Introduction to Model Spaces and their Operators (Cambridge Studies in Advanced Mathematics)

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