The purpose of this book is to make available to beginning graduate students, and to others, some core areas of analysis which serve as prerequisites for new developments in pure and applied areas. We begin with a presentation (Chapters 1 and 2) of a selection of topics from the theory of operators in Hilbert space, algebras of operators, and their corresponding spectral theory. This is a systematic presentation of interrelated topics from infinite-dimensional and non-commutative analysis; again, with view to applications. Chapter 3 covers a study of representations of the canonical commutation relations (CCRs); with emphasis on the requirements of infinite-dimensional calculus of variations, often referred to as Ito and Malliavin calculus, Chapters 4-6. This further connects to key areas in quantum physics.
Author(s): Palle Jorgensen, James Tian
Publisher: World Scientific Publishing
Year: 2021
Language: English
Pages: 252
City: Singapore
Contents
Preface
Acknowledgments
Abbreviations
1. Analysis in Hilbert Space: Linear Operators in Hilbert Space with Emphasis on the Case of Unbounded Operators
1.1 Basics of Hilbert space theory
1.1.1 Positive definite functions
1.1.2 Orthonormal bases
1.1.3 Orthogonal projections in Hilbert space, and their role in probability theory
1.1.4 Bounded operators in Hilbert space
1.2 Dirac’s approach
1.3 Connection to quantum mechanics
1.4 Probabilistic interpretation of Parseval’s formula for Hilbert space
1.5 The lattice structure of projections
1.6 Unbounded operators between different Hilbert spaces
1.7 Normal operators
1.8 Closable pairs of operators
1.9 Stone’s Theorem
1.10 Guide to the literature
2. Infinite-Dimensional Algebraic Systems: Lie Algebras, Algebras with Involution (∗-Algebras), and the Canonical Commutation Relations (CCRs)
2.1 Some history and comments on commutation relations
2.2 Infinite-dimensional analysis
2.3 Positivity and representations
2.4 An infinite-dimensional Lie algebra
2.5 Guide to the literature
3. Representation Theory, with Emphasis on the Case of the CCRs
3.1 The CCR-algebra, and the Fock representations
3.2 Symmetric Fock space and Malliavin derivative
3.3 Itô-integrals
3.3.1 Transforms induced by isometries V : H (K) → L2 (P) for different choices of p.d. kernels
3.3.2 The case K (A, B) = μ (A ∩ B)
3.3.3 Jointly Gaussian distributions
3.3.4 CND kernels
3.4 Guide to the literature
4. Gaussian Stochastic Processes: Gaussian Fields and Their Realizations
4.1 Analysis on reproducing Kernel Hilbert space (RKHS)
4.1.1 Stochastic analysis and positive definite kernels
4.1.2 Function spaces and Schwartz distributions
4.1.3 The isomorphism TK : L2 (K) → H (K)
4.1.4 Reversible kernels
4.2 Transforms and factorizations
4.2.1 Summary of Wiener processes
4.2.2 Generalized Carleson measures
4.2.3 Factorization of p.d. kernels
4.3 Numerical models
4.3.1 Complex-valued Gaussian processes
4.3.2 Hermite polynomials
4.3.3 Simple harmonic oscillator
4.3.4 Segal–Bargmann transforms
4.4 Gaussian Hilbert space
4.4.1 White noise analysis
4.5 Equivalence of pairs of Gaussian processes
4.6 Guide to the literature
5. Infinite-Dimensional Stochastic Analysis: White Noise Analysis and Generalized Itô Calculus
5.1 The Malliavin derivatives
5.2 A derivation on the algebra D
5.3 Infinite-dimensional Δ and ∇Φ
5.4 Guide to the literature
6. Representations of the CCRs Realized as Gaussian Fields and Malliavin Derivatives
6.1 Realization of the operators
6.2 The unitary group
6.3 The Fock-state, and representation of CCR, realized as Malliavin calculus
6.4 Conclusions: The general case
6.5 Guide to the literature
7. Intertwining Operators and Their Realizations in Stochastic Analysis
7.1 Representations from states, the GNS construction, and intertwining operators
7.2 Guide to the literature
8. Applications
8.1 Machine learning
8.2 Relative RKHSs
8.2.1 Deciding when the Dirac point functions δx are in H (K)
8.3 Guide to the literature
Appendex A. Some Biographical Sketches
Bibliography
Index
