Introduction to Model Spaces and their Operators

Description

The study of model spaces is a broad field with connections to complex analysis, operator theory, engineering and mathematical physics. This self-contained text is the ideal introduction for newcomers, quickly taking them through the history of the subject and then pointing towards areas of future research. Contents Preface Notation 1 Preliminaries 1.1 Measure and integral 1.2 Poisson integrals 1.3 Hilbert spaces and their operators 1.4 Notes 2 Inner functions 2.1 Disk automorphisms 2.2 Bounded analytic functions 2.3 Inner functions 2.4 Unimodular boundary limits 2.5 Angular derivatives 2.6 Frostman’s Theorem 2.7 Notes 3 Hardy spaces 3.1 Three approaches to the Hardy space 3.2 The Riesz projection 3.3 Factorization 3.4 A growth estimate 3.5 Associated classes of functions 3.6 Notes 3.7 For further exploration 4 Operators on the Hardy space 4.1 The shift operator 4.2 Toeplitz operators 4.3 A characterization of Toeplitz operators 4.4 The commutant of the shift 4.5 The backward shift 4.6 Difference quotient operator 4.7 Notes 4.8 For further exploration 5 Model spaces 5.1 Model spaces as invariant subspaces 5.2 Stability under conjugate analytic Toeplitz operators 5.3 Containment and lattice operations 5.4 A decomposition for Ku 5.5 Reproducing kernels 5.6 The projection Pu 5.7 Finite-dimensional model spaces 5.8 Density results 5.9 Takenaka–Malmquist–Walsh bases 5.10 Notes 5.11 For further exploration 6 Operators between model spaces 6.1 Littlewood Subordination Principle 6.2 Composition operators on model spaces 6.3 Unitary maps between model spaces 6.4 Multipliers of Ku 6.5 Multipliers between two model spaces 6.6 Notes 6.7 For further exploration 7 Boundary behavior 7.1 Pseudocontinuation 7.2 Cyclicity via pseudocontinuation 7.3 Analytic continuation 7.4 Boundary limits 7.5 Notes 8 Conjugation 8.1 Abstract conjugations 8.2 Conjugation on Ku 8.3 Inner functions in Ku 8.4 Generators of Ku 8.5 Cartesian decomposition 8.6 2 × 2 inner functions 8.7 Notes 9 The compressed shift 9.1 What is a compression? 9.2 The compressed shift 9.3 Invariant subspaces and cyclic vectors 9.4 The Sz.-Nagy–Foias¸ model 9.5 Functional calculus for Su 9.6 The spectrum of Su 9.7 The C*-algebra generated by Su 9.8 Notes 9.9 For further exploration 10 The commutant lifting theorem 10.1 Minimal isometric dilations 10.2 Existence and uniqueness 10.3 Strong convergence 10.4 An associated partial isometry 10.5 The commutant lifting theorem 10.6 The characterization of {Su}′ 10.7 Notes 11 Clark measures 11.1 The family of Clark measures 11.2 The Clark unitary operators 11.3 Spectral representation of the Clark operator 11.4 The Aleksandrov disintegration theorem 11.5 A connection to composition operators 11.6 Carleson measures 11.7 Isometric embeddings 11.8 Notes 11.9 For further exploration 12 Riesz bases 12.1 Minimal sequences 12.2 Uniformly minimal sequences 12.3 Uniformly separated sequences 12.4 The mappings Λ, V, and Γ 12.5 Abstract Riesz sequences 12.6 Riesz sequences in KB 12.7 Completeness problems 12.8 Notes 13 Truncated Toeplitz operators 13.1 The basics 13.2 A characterization 13.3 C-symmetric operators 13.4 The spectrum of Auϕ 13.5 An operator disintegration formula 13.6 Norm of a truncated Toeplitz operator 13.7 Notes 13.8 For further exploration References Index