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
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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 $r \geqq 2$.References
- Arne Beurling, On two problems concerning linear transformations in Hilbert space, Acta Math. 81 (1948), 239–255. MR 27954, DOI 10.1007/BF02395019
- Louis de Branges and James Rovnyak, Square summable power series, Holt, Rinehart and Winston, New York-Toronto-London, 1966. MR 0215065
- John von Neumann, Functional Operators. II. The Geometry of Orthogonal Spaces, Annals of Mathematics Studies, No. 22, Princeton University Press, Princeton, N. J., 1950. MR 0034514
- James Rovnyak, Ideals of square summable power series, Math. Mag. 33 (1959/60), 265–270. MR 123906, DOI 10.2307/3029797
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