A First Course in Spectral Theory

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. Contents 8 Preface 14 Chapter 1. Measure theory 18 1.1. σ-algebras and monotone classes 18 1.2. Measures and Carathéodory’s theorem 23 1.3. Borel σ-algebra on the real line and related spaces 27 1.4. Lebesgue integration 32 1.5. Lebesgue–Stieltjes measures on R 41 1.6. Product measures 47 1.7. Functions on σ-locally compact spaces 49 1.8. Regularity of measures 52 1.9. The Riesz–Markov theorem 55 1.10. Exercises 58 Chapter 2. Banach spaces 62 2.1. Norms and Banach spaces 62 2.2. The Banach space C(K) 65 2.3. L^{p} spaces 71 2.4. Bounded linear operators and uniform boundedness 76 2.5. Weak-* convergence and the separable Banach–Alaoglu theorem 82 2.6. Banach-space valued integration 85 2.7. Banach-space valued analytic functions 88 2.8. Exercises 91 Chapter 3. Hilbert spaces 94 3.1. Inner products 94 3.2. Subspaces and orthogonal projections 99 3.3. Direct sums of Hilbert spaces 105 3.4. Orthonormal sets and orthonormal bases 108 3.5. Weak convergence 114 3.6. Tensor products of Hilbert spaces 117 3.7. Exercises 121 Chapter 4. Bounded linear operators 124 4.1. The C*-algebra of bounded linear operators on H 124 4.2. Strong and weak operator convergence 127 4.3. Invertibility, spectrum, and resolvents 130 4.4. Polynomials of operators 135 4.5. Invariant subspaces and direct sums of operators 136 4.6. Compact operators 139 4.7. Exercises 142 Chapter 5. Bounded self-adjoint operators 146 5.1. A first look at self-adjoint operators 147 5.2. Spectral theorem for compact self-adjoint operators 153 5.3. Spectral measures 156 5.4. Spectral theorem on a cyclic subspace 158 5.5. Multiplication operators 160 5.6. Spectral theorem on the entire Hilbert space 163 5.7. Borel functional calculus 166 5.8. Spectral theorem for unitary operators 170 5.9. Exercises 172 Chapter 6. Measure decompositions 176 6.1. Pure point and continuous measures 177 6.2. Singular and absolutely continuous measures 179 6.3. Hausdorff measures on R 186 6.4. Matrix-valued measures 193 6.5. Exercises 195 Chapter 7. Herglotz functions 200 7.1. Möbius transformations 201 7.2. Schur functions and convergence 205 7.3. Carathéodory functions 207 7.4. The Herglotz representation 210 7.5. Growth at infinity and tail of the measure 213 7.6. Half-plane Poisson kernel and Stieltjes inversion 216 7.7. Pointwise boundary values 221 7.8. Meromorphic Herglotz functions 227 7.9. Exponential Herglotz representation 229 7.10. The Phragmén–Lindelöf method and asymptotic expansions 232 7.11. Matrix-valued Herglotz functions 233 7.12. Weyl matrices and Dirichlet decoupling 236 7.13. Exercises 239 Chapter 8. Unbounded self-adjoint operators 244 8.1. Graphs and adjoints 245 8.2. Resolvents and self-adjointness 248 8.3. Unbounded multiplication operators and direct sums 253 8.4. Spectral measures and the spectral theorem 255 8.5. Borel functional calculus 260 8.6. Absolutely continuous functions and derivatives on intervals 264 8.7. Self-adjoint extensions and symplectic forms 270 8.8. Exercises 279 Chapter 9. Consequences of the spectral theorem 284 9.1. Maximal spectral measure 285 9.2. Spectral projections 287 9.3. Spectral type and spectral decompositions 289 9.4. Ruelle–Amrein–Georgescu–Enss (RAGE) theorem 292 9.5. Essential and discrete spectrum; the min-max principle 295 9.6. Spectral multiplicity 300 9.7. Stone’s theorem 306 9.8. Fourier transform on R 307 9.9. Abstract eigenfunction expansions 310 9.10. Exercises 313 Chapter 10. Jacobi matrices 316 10.1. The canonical spectral measure and Favard’s theorem 317 10.2. Unbounded Jacobi matrices 322 10.3. Weyl solutions and m-functions 326 10.4. Transfer matrices and Weyl disks 330 10.5. Full-line Jacobi matrices 336 10.6. Eigenfunction expansion for full-line Jacobi matrices 339 10.7. The Weyl M-matrix 342 10.8. Subordinacy theory 345 10.9. A Combes–Thomas estimate and Schnol’s theorem 351 10.10. The periodic discriminant and the Marchenko–Ostrovski map 353 10.11. Direct spectral theory of periodic Jacobi matrices 364 10.12. Exercises 369 Chapter 11. One-dimensional Schrödinger operators 376 11.1. An initial value problem 378 11.2. Fundamental solutions and transfer matrices 384 11.3. Schrödinger operators with two regular endpoints 390 11.4. Endpoint behavior 396 11.5. Self-adjointness and separated boundary conditions 403 11.6. Weyl solutions and Green’s functions 407 11.7. Weyl solutions and m-functions 411 11.8. The half-line eigenfunction expansion 416 11.9. Weyl disks and applications 424 11.10. Asymptotic behavior of m-functions 432 11.11. The local Borg–Marchenko theorem 440 11.12. Full-line eigenfunction expansions 442 11.13. Subordinacy theory 446 11.14. Potentials bounded below in an L1_{}loc sense 450 11.15. A Combes–Thomas estimate and Schnol’s theorem 456 11.16. The periodic discriminant and the Marchenko–Ostrovski map 460 11.17. Direct spectral theory of periodic Schrödinger operators 467 11.18. Exercises 470 Bibliography 476 Notation Index 484 Index 486