An invitation to the Drury-Arveson space

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. Contents 1. Introduction 2. Several definitions of H 2 d 2.1. Hardy space preliminaries 2.2. Discovering the Drury-Arveson space 2.3. Power series description of the Drury-Arveson space 2.4. RKHS description of the Drury-Arveson space 2.5. Function theory description of the Drury-Arveson space 2.6. The Drury-Arveson space as a member of a scale of spaces 2.7. The non-commutative approach 3. Multipliers and operator theory 3.1. Multipliers 3.2. Function theory of multipliers 3.3. Dilation and von Neumann's inequality 3.4. The non-commutative approach to multipliers 3.5. The Toeplitz algebra 3.6. Functional calculus 4. Complete Pick spaces

Cyclicity in the Drury-Arveson space and other weighted Besov spaces

“…Some cyclicity results for the Drury-Arveson space H 2 d can be found in [40, Theorems 1.4 and 1.5]. We refer the reader to [26] for a general introduction to the Drury-Arveson space H 2 d . The purpose of this paper is to study the properties of both cyclic and non-cyclic functions f ∈ H 2 d which have no zeros in B d .…”