Treatise on the Shift Operator

Introductory Lecture. What This Book is About.- 1. Basic Objects.- 2. The Functional Model.- 3. The Details of the Plan.- 4. Concluding Remarks.- Lecture I. Invariant Subspaces.- 1. The Fundamental Theorem.- 2. The Inner-Outer Factorization.- 3. The Arithmetic of Inner Functions.- 4. The Adjoint Operators S*.- Supplements and Bibliographical Notes.- 5. Invariant Subspaces.- 6. The Shift of Arbitrary Multiplicity.- 7. Concluding Remarks.- Lecture II. Individual Theorems for the Operator S*.- 1. Pseudocontinuation of H2-Functions and S*-Cyclicity.- 2. Approximation by Rootspaces.- Supplements and Bibliographical Notes.- 3. More General Capacities.- 4. The Operator SE*.- 5. Concluding Remarks.- Lecture III. Compressions of the Shift and the Spectra of Inner Functions.- 1. The Spectrum of an Operator and the Spectrum of a Function.- 2. Functional Calculus and Derivation of Theorem LM.- 3. The Spectrum of the Operator ?(T).- Supplements and Bibliographical Notes.- 4. The Cyclic Vectors for the Operators T = PS|K and T*.- 5. A Calculus for Completely Non-Unitary Contractions.- 6. The Class C0.- 7. The Characteristic Function and the Spectrum.- 8. Concluding Remarks.- Lecture IV. Decomposition into Invariant Subspaces.- 1. Spectral Synthesis.- 2. Spectral Subspaces.- 3. Unicellular Operators.- Supplements and Bibliographical Notes.- 4. On Invariant Subspaces.- 5. Synthesis for C0-Operators.- 6. On Spectral Subspaces.- 7. Concerning Unicellular Operators.- 8. Concluding Remarks.- Lecture V. The Triangular Form of the Truncated Shift.- 1. Pure Point Spectrum.- 2. Continuous Singular Spectrum.- 3. Atomic Singular Spectrum.- 4. The General Case and Applications.- Supplements and Bibliographical Notes.- 5. Triangular Representations of More General Operators.- 6. ConcludingRemarks.- Lecture VI. Bases and Interpolation (Statement of the Problem).- 1. Riesz Bases.- 2. Interpolation.- 3. Spectral Projections and Unconditional Convergence.- Supplements and Bibliographical Notes.- 4. Bases of Subspaces.- 5. Bases of Eigenspaces.- 6. Concluding Remarks.- Lecture VII. Bases and Interpolation (Solution).- 1. Carleson Measures.- 2. Proof of the Theorem on Bases and Interpolation.- 3. Analysis of Carleson's Condition (C).- Supplements and Bibliographical Notes.- 4. Carleson Series.- 5. Remarks on Imbedding Theorems.- 6. Concluding Remarks.- Lecture VIII. Operator Interpolation and the Commutant.- 1. Interpolation by Bounded Analytic Functions.- 2. The Proof of Sarason's Theorem.- 3. Compact Operators in T??.- Supplements and Bibliographical Notes.- 4. The Multiplier Method and the Operator Calculus.- 5. Summation Bases.- 6. Hankel Operators and Angles Between Subspaces.- 7. Concluding Remarks.- Lecture IX. Generalized Spectrality and Interpolation of Germs of Analytic Functions.- 1. Generalized Spectrality.- 2. Non-Classical Interpolation in H? and Bases.- 3. The Rôle of the Uniform Minimality.- 4. Interpolation of Germs of Analytic Functions.- 5. Splitting and Blocking of Rootspaces.- 6. Spectrality and B0-Spectrality.- 7. Concluding Remarks.- Lecture X. Analysis of the Carleson-Vasyunin Condition.- 1. An Estimate for the Angle in Terms of Representing Measures.- 2. Bases of Rootspaces.- 3. Stolzian Spectrum.- 4. Singular Discrete Spectrum.- 5. Counterexamples.- 6. Concluding Remarks.- Lecture XI. On the Line and in the Halfplane.- 1. The Invariant Subspaces.- 2. Bases of Exponentials.- 3. Concluding Remarks.- Appendix 4. Essays on the Spectral Theory of Hankel and Toeplitz Operators.- (For detailed contents see page 300).- (Fordetailed contents see page 400).- List of Symbols.- Author Index.