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

The Drury–Arveson space H d 2 $$H_{d}^{2}$$ (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 —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 $$H_{d}^{2}$$ plays a universal role in operator theory (every pure, contractive Hilbert module is a quotient of an ampliation of H d 2 $$H_{d}^{2}$$ ) as well as in function theory (every irreducible complete Pick space is essentially a restriction of H d 2 $$H_{d}^{2}$$ 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.