CONTENTS xiii
2.3. The algebras H°° -f C and QC, and Compact Commutators 214
2.4. Invariant Subspaces and Kronecker's Theorem 216
2.5. Exercises and Further Results 218
2.6. Notes and Remarks 224
Chapter 3. Applications to Nevanlinna-Pick Interpolation 227
3.1. Model Operators 227
3.2. Schur and Nevanlinna-Pick Interpolation 231
3.3. Structure of Interpolating Functions and Rational Approximations 233
3.4. Exercises and Further Results 236
3.5. Notes and Remarks 239
Chapter 4. Essential Spectrum. The First Step: Elements of Toeplitz
Operators 243
4.1. Definition and Existence of the Symbol 243
4.2. Spectral Inclusions 246
4.3. The Fundamental Inversion Theorem 249
4.4. A Local Theory of Semicommutators 252
4.5. Fredholm Theory of the Toeplitz Algebra alg7#oo+c 256
4.6. Wiener-Hopf and Hankel Operators on the Real Line 261
4.7. Exercises and Further Results 262
4.8. Notes and Remarks 269
Chapter 5. Essential Spectrum. The Second Step: The Hilbert Matrix and
Other Hankel Operators 281
5.1. Piecewise Continuous Functions 281
5.2. The Schur Test 282
5.3. The Hilbert Matrix 283
5.4. The Main Theorem on the Essential Spectrum 288
5.5. Essentially Quasi-Nilpotent, and Essentially Self-Adjoint Hankel
Operators, and Other Corollaries 292
5.6. Exercises and Further Results 293
5.7. Notes and Remarks 302
Chapter 6. Hankel and Toeplitz Operators Associated
with Moment Problems 309
6.1. The Power Moment Problem 309
6.2. Hankel Operators Associated with a Measure 311
6.3. An Integral representation 314
6.4. The Trigonometric Moment Problem and Positive Toeplitz Forms 315
6.5. Exercises and Further Results 316
6.6. Notes and Remarks 324
Chapter 7. Singular Numbers of Hankel Operators 331
7.1. The Schmidt Decomposition 331
7.2. The Basic Adamyan-Arov-Krein Theorem 333
7.3. Multiplicative Properties of s-Functions 336
7.4. An Application to Interpolation by Meromorphic Functions:
The Schur-Takagi Problem 337
7.5. Exercises and Further Results 338
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