Mini-course on Drury-Arveson Space
The Drury-Arveson space H2d, also known as symmetric Fock space, is a Hilbert space of holomorphic functions on the Euclidean unit ball in Cd.
Its origins can be traced back to at least the 1970s, but the subject really gained momentum around the turn of the millennium.
The Drury-Arveson space is now widely considered to be the 'right' generalization of the Hardy space on the disc to the unit ball for many purposes.
The importance of the Drury-Arveson space can be seen from two distinct appearances of this space.
(1) Multivariable operator theory: In the study of contractions on Hilbert space, the unilateral shift plays a key role.
A natural multivariable generalization of contractions are commuting row contractions. In this setting, the tuple Mz=(Mz1,…,Mzd) of operators of multiplication by the coordinate functions on H2d plays the role of the unilateral shift. In particular, the natural analogues of von Neumann's inequality and of Sz.-Nagy's dilation theorem for commuting row contractions take place in the Drury-Arveson space.
(2) Complete Pick spaces: Complete Pick spaces are Hilbert function spaces satisfying a version of the Pick interpolation theorem. Important examples are the Hardy space and the Dirichlet space. The Drury-Arveson space is also a complete Pick space, but it is more than just an example.
A theorem of Agler and McCarthy shows that the Drury-Arveson space is a universal complete Pick space in the sense that every complete Pick space can essentially be identified with a restriction of the Drury-Arveson space.
In this mini course, I will provide an introduction to the Drury-Arveson space. Topics that will be covered include:
- different descriptions of the Drury-Arveson space
- row contractions, dilations and von Neumann's inequality
- pick spaces and universality of the Drury-Arveson space
- interpolating sequences