Table of Contents
Preface xi
Notation and Terminology xv
0.1 Notation xv
0.2 Terminology xvi
Part 1 Exercises ET AL. 1
Chapter 1 Normed Vector Spaces. Banach Spaces 3
1.1 Basics 3
1.1.1 Definitions and Examples 3
1.1.2 Operations on Banach Spaces 11
1.1.3 Convex Sets 13
1.1.4 Lp-Spaces 14
1.2 True or False: Questions 18
1.3 Exercises with Solutions 19
1.4 Tests 22
1.5 More Exercises 23
Chapter 2 Bounded Linear Operators on Banach Spaces 25
2.1 Basics 25
2.1.1 Basic Definitions 25
2.1.2 Dual Spaces 34
2.1.3 The Fourier Transform 36
2.1.4 Invertibility 38
2.1.5 Series in Banach Spaces 40
2.1.6 Neumann Series 44
2.1.7 Operator-Valued Analytic Functions 46
2.1.8 The Four Pillars of Functional Analysis 47
2.2 True or False: Questions 52
2.3 Exercises with Solutions 52
2.4 Tests 62
2.5 More Exercises 63
Chapter 3 Hilbert Spaces 67
3.1 Basics 67
3.1.1 Definitions and Examples 67
3.1.2 Modes of Convergence of Sequences 70
3.1.3 Orthogonality 72
3.1.4 The Closest Point Property and the Projection Theorem 74
3.1.5 Orthonormal Sots and Bases 75
3.1.6 Orthogonal Polynomials 78
3.1.7 Linear Operators on a Hilbert Space 79
3.1.8 Operator Topologies 84
3.2 True or False: Questions 86
3.3 Exercises with Solutions 87
3.4 Tests 96
3.5 More Exercises 96
Chapter 4 Classes of Linear Operators on Hilbert Spaces 99
4.1 Basics 99
4.1.1 Operator Theory Basics 99
4.1.2 Cartesian Form 103
4.1.3 Operator Matrices 104
4.1.4 A Word on the Invariant Subspace Problem 108
4.2 True or False: Questions 113
4.3 Exercises with Solutions 113
4.4 Tests 120
4.5 More Exercises 120
Chapter 5 Positive Operators. Square Root 123
5.1 Basics 123
5.1.1 Positive Operators 123
5.1.2 Invertibility Revisited 127
5.1.3 Square Root of Positive Operators 128
5.2 True or False: Questions 133
5.3 Exercises with Solutions 134
5.4 Tests 139
5.5 More Exercises 140
Chapter 6 Absolute Value. Polar Decomposition of an Operator 143
6.1 Basics 143
6.1.1 Definitions and Properties 143
6.1.2 The Absolute Value and the Product 146
6.1.3 The Absolute Value and the Sum 147
6.1.4 The Absolute Value and Other Inequalities 148
6.1.5 Polar Decomposition 148
6.2 True or False: Questions 150
6.3 Exercises with Solutions 151
6.4 Tests 155
6.5 More Exercises 156
Chapter 7 Spectrum of an Operator 159
7.1 Basics 159
7.1.1 Definitions and Properties 159
7.1.2 The Resolvent Function 162
7.1.3 Subsets of the Spectrum 163
7.1.4 Spectrum of the Product of Operators 165
7.1.5 On The Operator Equation AX - XB = C 168
7.1.6 Polynomial Calculus 171
7.2 True or False: Questions 173
7.3 Exercises with Solutions 173
7.4 Tests 179
7.5 More Exercises 180
Chapter 8 Spectral Radius. Numerical Range 183
8.1 Basics 183
8.1.1 Spectral Radius 183
8.1.2 Numerical Range 185
8.2 True or False: Questions 187
8.3 Exercises with Solutions 188
8.4 Tests 190
8.5 More Exercises 191
Chapter 9 Compact Operators 193
9.1 Basics 193
9.1.1 Compact Operators 193
9.1.2 Hilbert-Schmidt Operators 195
9.1.3 Spectrum and Compact Operators 197
9.2 True or False: Questions 198
9.3 Exercises with Solutions 199
9.4 Tests 202
9.5 More Exercises 203
Chapter 10 Closed Operators 207
10.1 Basics 207
10.1.1 Spectrum 211
10.1.2 Closedness of Products and Sums 212
10.1.3 A Word on Distributional Derivatives 213
10.2 True or False: Questions 216
10.3 Exercises with Solutions 216
10.4 Tests 221
10.5 More Exercises 222
Chapter 11 Functional Calculi 225
11.1 Basics 225
11.1.1 Functional Calculus for Self-adjoint Operators 225
11.1.2 Functional Calculus for Normal Operators 234
11.1.3 A Little Digression 237
11.1.4 The Fuglede-Putnam Theorem Revisited 238
11.1.5 The Beck-Putnam Theorem 240
11.2 True or False: Questions 241
11.3 Exercises with Solutions 242
11.4 Tests 249
11.5 More Exercises 250
Chapter 12 Hyponormal Operators 253
12.1 Basics 253
12.2 True or False: Questions 255
12.3 Exercises with Solutions 256
12.4 Tests 259
12.5 More Exercises 260
Chapter 13 Similarities of Operators 261
13.1 Important Theorems 261
13.1.1 The Berberian Theorem 261
13.1.2 The Williams Theorem 261
13.1.3 The Embry Theorem 262
13.2 Exercises with Solutions 262
13.3 Tests 265
13.4 More Exercises 265
Part 2 Solutions 267
Chapter 1 Normed Vector Spaces. Banach Spaces 269
1.2 True or False: Answers 269
1.3 Solutions to Exercises 270
1.4 Hints/Answers to Tests 292
Chapter 2 Bounded Linear Operators on Banach Spaces 293
2.2 True or False: Answers 293
2.3 Solutions to Exercises 295
2.4 Hints/Answers to Tests 335
Chapter 3 Hilbert Spaces 337
3.2 True or False: Answers 337
3.3 Solutions to Exercises 339
3.4 Hints/Answers to Tests 375
Chapter 4 Classes of Linear Operators on Hilbert Spaces 377
4.2 True or False: Answers 377
4.3 Solutions to Exercises 379
4.4 Hints/Answers to Tests 405
Chapter 5 Positive Operators. Square Root 407
5.2 True or False: Answers 407
5.3 Solutions to Exercises 409
5.4 Hints/Answers to Tests 431
Chapter 6 Absolute Value. Polar Decomposition of an Operator 433
6.2 True or False: Answers 433
6.3 Solutions to Exercises 434
6.4 Hints/Answers to Tests 449
Chapter 7 Spectrum of an Operator 453
7.2 True or False: Answers 453
7.3 Solutions to Exercises 454
7.4 Hints/Answers to Tests 484
Chapter 8 Spectral Radius. Numerical Range 487
8.2 True or False: Answers 487
8.3 Solutions to Exercises 489
8.4 Hints/Answers to Tests 499
Chapter 9 Compact Operators 501
9.2 True or False: Answers 501
9.3 Solutions to Exercises 502
9.4 Hints/ Answers to Tests 527
Chapter 10 Closed Operators 529
10.2 True or False: Answers 529
10.3 Solutions to Exercises 530
10.4 Hints/Answers to Tests 556
Chapter 11 Functional Calculi 559
11.2 True or False: Answers 559
11.3 Solutions to Exercises 562
11.4 Hints/Answers to Tests 587
Chapter 12 Hyponormal Operators 591
12.2 True or False: Answers 591
12.3 Solutions to Exercises 593
12.4 Hints/Answers to Tests 609
Chapter 13 Similarities of Operators 611
13.2 Solutions to Exercises 611
13.3 Hints/Answers to Tests 619
Bibliography 621
Index 633
