Invariant subspaces of weighted Bergman spaces in infinitely many variables

TL;DR

This paper classifies polynomially generated multiplier invariant subspaces of weighted Bergman spaces in infinitely many variables, providing a comprehensive understanding that extends to Hardy and finite-variable cases.

Contribution

It offers a complete classification of these invariant subspaces under unitary equivalence, covering both infinite and finite-variable settings.

Findings

Complete classification of invariant subspaces in infinite variables

Results unify Hardy and Bergman space cases

Applicable to finite-variable scenarios

Abstract

This paper is concerned with polynomially generated multiplier invariant subspaces of the weighted Bergman space $A_{\boldsymbol{\beta}}^2$ in infinitely many variables. We completely classify these invariant subspaces under the unitary equivalence. Our results not only cover cases of both the Hardy space $H^{2}(\mathbb{D}_{2}^{\infty})$ and the Bergman space $A^{2}(\mathbb{D}_{2}^{\infty})$ in infinitely many variables, but also apply in finite-variable setting.