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. More explicitly, 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.

Javad Mashreghi is a mathematician and author working in fields of function space theory, functional analysis and complex analysis. He is a professeur titulaire at Laval University and was recently the 35th President of the Canadian Mathematical Society (2020-2022). He is immensely involved in various aspects of North America's mathematical community, having served on numerous editorial, administrative and selection committees all across Canada and the U.S. (CMS, AMS, Fields Institute, CRM, AARMS, NSERC, FQRNT, NSF). He is the editor-in-chief of the Canadian Mathematical Bulletin (2020-2025) and Concrete Operators (2018-2022), and the Analysis Section Editor of the Proceedings of the American Mathematical Society (2020-2027). Among his awards and distinctions, there are the IEEE Prize Paper Award, 2021, the Fellow of the Canadian Mathematical Society, 2019, the Khwarizmi International Award, Research Prize of IROST, 2009, which he declined, the G. de B. Robinson Award, 2004. As for publications, he has more than 120 articles, 12 books (e.g., published by Cambridge, Oxford, AMS and Springer) and 10 conference proceedings.

Preface

Hardy Spaces

The Dirichlet space

Bergman space of the unit disc

Model Spaces

Operators on Function Spaces

Truncated Toeplitz Operators

Semigroups of weighted composition operators on spaces of holomorphic functions

The Corona Problem

A brief introduction to noncommutative function theory

An invitation to the Drury-Arveson space

References