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The Forelli problem concerning ideals in the disk algebra $ A({\bf D})$


Author: Raymond Mortini
Journal: Proc. Amer. Math. Soc. 95 (1985), 261-264
MSC: Primary 46J15; Secondary 30H05
DOI: https://doi.org/10.1090/S0002-9939-1985-0801335-2
MathSciNet review: 801335
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Abstract: Let $ Z(f)$ be the zero set of a function $ f \in A({\mathbf{D}})$ and $ Z(I) = { \cap _{f \in I}}Z(f)$ the zero set of an ideal $ I$ in $ A({\mathbf{D}})$. It is shown that in the disk algebra $ A({\mathbf{D}})$ every finitely generated ideal $ I$ has the weak Forelli property, i.e. there exists a function $ f \in I$ such that $ Z(f) \cap T = Z(I) \cap T$, where $ T$ is the boundary of the unit circle $ {\mathbf{D}}$. On the other hand, there exists a finitely generated ideal $ I$ in $ A({\mathbf{D}})$ such that $ Z(f) \ne Z(I)$ for each choice of $ f \in I$. This provides us with a negative answer to a problem of F. Forelli [1].


References [Enhancements On Off] (What's this?)

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  • [2] J. Garnett, Bounded analytic functions, Academic Press, New York, 1981. MR 83g 30037 MR 628971 (83g:30037)
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DOI: https://doi.org/10.1090/S0002-9939-1985-0801335-2
Article copyright: © Copyright 1985 American Mathematical Society

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