Embedding dimension of the Dirichlet space

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.

20 pages; minor changes

References in corpus (2)

• The Smirnov class for spaces with the complete Pick property
• Factorization in weak products of complete Pick spaces