Function Spaces, Theory and Applications (Fields Institute Communications, 87)

✍ Scribed by Ilia Binder (editor), Damir Kinzebulatov (editor), Javad Mashreghi (editor)

• Publisher: Springer
• Year: 2023
• Tongue: English
• Leaves: 487
• Edition: 1st ed. 2023
• 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 also have several essential applications in other fields of mathematics and engineering, e.g., robust control engineering, signal and image processing, and theory of communication. The most important Hilbert space of analytic functions is the Hardy class H2. However, its close cousins, e.g. the Bergman space A2, the Dirichlet space D, the model subspaces Kt, and the de Branges-Rovnyak spaces H(b), have also been the center of attention in the past two decades. Studying the Hilbert spaces of analytic functions and the operators acting on them, as well as their applications in other parts of mathematics or engineering were the main subjects of this program. During the program, the world leading experts on function spaces gathered and discussed the new achievements and future venues of research on analytic function spaces, their operators, and their applications in other domains.

With more than 250 hours of lectures by prominent mathematicians, a wide variety of topics were covered. More explicitly, there were mini-courses and workshops on Hardy Spaces, Dirichlet Spaces, Bergman Spaces, Model Spaces, Interpolation and Sampling, Riesz Bases, Frames and Signal Processing, Bounded Mean Oscillation, de Branges-Rovnyak Spaces, Operators on Function Spaces, Truncated Toeplitz Operators, Blaschke Products and Inner Functions, Discrete and Continuous Semigroups of Composition Operators, The Corona Problem, Non-commutative Function Theory, Drury-Arveson Space, and Convergence of Scattering Data and Non-linear Fourier Transform. At the end of each week, there was a high profile colloquium talk on the current topic. The program also contained two semester-long advanced courses on Schramm Loewner Evolution and Lattice Models and Reproducing Kernel Hilbert Space of Analytic Functions. The current volume features a more detailed version of some of the talks presented during the program.

✦ Table of Contents

Preface
Contents
Contributors
Absolute Continuity in Higher Dimensions
1 Introduction
2 Sobolev Spaces and Absolute Continuity on Lines
2.1 Sobolev Classes
2.2 Absolute Continuity on Lines
2.3 Mappings of Bounded and Finite Distortions
3 Lusin's (N) and (N-1)-Properties and Area Formulas
4 n-Absolute Continuity and n-Bounded Variation
4.1 n-Absolute Continuity
4.2 n-Bounded Variation
5 Modulus of Curve and Surface Families
5.1 Definition of Modulus
5.2 Distortion of Modulus Under Quasiregular Mappings
6 Absolute Continuity on Paths and on Surfaces
6.1 Absolute Continuity on Paths
6.2 Absolute Continuity on Surfaces
6.3 Absolute Continuity of Quasiconformal Mappings
7 Mappings of Finite Metric and Finite Area Distortions
7.1 Definitions
7.2 Metric Quasiconformality
8 Other Results and Illustrating Examples
8.1 Absolute Continuity in W1,n-1
8.2 Examples
8.3 Absolute Continuity of Finitely Lipschitz Mappings
8.4 Absolute Continuity of Sobolev Mappings
8.5 Open Problem
References
An Indefinite Analog of Sarason's Generalized Interpolation Theorem
1 Introduction and Preliminaries
1.1 The Inner Case and Sarason's Theorem
1.2 An Indefinite Generalization of Theorem 1.1
2 The Root Subspaces of T and T
3 The Operator Equation B(T)R=f(T)
4 Dual Results
5 Appendix
References
An Operator Theoretical Approach of Some Inverse Problems
1 Introduction
2 Framework, Notation, Definitions
2.1 With PDE, Examples
2.2 With Banach Spaces and Their Operators
2.3 Notation, Definitions
3 Inverse Boundary Value Problems
3.1 Harmonic Solutions, Hardy Spaces
3.1.1 Forward Problems
3.1.2 Inverse Problems
3.2 Conductivity PDE, Generalized Hardy Spaces
3.2.1 Forward Problem
3.2.2 Inverse Problems
4 Conclusion
References
Applications of the Automatic Additivity of Positive Homogeneous Order Isomorphisms Between Positive Definite Cones in C-Algebras
1 Introduction
2 Thompson Isometries
3 Preservers of Means
4 Preservers of the Norm of Means
References
Direct and Inverse Spectral Theorems for a Class of Canonical Systems with two Singular Endpoints
1 Introduction
2 The Two Basic Classes
2.1 The Class H of Hamiltonians
2.2 The Class M of Measures
3 Preliminaries from Indefinite Theory
3.1 Generalized Nevanlinna Functions and the Class N<∞(∞)
3.2 The Operator Associated with a Canonical System
3.3 General Hamiltonians
3.4 The Class H0
3.5 The Operator Model
3.6 The Basic Identification
4 Construction of the Spectral Measure
5 The Fourier Transform
5.1 Construction of a Fourier Transform
5.2 Computation of H as an Integral Transform
5.3 Computation of H-1 as an Integral Transform
5.4 The Connection Between the Point Mass at 0 and the Behaviour of H
6 Inverse Theorems
7 Sturm–Liouville Equations Without Potential: Singular 1/p
8 Sturm–Liouville Equations Without Potential: Singular w
9 Schrödinger Equations
References
Nevanlinna Domains and Uniform Approximation by Polyanalytic Polynomial Modules
1 Introduction
2 Nevanlinna Domains and Their Conformal Representations
3 Locally Nevanlinna Domains
4 g-Nevanlinna Domains
References
On Meromorphic Inner Functions in the Upper Half-Plane
1 Introduction
2 Some Properties of MIFs
3 Interpolation with Uniformly Bounded Derivatives
4 Uniqueness Results
5 MIFs and Toeplitz Operators
6 MIFs in Inverse Spectral Theory
7 Carleson's Theorem for the Non-linear Fourier Transform
References
On the Norm of the Hilbert Matrix
1 Introduction
1.1 Hilbert Matrix
1.2 Function Spaces
2 Boundedness of the Hilbert Matrix
3 Norm of the Hilbert Matrix on Weighted Bergman Spaces
3.1 Lower Bound
3.2 Upper Bound
References
Radial Limits of Holomorphic Functions in Cn or the Polydisc
1 Introduction
2 Radial Limits for a Handful of Function Spaces
3 Measure Theory
4 Holomorphic Approximation
5 Proof of Theorem 1.1 BoG,9:CL
6 Some Special Domains for Holomorphic Functions
6.1 Radial Domains
6.2 Strictly Starlike Domains
6.3 Vertical Line Domains
6.4 Upper Domains
7 Radial Limits of Entire Functions
References
Recent Developments in the Interplay Between Function Theory and Operator Theory for Block Toeplitz, Hankel, and Model Operators
1 Introduction
2 Preliminaries and Basic Theory
2.1 Basic Notions
2.2 Strong L2-Functions
2.3 The Douglas-Shapiro-Shields Factorization
2.4 Complementary Factors
2.5 Degree of Non-cyclicity
3 Matrix-Valued Function Theory
3.1 Tensored-Scalar Singularities
3.2 The Beurling Degree of an Inner Matrix Function
3.3 Abrahamse's Theorem for Matrix-Valued Symbols
3.4 H∞-Functional Calculus for the Compressions of the Shift
4 Operator-Valued Function Theory
4.1 Meromorphic Pseudo-continuations of Bounded Type
4.2 A Canonical Decomposition of Strong L2-Functions
4.3 Spectra of Model Operators
4.4 An Extension of Potapov's Factorization Theorem
4.5 Hyponormality and Subnormality of Toeplitz Operators
References
Sarason's Ha-plitz Product Problem
1 Introduction
2 The Berezin Transform
3 Toeplitz Products on H2 and A2
4 Toeplitz Products on F2
5 Products of Hankel Operators
6 Mixed Products of Ha-plitz Operators
7 Further Remarks
References
Sub-Hardy-Hilbert Spaces in the Non-commutative Unit Row Ball
1 Introduction
2 Background and Notation
2.1 Multipliers of Fock Space
2.2 Non-commutative Reproducing Kernel Hilbert Spaces
3 Column-Extreme Multipliers and the Sarason Outer Function
3.1 The Sarason Function
4 Free de Branges–Rovnyak Space and Smirnov Graph Analysis
5 Smirnov Column–Inner Pairs and Toeplitz Factorization
6 Application to Rational Multipliers
6.1 Preliminaries on NC Rational Functions
6.2 NC Rational Fejér–Riesz
References
The Relationship of the Gaussian Curvature with the Curvature of a Cowen-Douglas Operator
1 Introduction
2 Remarks on Curvature Inequality
3 A limit Computation
References
Weighted Polynomial Approximation on the Cubes of the Nonzero Integers
1 Introduction
2 Initial Simplifications of the Problem
3 Use of de Branges' Theorem
4 Formation of a Majorant for logW(n3)
5 Examination of the Majorant
6 Estimate of a Dirichlet Integral
7 Further Examination of the Majorant
8 Beginning of the Discussion of Sufficiency
9 A Parameter and Its Use
10 Partition of the Function G
11 Change in Notation and a Least Superharmonic Majorant
12 Use of the Least Superharmonic Majorant
13 Conclusion
14 Addendum
References
Index