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.

Keywords: Hardy submodules, ideals, charactistic space

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