Topological Hilbert Nullstellensatz for Bergman Spaces

TL;DR

This paper characterizes polynomial and analytic function ideals closed in Bergman spaces on certain domains, advancing the understanding of invariant subspaces of multiplication operators in several variables.

Contribution

It provides the first characterizations of closed ideals in Bergman spaces in two variables, proving analogues of a conjecture by Douglas and Paulsen.

Findings

Characterization of closed polynomial ideals in Bergman spaces

Proof of Bergman space analogues of Douglas and Paulsen's conjecture

Advancement in classification of invariant subspaces in several variables

Abstract

The results in the paper are related to the classification problem for invariant subspaces of multiplication operators in several variables. The main results consist of characterizations, in the two dimensional case, of ideals of polynomials and analytic functions which are closed in the relative topology induced by Bergman spaces on certain domains. This provides proofs of the Bergman space analogues of a conjecture of R. G. Douglas and V. Paulsen.