Treatise on the Shift Operator: Spectral Function Theory

ISBN-13: 9783642701535 / Angielski / Miękka / 2011 / 491 str.

N.K. Nikol'skii;S.V. Hruscev;V.V. Peller

Opis

This book is an elementary introduction to non-classical spectral theory. Mter the basic definitions and a reduction to the study of the functional model the discussion will be centered around the simplest variant of such a model which, formally speaking, comprises only the class of contraction operators with a one dimensional rank of non-unitarity (rank(I - T*T) = rank(I - TT*) = 1). The main emphasis is on the technical side of the subject, the book being mostly devoted to a development of the analytical machinery of spectral theory rather than to this discipline itself. The functional model of Sz. -Nagy and Foia re duces the study of general operators to an investigation of the . compression T=PSIK of the shift operator S, Sf = zf, onto coinvariant subspaces (i. e. subspaces in variant with respect to the adjoint shift S*). In the main body of the book (the "Lectures" in the proper meaning of the word) this operator acts on the Hardy space H2 and is itself a part of the operator of multiplication by the independent variable in the space L2 (in the case at hand L2 means L2(lf), If being the unit circle), this operator again being fundamental for classical spectral theory."

Spis treści

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. Concluding Remarks.- 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).- (For detailed contents see page 400).- List of Symbols.- Author Index.