Matrix Completions, Moments, and Sums of Hermitian Squares

Preface ix
Chapter 1. Cones of Hermitian matrices and trigonometric polynomials 1
1.1 Cones and their basic properties 1
1.2 Cones of Hermitian matrices 6
1.3 Cones of trigonometric polynomials 13
1.4 Determinant and entropy maximization 41
1.5 Semidefinite programming 47
1.6 Exercises 51
1.7 Notes 65

Chapter 2. Completions of positive semidefinite operator matrices 69
2.1 Positive definite completions: the banded case 69
2.2 Positive definite completions: the chordal case 76
2.3 Positive definite completions: the Toeplitz case 83
2.4 The Schur complement and Fejér-Riesz factorization 98
2.5 Schur parameters 115
2.6 The central completion, maximum entropy, and inheritance principle 124
2.7 The Hamburger moment problem and spectral factorization on the real line 138
2.8 Linear prediction 156
2.9 Exercises 157
2.10 Notes 170

Chapter 3. Multivariable moments and sums of Hermitian squares 175
3.1 Positive Carathéodory interpolation on the polydisk 176
3.2 Inverses of multivariable Toeplitz matrices and Christoffel-Darboux formulas 187
3.3 Two-variable moment problem for Bernstein-Szegö measures 198
3.4 Fejér-Riesz factorization and sums of Hermitian squares 208
3.5 Completion problems for positive semidefinite functions on amenable groups 211
3.6 Moment problems on free groups 214
3.7 Noncommutative factorization 223
3.8 Two-variable Hamburger moment problem 228
3.9 Bochner's theorem and an application to autoregressive stochastic processes 235
3.10 Exercises 240
3.11 Notes 250

Chapter 4. Contractive analogs 257
4.1 Contractive operator-matrix completions 258
4.2 Linearly constrained completion problems 269
4.3 The operator-valued Nehari and Carathéodory problems 275
4.4 Nehari's problem in two variables 286
4.5 Nehari and Carathéodory problems for functions on compact groups 292
4.6 The Nevanlinna-Pick problem 299
4.7 The operator Corona problem 308
4.8 Joint operator/Hilbert-Schmidt norm control extensions 314
4.9 An L1 extension problem for polynomials 317
4.10 Superoptimal completions 321
4.11 Superoptimal approximations of analytic functions 329
4.12 Model matching 339
4.13 Exercises 341
4.14 Notes 352

Chapter 5. Hermitian and related completion problems 361
5.1 Hermitian completions 361
5.2 Ranks of completions 374
5.3 Minimal negative and positive signature 383
5.4 Inertia of Hermitian matrix expressions 401
5.5 Bounds for eigenvalues of Hermitian completions 407
5.6 Bounds for singular values of completions of partial triangular matrices 413
5.7 Moment problems for real measures on the unit circle 418
5.8 Euclidean distance matrix completions 426
5.9 Normal completions 433
5.10 Application to minimal representation of discrete systems 442
5.11 The separability problem in quantum information 443
5.12 Exercises 451
5.13 Notes 470

Bibliography 475
Subject Index 513
Notation Index 517