Multilinear Operator Integrals: Theory and Applications

Preface......Page 6
Acknowledgements......Page 7
Contents......Page 8
1 Introduction......Page 11
 2.1 Spaces of Functions......Page 16
 2.2 Divided Differences......Page 19
 2.3 Linear Operators......Page 20
 2.4 Schatten-von Neumann Classes......Page 21
 2.5 Product of Spectral Measures......Page 24
 2.6 Classical Noncommutative Lp-Spaces and Weak Lp-Spaces......Page 27
 2.7 The Haagerup Lp-Space......Page 31
 2.8 Symmetrically Normed Ideals......Page 33
 2.9 Traces on L1,∞(M,τ)......Page 35
 2.10 Banach Spaces and Spectral Operators......Page 38
 2.11 Differentiability of Maps on Banach Spaces......Page 41
 3.1 Double Operator Integrals on Finite Matrices......Page 44
  3.1.2 Relation to Finite-Dimensional Schur Multipliers......Page 45
   Algebraic Properties......Page 46
   Norm Estimates......Page 47
   Perturbation Formula......Page 48
   Continuity......Page 49
   Birman–Solomyak's Approach......Page 50
   Pavlov's Approach......Page 51
  3.2.2 Relation to Schur Multipliers on S2......Page 52
   Algebraic Properties......Page 53
  3.3.1 Daletskii-Krein's Approach......Page 54
  3.3.2 Extension from the Double Operator Integral on S2......Page 55
   Demonstration in a Simple Case......Page 56
   General Case......Page 57
   Definition......Page 58
  3.3.4 Approach Without Separation of Variables......Page 59
   Norm Estimates......Page 60
   Perturbation Formula......Page 61
   Continuity......Page 62
  3.3.6 Symbols of Bounded Double Operator Integrals......Page 63
  3.3.7 Transference Principle......Page 67
 3.5 Double Operator Integrals on Noncommutative Lp-Spaces......Page 69
  3.5.3 Approach Without Separation of Variables......Page 70
   Norm Estimates......Page 71
 3.6 Double Operator Integrals on Banach Spaces......Page 72
  4.1.1 Definition......Page 74
  4.1.2 Relation to Multilinear Schur Multipliers......Page 75
   Algebraic Properties......Page 76
   Norm Estimates......Page 77
   Perturbation Formula......Page 78
   Continuity......Page 80
   Reduction to Identical Spectral Measures......Page 82
  4.2.1 Pavlov's Approach......Page 83
  4.2.2 Coine-Le Merdy-Sukochev's Approach......Page 84
  4.3.1 Approach via Separation of Variables......Page 86
  4.3.2 Approach Without Separation of Variables......Page 88
   Algebraic Properties......Page 90
   Norm Estimates......Page 92
   Perturbation Formula......Page 100
   Continuity......Page 101
  4.3.4 Nonself-adjoint Case......Page 102
  4.3.5 Change of Variables......Page 108
  4.4.2 Approach Without Separation of Variables......Page 109
   Algebraic Properties......Page 110
   Norm Estimates......Page 113
   Perturbation Formula......Page 114
   Hölder-Type Estimates......Page 117
 5.1 Operator Lipschitz Functions......Page 122
  5.1.1 Commutator and Lipschitz Estimates in S2......Page 123
  5.1.2 Commutator and Lipschitz Estimates in Sp and B(H)......Page 124
  5.1.3 Commutator and Lipschitz Estimates: Nonself-adjoint Case......Page 128
  5.1.4 Lipschitz Type Estimates in NoncommutativeLp-Spaces......Page 130
  5.1.5 Lipschitz Type Estimates in Banach Spaces......Page 131
   Operator I-Lipschitz and Commutator I-Bounded Functions......Page 132
   I-Stable and Commutator I-Stable Functions......Page 134
   A-Lipschitz Functions on Semisimple Hermitian Banach *-Algebras A......Page 135
 5.2 Operator Hölder Functions......Page 137
  5.3.1 Differentiation of Matrix Functions......Page 138
  5.3.2 Differentiation in B(H) Along Multiplicative Paths of Unitaries......Page 141
  5.3.3 Differentiation in B(H) and S1 Along Linear Paths of Self-adjoints......Page 144
  5.3.4 Differentiation in Sp Along Linear Pathsof Self-adjoints......Page 146
  5.3.5 Differentiation of Functions of Contractive and Dissipative Operators......Page 149
  5.3.6 Differentiation in Noncommutative Lp-Spaces......Page 150
  5.3.7 Gâteaux and Fréchet I-Differentiable Functions......Page 151
   I-Lipschitz and Gâteaux I-Differentiable Functions......Page 152
   Fréchet I-Differentiable Functions......Page 153
  5.4.1 Taylor Remainders of Matrix Functions......Page 154
   Self-adjoint Case......Page 158
   Unitary Case......Page 161
  5.4.3 Taylor Remainders for Unsummable Perturbations......Page 162
 5.5 Spectral Shift......Page 163
   Summable Perturbations......Page 164
   Unsummable Perturbations......Page 168
  5.5.2 Spectral Shift Function for Nonself-adjoint Operators......Page 171
  5.5.3 Spectral Shift Measure in the Setting of von Neumann Algebras......Page 174
 5.6 Spectral Flow......Page 175
 5.7 Quantum Differentiability......Page 179
 5.8 Differentiation of Noncommutative Lp-Norms......Page 181
References......Page 187
Index......Page 197