Operator theory and function theory in Drury-Arveson space and its quotients

TL;DR

This survey explores the operator and function theory of the Drury-Arveson space, highlighting its fundamental role in multivariable operator theory and Pick interpolation, and providing a comprehensive overview of its properties and applications.

Contribution

It offers a comprehensive overview of the operator and function theoretic aspects of the Drury-Arveson space and emphasizes its universal role in multivariable operator theory and interpolation.

Findings

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

Provides a panoramic view of operator and function theoretic aspects

Connects the space to Pick interpolation theory

Abstract

The Drury-Arveson space $H^2_d$, also known as symmetric Fock space or the $d$-shift space, is a Hilbert function space that has a natural $d$-tuple of operators acting on it, which gives it the structure of a Hilbert module. This survey aims to introduce the Drury-Arveson space, to give a panoramic view of the main operator theoretic and function theoretic aspects of this space, and to describe the universal role that it plays in multivariable operator theory and in Pick interpolation theory.