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Proceedings of the American Mathematical Society
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
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 $ \vert\vert B(z)f(z)\vert\vert = \vert\vert f(z)\vert\vert$, 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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DOI: http://dx.doi.org/10.1090/S0002-9939-1973-0312254-4
PII: S 0002-9939(1973)0312254-4
Keywords: Hilbert space of formal power series, ideal, formal product, Beurling's theorem, projection, commutativity relations
Article copyright: © Copyright 1973 American Mathematical Society