Introduction to Model Spaces and their Operators

✍ Scribed by Stephan Ramon Garcia, Javad Mashreghi, William T. Ross

• Publisher: Cambridge University Press
• Year: 2016
• Tongue: English
• Leaves: 339
• Series: Cambridge Studies in Advanced Mathematics (Book 148)
• Edition: 1
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

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

✦ Subjects

Mathematics;Calculus; Complex variable