Operator Theory and Function Theory in Drury-Arveson Space and Its Quotients

Michael Hartz () and Orr Moshe Shalit ()
Additional contact information
Michael Hartz: Universität des Saarlandes, Fachrichtung Mathematik
Orr Moshe Shalit: Technion Israel Institute of Technology

Chapter 43 in Operator Theory, 2026, pp 1259-1338 from Springer

Abstract: Abstract The Drury-Arveson space H d 2 $$H^{2}_{d}$$ (also known as symmetric Fock space or the d-shift space) is the reproducing kernel Hilbert space on the unit ball of ℂ d $$\mathbb {C}^d$$ with the kernel k ( z , w ) = ( 1 − 〈 z , w 〉 ) −1 $$k(z,w) = (1 - \langle z, w \rangle )^{-1}$$ . The operators M z i : f ( z ) ↦ z i f ( z ) $$M_{z_i} : f(z) \mapsto z_i f(z)$$ , arising from multiplication by the coordinate functions z 1 , … , z d $$z_1, \ldots , z_d$$ , form a commuting d-tuple M z = ( M z 1 , … , M z d ) $$M_z = (M_{z_1}, \ldots , M_{z_d})$$ . The d-tuple M z $$M_z$$ —which is called the d-shift—gives the Drury-Arveson space the structure of a Hilbert module. This Hilbert module is arguably the correct multivariable generalization of the Hardy space on the unit disc H 2 ( 𝔻 ) $$H^2(\mathbb {D})$$ . It turns out that the Drury-Arveson space H d 2 $$H^2_d$$ plays a universal role in operator theory (every pure, contractive Hilbert module is a quotient of an ampliation of H d 2 $$H^2_d$$ ) as well as in function theory (every irreducible complete Pick space is essentially a restriction of H d 2 $$H^2_d$$ to a subset of the ball). These universal properties resulted in the Drury-Arveson space being the subject of extensive studies, and the theory of the Drury-Arveson is today broad and deep. 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.

Keywords: Drury-arveson space; Symmetric fock space; Row contraction; D-contraction; Interpolation; Reproducing kernel Hilbert space; Multipliers; Dilation; Multivariable operator theory; Model theory; Hilbert module; Complete Pick space; Besov-Sobolev space (search for similar items in EconPapers)
Date: 2026
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-032-16356-1_60

Ordering information: This item can be ordered from
http://www.springer.com/9783032163561

DOI: 10.1007/978-3-032-16356-1_60

Access Statistics for this chapter

More chapters in Springer Books from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().