Ideaux fermes d'algebres de Beurling analytiques sur le bidisque

TL;DR

This paper characterizes the closed ideals in certain Beurling algebras of holomorphic functions on the bidisc, identifying when ideals generated by a function match those generated by its zero set.

Contribution

It provides a complete description of ideal generation in Beurling algebras on the bidisc, extending classical results to weighted holomorphic function spaces.

Findings

Characterization of functions generating ideals equal to their zero set ideals

Identification of conditions for ideal equality in Beurling algebras

Extension of ideal theory to weighted bidisc holomorphic functions

Abstract

We study the closed ideal in the Beurling algebras $\mathcal{A}^{+}_{\alpha,\beta}$ of holomorphic function $f$ in the bidisc such that $\sum_{n,m\geq 0}|\hat{f}(n,m)|(1+n)^{\alpha}(1+m)^\beta<+\infty$. We determine the function $f\in\mathcal{A}^{+}_{\alpha,\beta}$ such that the ideals generated by $f$ coincide with the ideal generated by their zeros set.