Cover
Title Page
Copyright Page
Preface
Contents
Introduction
Abstract theory
Applications
Preliminaries
A. Operators on finite dimensional linear spaces
A.1. Finite dimensional linear spaces
A.2. Operators and matrices
A.3. Traces
A.4. Determinants
A.5. Eigenvalues
B. Spaces and operators
B.1. Operators on quasi-Banach spaces
B.2. Operators on Banach spaces
B.3. Duality
B.4. Finite operators on Banach spaces
C. Sequence and function spaces
C.1. Classical sequence spaces
C.2. Direct sums of Banach spaces
C.3. Classical function spaces
C.4. The metric extension property
D. Operator ideals
D.1. Quasi-Banach operator ideals
D.2. Some examples of operator ideals
D.3. Extensions of operator ideals
E. Tensor products
E.1. Algebraic tensor products
E.2. Banach tensor products
E.3. Tensor stability of operator ideals
F. Interpolation theory
F.1. Intermediate spaces
F.2. Interpolation methods
F.3. Real interpolation
F.4. Interpolation of quasi-Banach operator ideals
G. Inequalities
G.1. The inequality of means
G.2. The Khintchine inequality
G.3. Asymptotic estimates
Chapter 1. Absolutely summing operators
1.1. Summable sequences
1.2. Absolutely (r, s)-summing operators
1.3. Absolutely r-summing operators
1.4. HilbertโSchmidt operators
1.5. Absolutely 2-summing operators
1.6. Diagonal operators
1.7. Nuclear operators
Chapter 2. s-Numbers
2.1. Lorentz sequence spaces
2.2. Axiomatic theory of s-numbers
2.3. Approximation numbers
2.4. Gel'fand and Weyl numbers
2.5. Kolmogorov and Chang numbers
2.6. Hilbert numbers
2.7. Absolutely (r, 2)-summing operators
2.8. Generalized approximation numbers
2.9. Diagonal operators
2.10. Relationships between various s-numbers
2.11. Schattenโvon Neumann operators
Chapter 3. Eigenvalues
3.1. The Riesz decomposition
3.2. Riesz operators
3.3. Related operators
3.4. The eigenvalue type of operator ideals
3.5. Eigenvalues of Schattenโvon Neumann operators
3.6. Eigenvalues of s-type operators
3.7. Eigenvalues of absolutely summing operators
3.8. Eigenvalues of nuclear operators
3.9. The eigenvalue type of sums of operator ideals
Chapter 4. Traces and determinants
4.1. Fredholm resolvents
4.2. Traces
4.3. Determinants
4.4. Fredholm denominators
4.5. Regularized Fredholm denominators
4.6. The relationship between traces and determinants
4.7. Traces and determinants of nuclear operators
4.8. Entire functions
Chapter 5. Matrix operators
5.1. Examples of finite matrices
5.2. Examples of infinite matrices
5.3. HilleโTamarkin matrices
5.4. Besov matrices
5.5. Traces and determinants of matrices
Chapter 6. Integral operators
6.1. Continuous kernels
6.2. HilleโTamarkin kernels
6.3. Weakly singular kernels
6.4. Besov kernels
6.5. Fourier coefficients
6.6. Traces and determinants of kernels
Chapter 7. Historical survey
7.1. Classical background
7.1.1. Determinants
7.1.2. Matrices
7.1.3. Vector spaces
7.1.4. Eigenvalues
7.1.5. Traces
7.1.6. Canonical forms of matrices
7.1.7. Axiomatic approach
7.2. Spaces
7.2.1. Sequence spaces
7.2.2. Function spaces
7.2.3. Banach spaces
7.2.4. Interpolation theory
7.3. Operators
7.3.1. Infinite matrices
7.3.2. Integral operators
7.3.3. Abstract operators
7.3.4. Operator ideals on Hilbert spaces
7.3.5. Operator ideals on Banach spaces
7.3.6. s-Numbers
7.3.7. Absolutely summing operators
7.3.8. Nuclear operators
7.4. Eigenvalues
7.4.1. Riesz theory
7.4.2. Related operators
7.4.3. Eigenvalues and s-numbers
7.4.4. Eigenvalues of nuclear operators
7.4.5. Eigenvalues of absolutely summing operators
7.4.6. Summary
7.5. Determinants
7.5.1. Determinants of infinite matrices
7.5.2. Determinants of integral operators
7.5.3. Determinants of abstract operators
7.5.4. Eigenvalues and zeros of entire functions
7.6. Traces
7.6.1. Operator ideals with a trace
7.6.2. The trace formula
7.7. Applications
7.7.1. Diagonal and embedding operators
7.7.2. s-Numbers of diagonal operators
7.7.3. s-Numbers of embedding operators
7.7.4. Eigenvalues of infinite matrices
7.7.5. Eigenvalues of integral operators
7.7.6. Eigenvalues of differential operators
7.7.7. Fourier coefficients
7.7.8. Practical applications
Appendix
Open problems
Epilogue
Bibliography
A) Textbooks and monographs
B) Research papers
Index
List of special symbols
1. Banach and Hilbert spaces
2. Operators
3. s-Numbers
4. Operator ideals
5. Sequences and matrices
6. Spaces of sequences and matrices
7. Functions and kernels
8. Spaces of functions and kernels