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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