50,58J30,46L87,46G12,46H10,47L25,26A16
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Preface
A multilinear operator integral is a powerful tool in noncommutative analysis
and its applications. Theory underlying multilinear operator integration has been
developing since the 1950s, with a number of amazing advancements made in
recentyears.Thefieldhasaccumulatedmanydeeptheoreticalresultsandimportant
applications, but no book on this beautiful and important subject appeared in the
literature. This book provides a brief yet comprehensive treatment of multilinear
operator integral techniques and their applications, partially filling the gap in the
literature.Theexpositionisstructuredtobesuitableforbothatopicscourseanda
researchaidonmethods,results,andapplicationsofmultilinearoperatorintegrals.
We survey on earlier ideas and contributions to the field and then present in
greaterdetailthebestup-to-dateresultsandmodernmethods.Thecontentincludes
mostpractical,refinedconstructionsofmultipleoperatorintegralsandfundamental
technicalresultsalongwithmajorapplicationsofthistooltosmoothnessproperties
of operator functions (Lipschitz continuity, Hölder continuity, differentiability),
approximation of operator functions, spectral shift functions, spectral flow in the
settingofnoncommutativegeometry,quantumdifferentiability,anddifferentiability
ofnoncommutativeLp norms.Wedemonstrateideasandincludeproofsinsimpler
cases, while highly technical proofs are outlined and supplemented with a list of
references.Wealsostateselectedopenproblemsinthefield.
Albuquerque,NM,USA AnnaSkripka
Tashkent,Uzbekistan AnnaTomskova
September2019
v
Acknowledgements
TheauthorsthankFedorSukochevforinspirationtowriteanoverviewofmultilin-
earoperatorintegration,whichhasultimatelygrownintothisbook.Theauthorsare
also gratefulto the three refereesfortheir valuablecommentsandsuggestions.In
particular,Theorems5.1.12and5.1.13,Sects.5.1.6and5.3.7,andtheexampleafter
Theorem3.3.11weresuggestedbythereferees.
ResearchofthefirstauthorwassupportedinpartbyNSFgrantDMS-1554456.
vii
Contents
1 Introduction .................................................................. 1
2 NotationsandPreliminaries................................................ 7
2.1 SpacesofFunctions ................................................... 7
2.2 DividedDifferences ................................................... 10
2.3 LinearOperators....................................................... 11
2.4 Schatten-vonNeumannClasses....................................... 12
2.5 ProductofSpectralMeasures......................................... 15
2.6 ClassicalNoncommutativeLp-SpacesandWeakLp-Spaces....... 18
2.7 TheHaagerupLp-Space .............................................. 22
2.8 SymmetricallyNormedIdeals ........................................ 24
2.9 TracesonL1,∞(M,τ) ................................................ 26
2.10 BanachSpacesandSpectralOperators............................... 29
2.11 DifferentiabilityofMapsonBanachSpaces......................... 32
3 DoubleOperatorIntegrals.................................................. 35
3.1 DoubleOperatorIntegralsonFiniteMatrices........................ 35
3.1.1 Definition...................................................... 36
3.1.2 RelationtoFinite-DimensionalSchurMultipliers........... 36
3.1.3 PropertiesofFiniteDimensionalDoubleOperator
Integrals........................................................ 37
3.2 DoubleOperatorIntegralsonS2 ..................................... 41
3.2.1 Definition...................................................... 41
3.2.2 RelationtoSchurMultipliersonS2.......................... 43
3.2.3 BasicPropertiesofDoubleOperatorIntegralsonS2 ....... 44
3.3 DoubleOperatorIntegralsonSchattenClassesandB(H).......... 45
3.3.1 Daletskii-Krein’sApproach................................... 45
3.3.2 ExtensionfromtheDoubleOperatorIntegralonS2 ........ 46
3.3.3 ApproachviaSeparationofVariables........................ 47
3.3.4 ApproachWithoutSeparationofVariables .................. 50
3.3.5 PropertiesofDoubleOperatorIntegralsonSp
andB(H)...................................................... 51
ix
x Contents
3.3.6 SymbolsofBoundedDoubleOperatorIntegrals ............ 54
3.3.7 TransferencePrinciple ........................................ 58
3.4 Nonself-adjointCase .................................................. 60
3.5 DoubleOperatorIntegralsonNoncommutativeLp-Spaces......... 60
3.5.1 ExtensionfromtheDoubleOperatorIntegral
onL2(M,τ)................................................... 61
3.5.2 ApproachviaSeparationofVariables........................ 61
3.5.3 ApproachWithoutSeparationofVariables .................. 61
3.5.4 PropertiesofDoubleOperatorIntegralsonLp(M,τ)...... 62
3.6 DoubleOperatorIntegralsonBanachSpaces........................ 63
4 MultipleOperatorIntegrals................................................ 65
4.1 MultipleOperatorIntegralsonFiniteMatrices ...................... 65
4.1.1 Definition...................................................... 65
4.1.2 RelationtoMultilinearSchurMultipliers.................... 66
4.1.3 PropertiesofFiniteDimensionalMultipleOperator
Integrals........................................................ 67
4.1.4 EstimatesofMultipleOperatorIntegralsviaDouble
OperatorIntegrals............................................. 74
4.2 MultipleOperatorIntegralsonS2 .................................... 74
4.2.1 Pavlov’sApproach ............................................ 74
4.2.2 Coine-LeMerdy-Sukochev’sApproach...................... 75
4.3 MultipleOperatorIntegralsonSchattenClassesandB(H)......... 77
4.3.1 ApproachviaSeparationofVariables........................ 77
4.3.2 ApproachWithoutSeparationofVariables .................. 79
4.3.3 PropertiesofMultipleOperatorIntegralsonSp
andB(H)...................................................... 81
4.3.4 Nonself-adjointCase.......................................... 93
4.3.5 ChangeofVariables........................................... 99
4.4 MultipleOperatorIntegralsonNoncommutativeandWeak
Lp-Spaces.............................................................. 100
4.4.1 ApproachviaSeparationofVariables........................ 100
4.4.2 ApproachWithoutSeparationofVariables .................. 100
4.4.3 PropertiesofMultipleOperatorIntegrals
onLp,∞(M,τ)................................................ 101
5 Applications .................................................................. 113
5.1 OperatorLipschitzFunctions ......................................... 113
5.1.1 CommutatorandLipschitzEstimatesinS2.................. 114
5.1.2 CommutatorandLipschitzEstimatesinSp andB(H) ..... 115
5.1.3 CommutatorandLipschitzEstimates:Nonself-adjoint
Case............................................................ 119
5.1.4 LipschitzTypeEstimatesinNoncommutative
Lp-Spaces ..................................................... 121
5.1.5 LipschitzTypeEstimatesinBanachSpaces ................. 122
5.1.6 OperatorI-LipschitzFunctions .............................. 123
Contents xi
5.2 OperatorHölderFunctions............................................ 128
5.3 DifferentiationofOperatorFunctions................................ 129
5.3.1 DifferentiationofMatrixFunctions.......................... 129
5.3.2 DifferentiationinB(H)AlongMultiplicativePaths
ofUnitaries .................................................... 132
5.3.3 DifferentiationinB(H)andS1 AlongLinearPaths
ofSelf-adjoints ................................................ 135
5.3.4 DifferentiationinSp AlongLinearPaths
ofSelf-adjoints ................................................ 137
5.3.5 Differentiationof Functionsof Contractiveand
DissipativeOperators ......................................... 140
5.3.6 DifferentiationinNoncommutativeLp-Spaces.............. 141
5.3.7 GâteauxandFréchetI-DifferentiableFunctions............ 142
5.4 TaylorApproximationofOperatorFunctions........................ 145
5.4.1 TaylorRemaindersofMatrixFunctions ..................... 145
5.4.2 TaylorRemaindersforPerturbationsinSp andB(H)...... 149
5.4.3 TaylorRemaindersforUnsummablePerturbations ......... 153
5.5 SpectralShift........................................................... 154
5.5.1 SpectralShiftFunctionforSelf-adjointOperators .......... 155
5.5.2 SpectralShiftFunctionforNonself-adjointOperators...... 162
5.5.3 SpectralShiftMeasureintheSettingofvonNeumann
Algebras ....................................................... 165
5.6 SpectralFlow .......................................................... 166
5.7 QuantumDifferentiability............................................. 170
5.8 DifferentiationofNoncommutativeLp-Norms...................... 172
References......................................................................... 179
Index............................................................................... 189
