Ideals of Square Summable Power Series in Several Variables

Abstract

Let $\mathcal {C}(z)$ be the Hilbert space of formal power series in ${z_1}, \cdots ,{z_r}(r \geqq 1)$. An ideal of $\mathcal {C}(z)$ is a vector subspace $\mathcal {M}$ of $\mathcal {C}(z)$ which contains ${z_1}f(z), \cdots ,{z_r}f(z)$ whenever it contains $f(z)$. If $B(z)$ is a formal power series such that $B(z)f(z)$ belongs to $\mathcal {C}(z)$ and $||B(z)f(z)|| = ||f(z)||$, then the set $\mathcal {M}(B)$ of all products $B(z)f(z)$ is a closed ideal of $\mathcal {C}(z)$. In the case $r = 1$, Beurling showed that every closed ideal is of this form for some such $B(z)$. Here we give conditions under which a closed ideal is of the form $\mathcal {M}(B)$ for $r \geqq 2$.