Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Michael Hartz, Orr Shalit

TL;DR

The Drury-Arveson space $H^2_d$ is presented as a universal setting for multivariable operator theory and function theory, linking Hilbert module methods with reproducing-kernel function theory. The survey develops a dual perspective: as a function space on the ball with a universal kernel $k(z,w)= rac{1}{1-racket{z,w}}$ and as a symmetric Fock-space construction embedded in noncommutative operator theory, yielding a rich dilation and interpolation framework. Core contributions include the universal role of the $d$-shift in model theory for $d$-contractions, a detailed analysis of the multiplier algebra $ M_d$, Beurling-type theorems, and the complete Pick property, along with extensive theory on submodules, quotient modules, curvature invariants, essential normality, and isomorphism problems for complete Pick algebras. The work integrates operator-theoretic and function-theoretic methods to derive dilation results, interpolation on varieties, and corona-type theorems, with significant implications for multivariable operator theory and complex geometry.

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.

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

TL;DR

The Drury-Arveson space is presented as a universal setting for multivariable operator theory and function theory, linking Hilbert module methods with reproducing-kernel function theory. The survey develops a dual perspective: as a function space on the ball with a universal kernel and as a symmetric Fock-space construction embedded in noncommutative operator theory, yielding a rich dilation and interpolation framework. Core contributions include the universal role of the -shift in model theory for -contractions, a detailed analysis of the multiplier algebra , Beurling-type theorems, and the complete Pick property, along with extensive theory on submodules, quotient modules, curvature invariants, essential normality, and isomorphism problems for complete Pick algebras. The work integrates operator-theoretic and function-theoretic methods to derive dilation results, interpolation on varieties, and corona-type theorems, with significant implications for multivariable operator theory and complex geometry.

Abstract

The Drury-Arveson space , also known as symmetric Fock space or the -shift space, is a Hilbert function space that has a natural -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.

Paper Structure

This paper contains 103 sections, 86 theorems, 163 equations.

Table of Contents

  1. Introduction
  2. Notation and terminology
  3. Basic notation
  4. Tuples of operators
  5. Commuting and essentially normal tuples
  6. Row contractions and $d$-contractions
  7. Defect operator and defect space
  8. Hilbert modules
  9. Hilbert function spaces
  10. Drury-Arveson space as a function space
  11. $H^2_d$ as a graded completion of the polynomials
  12. $H^2_d$ as a Hilbert function space
  13. The multiplier algebra of $H^2_d$
  14. The $d$-shift
  15. Homogeneous decomposition of functions
  16. ...and 88 more sections

Key Result

Lemma 3.5.2

For all $n=0,1,\ldots,$, the integral converges in the strong operator topology to $f_n$. In particular, $\|f_n\|_{\mathcal{M}_d} \leq \|f\|_{\mathcal{M}_d}$.

Theorems & Definitions (138)

  • Lemma 3.5.2
  • Theorem 3.5.5
  • Theorem 3.6.1
  • Lemma 3.6.2
  • proof
  • Theorem 3.7.1
  • proof
  • Theorem 3.8.1
  • Theorem 3.8.3: BTV01, Theorem 2.1; EP02, Theorem 1.3
  • Theorem 3.9.2
  • ...and 128 more