# Journal of Operator Theory

Volume 80, Issue 2, Fall 2018 pp. 375-397.

Submodules of the Hardy module in infinitely
many variables

**Authors**:
Hui Dan (1), Kunyu Guo (2), and Hansong Huang (3)

**Author institution:** (1) School of Mathematical Sciences,
Fudan University, Shanghai, 200433, China

(2) School of Mathematical Sciences, Fudan University, Shanghai, 200433,
China

(3) Department of Mathematics, East China
University of Science and Technology, Shanghai, 200237,
China

**Summary: ** This paper is concerned with polynomially
generated submodules of the
Hardy module $H^2(\mathbb{D}_2^{\infty})$. Since the polynomial ring
$\mathcal{P}_{\infty}$ in infinitely many variables
is not Noetherian, some standard tricks for finitely many variables
fail to work.
Therefore, we need to introduce new techniques to the situation of
infinitely many variables. It is shown that some classical results of
$H^2(\mathbb{D}^n)$ remain valid for infinitely many variables.
However, some new phenomena indicate that the Hardy module
$H^2(\mathbb{D}_2^{\infty})$ diverges
considerably from the case in finitely many variables.

**DOI: **http://dx.doi.org/10.7900/jot.2017oct16.2187

**Keywords: ** Hardy module, infinitely many variables, closed ideals,
Ahern-Clark's theorem

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