Embedding dimension of the Dirichlet space

TL;DR

This paper proves that the Dirichlet space cannot be embedded into a finite-dimensional ball in a way that preserves its multiplier algebra structure, requiring an infinite-dimensional setting.

Contribution

It establishes that the embedding dimension for the Dirichlet space into the Drury-Arveson space must be infinite, even under surjectivity conditions on multiplier algebras.

Findings

Embedding dimension d must be infinite for Dirichlet space.

Finite-dimensional embeddings cannot preserve multiplier algebra structure.

Dirichlet space is fundamentally infinite-dimensional in this context.

Abstract

The classical Dirichlet space is a complete Pick space, hence by a theorem of Agler and McCarthy, there exists an embedding $b$ of the unit disc into a $d$-dimensional ball such that composition with $b$ realizes the Dirichlet space as a quotient of the Drury-Arveson space. We show that $d =\infty$ is necessary, even if we only demand that composition with $b$ induces a surjective map between the multiplier algebras.