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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A Wiener inversion-type theorem


Author: James R. Holub
Journal: Proc. Amer. Math. Soc. 97 (1986), 399-402
MSC: Primary 46J10
MathSciNet review: 840618
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Abstract: Let $ W(D) = \{ f(z) = \Sigma _{n = 0}^\infty {a_n}{z^n}\vert\;\vert\vert f\vert{\vert _1} = \Sigma _{n = 0}^\infty \vert{a_n}\vert < + \infty \} $, $ f(z)$ a function in $ W(D)$ for which $ f(0) = 1$, and $ {M_f}$ the operator of multiplication by $ f(z)$ on $ W(D)$. It is shown that if $ k$ and $ m$ are integers for which $ 0 \leq m \leq k - 1$ and $ X_k^m$ is the closed subspace of $ W(D)$ spanned by $ \{ {z^{nk + i}}\vert n = 0,1, \ldots ;i = 0,1, \ldots ,m\} $, then $ {M_f}$ is bounded below on $ X_k^m \Leftrightarrow f(z)$ does not have $ k - m$ distinct zeros in any set of the form $ \{ {w^i}{z_0}\vert \leq i \leq k - 1;\vert{z_0}\vert = 1\} $, where $ w$ is a primitive $ k$th root of unity.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1986-0840618-8
PII: S 0002-9939(1986)0840618-8
Keywords: Wiener disc algebra, multiplication operator, inversion theorem, basic sequence, shift basic sequence
Article copyright: © Copyright 1986 American Mathematical Society