Journal of Operator Theory

Volume 41, Issue 1, Winter 1999 pp. 127-138.

Algebraic reduction for Hardy submodules over polydisk algebras

Authors: Kunyu Guo
Author institution: Department of Mathematics, Fudan University, Shanghai, 200433, P.R. China

Summary: For a Hardy submodule

$M$ of

$H^2({\bbb D}^n)$ , assume that

$M\cap\cal C$ (or

$M \cap{\cal R}$ ) is dense in

$M$ , where

$\cal C$ (or

${\cal R}$ ) is the ring of all polynomials (or

${\cal R}$ is a Noetherian subring of

$\Hol({\overline{{\bbb D}}}^n)$ contai ning

$\cal C$ ). We describe those finite codimensional submodules of

$M$ by considering zero var ieties. The codimension formulas related to zero varieties, and some algebraic reduction theorems are obtained. These results can be regarded as generalizations of the result of Ahern-Clark ([2]). Finally, we point out that the results in this paper extend with essentially n o change to any reproducing Hilbert

$A(\Omega)$ -module

$H$ which satisfies certain techn ical hypotheses.

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