An invitation to the Drury-Arveson space

TL;DR

This paper introduces the Drury-Arveson space, a higher-dimensional generalization of the Hardy space, highlighting its significance in operator and function theory through an accessible overview and various approaches.

Contribution

It provides an introductory overview of the Drury-Arveson space, emphasizing its roles and different perspectives without covering the entire subject comprehensively.

Findings

Highlights the universal role of the Drury-Arveson space in operator and function theory

Showcases different approaches to understanding the space

Provides an accessible introduction for further study

Abstract

This is an extended version of a three part mini course on the Drury-Arveson space given as part of the Focus Program on Analytic Function Spaces and their Applications, hosted by the Fields Institute and held remotely. The Drury-Arveson space, also known as symmetric Fock space, is a natural generalization of the classical Hardy space on the unit disc to the unit ball in higher dimensions. It plays a universal role both in operator theory and in function theory. These notes give an introduction to the Drury-Arveson space. They are not intended to give a comprehensive overview of the entire subject, but rather aim to explain some of the contexts in which the Drury-Arveson space makes an appearance and to showcase the different approaches to this space.