The central topic of this book is the spectral theory of bounded and unbounded self-adjoint operators on Hilbert spaces. After introducing the necessary prerequisites in measure theory and functional analysis, the exposition focuses on operator theory and especially the structure of self-adjoint operators. These can be viewed as infinite-dimensional analogues of Hermitian matrices; the infinite-dimensional setting leads to a richer theory which goes beyond eigenvalues and eigenvectors and studies self-adjoint operators in the language of spectral measures and the Borel functional calculus. The main approach to spectral theory adopted in the book is to present it as the interplay between three main classes of objects: self-adjoint operators, their spectral measures, and Herglotz functions, which are complex analytic functions mapping the upper half-plane to itself. Self-adjoint operators include many important classes of recurrence and differential operators; the later part of this book is dedicated to two of the most studied classes, Jacobi operators and one-dimensional Schrödinger operators.
This text is intended as a course textbook or for independent reading for graduate students and advanced undergraduates. Prerequisites are linear algebra, a first course in analysis including metric spaces, and for parts of the book, basic complex analysis. Necessary results from measure theory and from the theory of Banach and Hilbert spaces are presented in the first three chapters of the book. Each chapter concludes with a number of helpful exercises.
Author(s): Milivoje Lukić
Series: Graduate Studies in Mathematics 226
Edition: 1
Publisher: American Mathematical Society
Year: 2022
Language: English
Pages: 472
City: Providence, Rhode Island
Tags: Spectral Theory, Hilbert Spaces, Linear Operators, Self-adjoint, Spectral Measures, Herglotz Functions, Spectral Theorem, Jacobi Matrices, One-dimensional Schrödinger Operators
Contents
Preface
Chapter 1. Measure theory
1.1. ?-algebras and monotone classes
1.2. Measures and Carathéodory’s theorem
1.3. Borel ?-algebra on the real line and related spaces
1.4. Lebesgue integration
1.5. Lebesgue–Stieltjes measures on ℝ
1.6. Product measures
1.7. Functions on ?-locally compact spaces
1.8. Regularity of measures
1.9. The Riesz–Markov theorem
1.10. Exercises
Chapter 2. Banach spaces
2.1. Norms and Banach spaces
2.2. The Banach space ?(?)
2.3. ?^{?} spaces
2.4. Bounded linear operators and uniform boundedness
2.5. Weak-* convergence and the separable Banach–Alaoglu theorem
2.6. Banach-space valued integration
2.7. Banach-space valued analytic functions
2.8. Exercises
Chapter 3. Hilbert spaces
3.1. Inner products
3.2. Subspaces and orthogonal projections
3.3. Direct sums of Hilbert spaces
3.4. Orthonormal sets and orthonormal bases
3.5. Weak convergence
3.6. Tensor products of Hilbert spaces
3.7. Exercises
Chapter 4. Bounded linear operators
4.1. The ?*-algebra of bounded linear operators on ℋ
4.2. Strong and weak operator convergence
4.3. Invertibility, spectrum, and resolvents
4.4. Polynomials of operators
4.5. Invariant subspaces and direct sums of operators
4.6. Compact operators
4.7. Exercises
Chapter 5. Bounded self-adjoint operators
5.1. A first look at self-adjoint operators
5.2. Spectral theorem for compact self-adjoint operators
5.3. Spectral measures
5.4. Spectral theorem on a cyclic subspace
5.5. Multiplication operators
5.6. Spectral theorem on the entire Hilbert space
5.7. Borel functional calculus
5.8. Spectral theorem for unitary operators
5.9. Exercises
Chapter 6. Measure decompositions
6.1. Pure point and continuous measures
6.2. Singular and absolutely continuous measures
6.3. Hausdorff measures on ℝ
6.4. Matrix-valued measures
6.5. Exercises
Chapter 7. Herglotz functions
7.1. Möbius transformations
7.2. Schur functions and convergence
7.3. Carathéodory functions
7.4. The Herglotz representation
7.5. Growth at infinity and tail of the measure
7.6. Half-plane Poisson kernel and Stieltjes inversion
7.7. Pointwise boundary values
7.8. Meromorphic Herglotz functions
7.9. Exponential Herglotz representation
7.10. The Phragmén–Lindelöf method and asymptotic expansions
7.11. Matrix-valued Herglotz functions
7.12. Weyl matrices and Dirichlet decoupling
7.13. Exercises
Chapter 8. Unbounded self-adjoint operators
8.1. Graphs and adjoints
8.2. Resolvents and self-adjointness
8.3. Unbounded multiplication operators and direct sums
8.4. Spectral measures and the spectral theorem
8.5. Borel functional calculus
8.6. Absolutely continuous functions and derivatives on intervals
8.7. Self-adjoint extensions and symplectic forms
8.8. Exercises
Chapter 9. Consequences of the spectral theorem
9.1. Maximal spectral measure
9.2. Spectral projections
9.3. Spectral type and spectral decompositions
9.4. Ruelle–Amrein–Georgescu–Enss (RAGE) theorem
9.5. Essential and discrete spectrum; the min-max principle
9.6. Spectral multiplicity
9.7. Stone’s theorem
9.8. Fourier transform on ℝ
9.9. Abstract eigenfunction expansions
9.10. Exercises
Chapter 10. Jacobi matrices
10.1. The canonical spectral measure and Favard’s theorem
10.2. Unbounded Jacobi matrices
10.3. Weyl solutions and ?-functions
10.4. Transfer matrices and Weyl disks
10.5. Full-line Jacobi matrices
10.6. Eigenfunction expansion for full-line Jacobi matrices
10.7. The Weyl ?-matrix
10.8. Subordinacy theory
10.9. A Combes–Thomas estimate and Schnol’s theorem
10.10. The periodic discriminant and the Marchenko–Ostrovski map
10.11. Direct spectral theory of periodic Jacobi matrices
10.12. Exercises
Chapter 11. One-dimensional Schrödinger operators
11.1. An initial value problem
11.2. Fundamental solutions and transfer matrices
11.3. Schrödinger operators with two regular endpoints
11.4. Endpoint behavior
11.5. Self-adjointness and separated boundary conditions
11.6. Weyl solutions and Green’s functions
11.7. Weyl solutions and ?-functions
11.8. The half-line eigenfunction expansion
11.9. Weyl disks and applications
11.10. Asymptotic behavior of ?-functions
11.11. The local Borg–Marchenko theorem
11.12. Full-line eigenfunction expansions
11.13. Subordinacy theory
11.14. Potentials bounded below in an ?¹_{}??? sense
11.15. A Combes–Thomas estimate and Schnol’s theorem
11.16. The periodic discriminant and the Marchenko–Ostrovski map
11.17. Direct spectral theory of periodic Schrödinger operators
11.18. Exercises
Bibliography
Notation Index
Index
