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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Ideals of square summable power series in several variables


Author: James Radlow
Journal: Proc. Amer. Math. Soc. 38 (1973), 293-297
MSC: Primary 46E99; Secondary 32A05
DOI: https://doi.org/10.1090/S0002-9939-1973-0312254-4
MathSciNet review: 0312254
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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 $r \geqq 2$.


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Keywords: Hilbert space of formal power series, ideal, formal product, Beurling’s theorem, projection, commutativity relations
Article copyright: © Copyright 1973 American Mathematical Society