Orthonormal Systems and Banach Space Geometry

Contents
Preface
Introduction
0. Preliminaries
0.1 Banach spaces and operators
0.2 Finite dimensional spaces and operators
0.3 Classical sequence spaces
0.4 Classical function spaces
0.5 Lorentz spaces
0.6 Interpolation methods
0.7 Summation operators
0.8 Finite representability and ultrapowers
0.9 Extreme points
0.10 Various tools
1. Ideal norms and operator ideals
1.1 Ideal norms
1.2 Operator ideals
1.3 Classes of Banach spaces
2. Ideal norms associated with matrices
2.1 Matrices
2.2 Parseval ideal norms and 2-summing operators
2.3 Kwapien ideal norms and Hilbertian operators
2.4 Ideal norms associated with Hilbert matrices
3. Ideal norms associated with orthonormal systems
3.1 Orthonormal systems
3.2 Khintchine constants
3.3 Riemann ideal norms
3.4 Dirichlet ideal norms
3.5 Orthonormal systems with special properties
3.6 Tensor products of orthonormal systems
3.7 Type and cotype ideal norms
3.8 Characters on compact Abelian groups
3.9 Discrete orthonormal systems
3.10 Some universal ideal norms
3.11 Parseval ideal norms
4. Rademacher and Gauss ideal norms
4.1 Rademacher functions
4.2 Rademacher type and cotype ideal norms
4.3 Operators of Rademacher type
4.4 B-convexity
4.5 Operators of Rademacher cotype
4.6 MP-convexity
4.7 Gaussian random variables
4.8 Gauss versus Rademacher
4.9 Gauss type and cotype ideal norms
4.10 Operators of Gauss type and cotype
4.11 Sidon constants
4.12 The Dirichlet ideal norms δ(R_n, R_n) and δ(G_n, G_n)
4.13 Inequalities between δ(R_n, R_n) and arrho(R_n, J_n)
4.14 The vector-valued Rademacher projection
4.15 Parseval ideal norms and γ-summing operators
4.16 The Maurey-Pisier theorem
5. Trigonometric ideal norms
5.1 Trigonometric functions
5.2 The Dirichlet ideal norms δ(E_n, E_n)
5.3 Hilbert matrices and trigonometric systems
5.4 The vector-valued Hilbert transform
5.5 Fourier type and cotype ideal norms
5.6 Operators of Fourier type
5.7 Operators of Fourier cotype
5.8 The vector-valued Fourier transform
5.9 Fourier versus Gauss and Rademacher
6. Walsh ideal norms
6.1 Walsh functions
6.2 Walsh type and cotype ideal norms
6.3 Operators of Walsh type
6.4 Walsh versus Rademacher
6.5 Walsh versus Fourier
7. Haar ideal norms
7.1 Martingales
7.2 Dyadic martingales
7.3 Haar functions
7.4 Haar type and cotype ideal norms
7.5 Operators of Haar type
7.6 Super weakly compact operators
7.7 Martingale type ideal norms
7.8 J-convexity
7.9 Uniform q-convexity and uniform p-smoothness
7.10 Uniform convexity and uniform smoothness
8. U nconditionality
8.1 Unconditional Riemann ideal norms
8.2 Unconditional Dirichlet ideal norms
8.3 Random unconditionality
8.4 Fourier unconditionality
8.5 Haar unconditionality /UMD
8.6 Random Haar unconditionality
8.7 The Dirichlet ideal norms δ(W_n, W_n)
8.8 The Burkholder-Bourgain theorem
9. Miscellaneous
9.1 Interpolation
9.2 Schatten-von Neumann spaces
9.3 Ideal norms of finite rank operators
9.4 Orthogonal polynomials
9.5 History
9.6 Epilogue
Summaries
List of symbols
Conventions
Miscellaneous
Banach spaces
Operators
Orthonormal systems
Norms of n-tuples
Parameters associated with orthonormal systems
Ideal norms
Operator ideals
Classes of Banach spaces
Bibliography
A
B
C
D
E
F・G
H
J・K
L
M
N・O
P
R
S
T・V
W・Y・Z
Index