Table of Contents
Preface vii
1 Introduction 1
1.1 Vector measures 6
1.2 Integration of scalar functions with respect to a vector valued measure 10
1.3 Integration of vector valued functions with respect to a scalar measure 12
1.3.1 The Pettis integral 13
1.3.2 The Bochner integral 14
1.4 Tensor products 15
1.4.1 Injective and projective tensor products 17
1.4.2 Grothendieck's inequality 21
1.5 Semivariation 24
1.5.1 Semivariation in Lp-spaces 26
1.5.2 Semi variation of positive operator valued measures 32
1.6 Bilinear integration after Bar tie and Dobrakov 35
2 Decoupled bilinear integration 41
2.1 Bilinear integration in tensor products 44
2.2 Order bounded measures 53
2.3 The bilinear Fubini theorem 54
2.4 Kxamples of bilinear integrals 59
3 Operator traces 71
3.1 Trace class operators 72
3.2 The Hardy-Littlewood maximal operator 74
3.3 The Banach function space of traceable functions 75
3.4 Traceable operators on Banach function spaces 83
3.4.1 Lusin filtrations 91
3.4.2 Connection with other generalised traces 94
3.5 Hermitian positive operators 94
4 Stochastic integration 101
4.1 Background on probability and discrete processes 101
4.1.1 Conditional probability and expectation 104
4.1.2 Discrete Martingales 106
4.1.3 Discrete stopping times 109
4.2 Stochastic processes 111
4.3 Brownian motion 112
4.3.1 Some properties of Brownian paths 114
4.4 Stochastic integration of vector valued processes 115
5 Scattering theory 123
5.1 Time-dependent scattering theory 123
5.2 Stationary state scattering theory 125
5.3 Time-dependent scattering theory for bounded Hamiltonians and potentials 128
5.1 Bilinear integrals in scattering theory 131
5.5 Application to the Lippmann-Sehwinger equations 138
6 Random evolutions 143
6.1 Evolution processes 143
6.2 Measurable functions 148
6.3 Progressive measurability 150
6.4 Operator bilinear integration 157
6.5 Random evolutions 167
7 CLR inequality 171
7.1 Asymptotic estimates for bound states 171
7.2 Lattice traces for positive operators 175
7.3 The CLR inequality for dominated semigroups 184
8 Operator equations 191
8.1 Operator equations 193
8.2 Double operator integrals 202
8.3 Traces of double operator integrals 209
8.3.1 Schur multipliers and Grothendieck's inequality 212
8.3.2 Schur multipliers on measure spaces 214
8.4 The spectral shift function 221
Bibliography 229
Index 237
