Lectures on Analytic Function Spaces and their Applications (Fields Institute Monographs, 39)

✍ Scribed by Javad Mashreghi (editor)

• Publisher: Springer
• Year: 2023
• Tongue: English
• Leaves: 426
• Category: Library

✦ Synopsis

The focus program on Analytic Function Spaces and their Applications took place at Fields Institute from July 1st to December 31st, 2021. Hilbert spaces of analytic functions form one of the pillars of complex analysis. These spaces have a rich structure and for more than a century have been studied by many prominent mathematicians. They have essential applications in other fields of mathematics and engineering. The most important Hilbert space of analytic functions is the Hardy class H2. However, its close cousins―the Bergman space A2, the Dirichlet space D, the model subspaces Kt, and the de Branges-Rovnyak spaces H(b)―have also garnered attention in recent decades. Leading experts on function spaces gathered and discussed new achievements and future venues of research on analytic function spaces, their operators, and their applications in other domains.
With over 250 hours of lectures by prominent mathematicians, the program spanned a wide variety of topics. Moreexplicitly, there were courses and workshops on Interpolation and Sampling, Riesz Bases, Frames and Signal Processing, Bounded Mean Oscillation, de Branges-Rovnyak Spaces, Blaschke Products and Inner Functions, and Convergence of Scattering Data and Non-linear Fourier Transform, among others. At the end of each week, there was a high-profile colloquium talk on the current topic. The program also contained two advanced courses on Schramm Loewner Evolution and Lattice Models and Reproducing Kernel Hilbert Space of Analytic Functions.
This volume features the courses given on Hardy Spaces, Dirichlet Spaces, Bergman Spaces, Model Spaces, Operators on Function Spaces, Truncated Toeplitz Operators, Semigroups of weighted composition operators on spaces of holomorphic functions, the Corona Problem, Non-commutative Function Theory, and Drury-Arveson Space. This volume is a valuable resource for researchers interested in analytic function spaces.

✦ Table of Contents

Preface
Contents
Contributors
1 Hardy Spaces
1.1 Preliminaries
1.1.1 Boundary Values
1.1.2 Convolution
1.1.3 Approximate Identities
1.2 An Overview of Hardy Spaces
1.2.1 The Banach Algebra H∞(D)
1.2.2 The Hardy-Hilbert Space H2(D)
1.2.3 The Banach Space Hp(D)
1.3 Analysis and Synthesis of Functions in Hardy Spaces
1.3.1 Synthesis: Various Convergence Types
1.3.1.1 A General Construction
1.3.1.2 L∞-Norm Convergence of Dilates: A Special Construction in h∞(D)
1.3.1.3 Weak-Convergence of Dilates: Construction in h1(D)
1.3.1.4 Lp-Norm Convergence of Dilates: Construction in hp(D)
1.3.1.5 Weak-Convergence of Dilates: Construction in h∞(D)
1.3.1.6 L2-Norm Convergence of Dilates: Construction in h2(D)
1.3.2 Analysis: Poisson Representations
1.3.2.1 Poisson Representation in h(D)
1.3.2.2 Poisson Representation in h∞(D)
1.3.2.3 Poisson Representation in hp(D), (1<p<∞)
1.3.2.4 Poisson Representation in h1(D)
1.3.2.5 Poisson Representation in Hp(D), 1 < p ∞
1.3.2.6 Poisson Representations in H1(D)
1.4 An Overview of Representation Theorems
1.4.1 Harmonic Hardy Spaces
1.4.2 Analytic Hardy Spaces
References
2 The Dirichlet Space
2.1 Introduction
2.1.1 What is the Dirichlet Space?
2.1.2 History and Motivation
2.1.3 What to Study?
2.1.4 Where to Find Out More About D?
2.2 Capacity
2.2.1 Energy
2.2.2 Capacity of Compact Sets
2.2.3 Capacity of General Sets
2.2.4 Equilibrium Measures
2.3 Boundary Behavior
2.3.1 Preliminary Remarks
2.3.2 Beurling's Theorem
2.3.3 Capacitary Weak-Type and Strong-Type Inequalities
2.3.4 Douglas' Formula
2.3.5 Exponential Approach Region
2.3.6 Carleson's Formula
2.3.7 Some Further Developments
2.4 Zeros
2.4.1 Preliminary Remarks
2.4.2 The Three Cases
2.4.3 Boundary Zero Sets
2.4.4 Arguments of Zero Sets
2.4.5 Some Further Developments
2.5 Multipliers
2.5.1 Preliminary Remarks
2.5.2 Multipliers
2.5.3 Carleson Measures
2.5.4 Characterization of Carleson Measures
2.5.5 Multipliers and Reproducing Kernels
2.5.6 Pick Interpolation
2.5.7 Interpolating Sequences
2.5.8 Factorization Theorems
2.5.9 Some Further Developments
2.6 Conformal Invariance
2.6.1 Preliminary Remarks
2.6.2 Characterization of D via Möbius Invariance
2.6.3 Composition Operators
2.6.4 Weighted Composition Operators
2.6.5 Some Further Developments
2.7 Weighted Dirichlet Spaces
2.7.1 The Dα Spaces
2.7.2 The Dμ Spaces
2.7.3 Properties of Dμ (Richter–Sundberg 2:MR1116495)
2.7.4 Hadamard Multipliers
2.7.5 Special Cases of Hadamard Multipliers
2.7.6 Some Further Developments
2.8 Shift-Invariant Subspaces
2.8.1 Preliminary Remarks
2.8.2 The Shift Operator on Dμ
2.8.3 Invariant Subspaces of (Mz,D)
2.8.4 Cyclic Invariant Subspaces
2.8.5 Cyclic Invariant Subspaces and Boundary Zeros
2.8.6 Brown–Shields Conjecture
2.8.7 Some Further Developments
References
3 Bergman Space of the Unit Disc
3.1 Origins
3.2 Aspects of the Basic Theory
3.2.1 Bergman Versus Hardy Spaces
3.2.2 Weighted Bergman Spaces
3.2.3 Disc Automorphisms and Change of Variables
3.2.4 Pointwise Bounds and Reproducing Property
3.2.5 Reproducing Kernel and the Bergman Projection
3.2.6 Duality
3.3 Zero Sets
3.4 Invariant Subspaces of Infinite Index
3.4.1 From Infinite Index to the Invariant Subspace Problem
3.4.2 The Index of Invariant Subspaces
3.4.3 Spectral Synthesis and Dominating Sets
3.4.4 Mz*|I() Can Be Similar to a Diagonal Normal Operator
3.4.5 An Interpolating Sequence That Is Dominating
3.4.6 Further Results About Invariant Subspaces with High Index
3.5 Bergman Inner Functions
3.5.1 An Extremal Problem
3.5.2 Contractive Divisors Via the Biharmonic Function
3.5.3 The Reproducing Kernel Approach
3.5.4 Further Contractive Divisor Results
3.6 Appendix
3.6.1 The Schur Test
3.6.2 Carleson's Interpolation Theorem
3.6.3 Positive Definite Functions
References
4 Model Spaces
4.1 Introduction
4.2 Preliminaries
4.2.1 The Hardy Space
4.2.2 Inner Functions
4.2.3 Canonical Factorization
4.2.4 Bounded Type
4.3 Model Spaces
4.3.1 Basic Properties
4.3.2 Reproducing Kernels
4.3.3 Finite-Dimensional Model Spaces
4.3.4 Stability Under Co-Analytic Toeplitz Operators
4.4 The Compressed Shift
4.4.1 The Livšic–Möller Theorem
4.4.2 Model Operators
4.5 Density Results
4.6 Bases for Model Spaces
4.6.1 Takenaka–Malmquist–Walsh Basis
4.6.2 Riesz Bases
4.7 Continuability
4.7.1 Analytic Continuation
4.7.2 Pseudocontinuation
4.8 Conjugation on Model Spaces
4.8.1 Conjugations
4.8.2 The Model Conjugation
4.8.3 Associated Inner Functions
4.8.4 Generators of Ku
4.9 Aleksandrov–Clark Theory
4.9.1 Herglotz' Theorem and Clark Measures
4.9.2 Clark Operators
4.9.3 Deeper Results
4.10 Explicit Description of Kup
4.11 Quaternionic Structure of 2 2 Inner Functions
4.12 Conclusion
References
5 Operators on Function Spaces
5.1 Introduction
5.2 Hilbert Spaces of Functions
5.2.1 General Operator Theory Concepts
5.2.2 The Lebesgue Spaces
5.2.3 The Hardy Spaces
5.3 The Volterra Operator
5.3.1 Basic Properties
5.3.2 Spectrum
5.4 The Numerical Range
5.4.1 Commutant
5.4.2 Square Root
5.4.3 Invariant Subspaces
5.4.4 Complex Symmetric Properties
5.4.5 Generalized Volterra Operator
5.5 The Cesàro Operator
5.5.1 Basic Properties
5.5.2 Spectrum
5.6 The Numerical Range
5.6.1 Subnormal
5.6.2 The Commutant
5.6.3 Invariant Subspaces
5.6.4 Square Root
5.6.5 Generalized Cesaro Operators
5.7 Toeplitz Operators
5.7.1 Multiplication Operators on L2(T)
5.7.2 The Bilateral Shift
5.7.3 Multiplication Operators on H2
5.7.4 The Unilateral Shift
5.7.5 Toeplitz Operators on H2(D)
5.7.6 Toeplitz Operators on H2(C+)
5.7.7 Toeplitz Operators on Other Spaces
5.8 Hankel Operators
5.8.1 Hankel Matrices
5.8.2 Hankel Operators
5.8.3 The Norm of a Hankel Operator
5.8.4 Bounded Mean Oscillation
5.9 The Hilbert Matrix Again
5.9.1 Another Setting for Hankel Operators
5.9.2 Back to Multiplication Operators
5.10 Fourier and Hilbert Transforms
5.10.1 Plancherel's Theorem
5.10.2 The Spectrum of F
5.10.3 The Hilbert Transform
5.10.4 Spectrum of H
5.11 Further Explorations
References
6 Truncated Toeplitz Operators
6.1 Introduction
6.2 Preliminaries
6.2.1 Function Spaces, Multiplication Operators and Their Cognates
6.2.2 Model Spaces and One Component Inner Functions
6.2.3 Carleson Measures for the Hardy Spaces and for the Model Spaces
6.3 Truncated Toeplitz Operators
6.3.1 Definition of Truncated Toeplitz Operators
6.3.2 An Equivalent Definition and Some Basic Properties
6.4 Why Studying Truncated Toeplitz Operators?
6.4.1 The Sz.-Nagy–Foias Model
6.4.2 The Commutant of S
6.4.3 The Nevanlinna–Pick Interpolation Problem
6.4.4 A Link with Truncated Wiener-Hopf Operators
6.5 The Class of Symbols for a Truncated Toeplitz Operator
6.6 Algebraic Characterization of Truncated Toeplitz Operators
6.7 Complex Symmetric Operators
6.7.1 Truncated Toeplitz Operators Are Complex Symmetric
6.7.2 Another Characterization of Truncated Toeplitz Operators and New Examples
6.8 Norm of a Truncated Toeplitz Operator
6.9 Spectral Properties of Aφ
6.10 Finite Rank Truncated Toeplitz Operators
6.11 Compact Truncated Toeplitz Operators
6.12 Problem of the Existence of a Bounded Symbol
References
7 Semigroups of Weighted Composition Operators on Spaces of Holomorphic Functions
7.1 Introduction
7.2 Background
7.2.1 Strongly Continuous Semigroups of Operators: Definition and Characterization
7.2.2 Analytic Semiflows on a Domain and Models for Semiflows on D
7.2.3 Models for Analytic Flows on D
7.2.4 C0-Semigroups of Composition Operators
7.2.5 Spaces on Which Semigroups Are Not C0
7.3 Motivation
7.3.1 Universal Operators
7.3.2 Isometries
7.3.3 Change of Domain
7.4 Asymptotic Behaviour of Tn or Tt
7.4.1 The Discrete Unweighted Case
7.4.1.1 Denjoy–Wolff Theory
7.4.2 The Continuous Unweighted Case
7.4.3 Weighted Composition Operators
7.4.4 Isometry and Similarity to Isometry
7.4.5 Generators
7.5 Compactness and Analyticity
7.5.1 Immediate and Eventual Compactness
7.5.2 Compact Analytic Semigroups
7.6 An Outlook on C+ and C
7.6.1 The Right Halfplane C+
7.6.2 The Complex Plane C
7.6.2.1 Composition Operators
7.6.2.2 Weighted Composition Operators
References
8 The Corona Problem
8.1 Introduction
8.2 Banach-Valued -Equations on the Disk
8.2.1 Interpolating Sequences for H∞
8.2.2 Main Result
8.3 Proof of Theorem 8.2.2
8.3.1 Particular Case
8.3.2 General Case
8.4 Carleson's Corona Theorem
8.5 Structure of the Maximal Ideal Space of H∞
8.5.1 Gleason Parts
8.5.2 Structure of Ma
8.5.3 Structure of Ms
8.6 Banach-Valued Corona Problem
8.7 Operator Completion Problem for H∞
References
9 A Brief Introduction to Noncommutative Function Theory
9.1 Introduction
9.2 nc Sets and nc Functions
9.2.1 Examples
9.2.2 nc Functions
9.3 Locally Bounded nc Functions Are Differentiable
9.4 Topologies and Further Remarks
9.5 Square-Summable nc Power Series
9.6 Reproducing Kernels
9.6.1 The Connection with the Drury-Arveson Space
9.7 nc Rational Functions
9.8 The d-Shift and Multipliers of H2nc
9.9 Invariant Subspaces
9.10 Further Topics
9.10.1 Nevanlinna-Pick Interpolation
9.10.2 nc Measures and Cauchy Integrals
References
10 An Invitation to the Drury–Arveson Space
10.1 Introduction
10.2 Several Definitions of Drury–Arveson Space
10.2.1 Hardy Space Preliminaries
10.2.2 Discovering the Drury–Arveson Space
10.2.3 Power Series Description of the Drury–Arveson Space
10.2.4 RKHS Description of the Drury–Arveson Space
10.2.5 Function Theory Description of the Drury–Arveson Space
10.2.6 The Drury–Arveson Space as a Member of a Scale of Spaces
10.2.7 The Non-commutative Approach
10.3 Multipliers and Operator Theory
10.3.1 Multipliers
10.3.2 Function Theory of Multipliers
10.3.3 Dilation and von Neumann's Inequality
10.3.4 The Non-commutative Approach to Multipliers
10.3.5 The Toeplitz Algebra
10.3.6 Functional Calculus
10.4 Complete Pick Spaces
10.4.1 Pick's Theorem and Complete Pick Spaces
10.4.2 Characterizing Complete Pick Spaces
10.4.3 Universality of the Drury–Arveson Space
10.5 Selected Topics
10.5.1 Maximal Ideal Space and Corona Theorem
10.5.2 Interpolating Sequences
10.5.3 Weak Products and Hankel Operators
References
Index