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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Ideals of square summable power series in several variables
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by James Radlow
Proc. Amer. Math. Soc. 38 (1973), 293-297
DOI: https://doi.org/10.1090/S0002-9939-1973-0312254-4

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 $r \geqq 2$.
References
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Bibliographic Information
  • © Copyright 1973 American Mathematical Society
  • 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