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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Outer functions in function algebras on the bidisc

Author: Håkan Hedenmalm
Journal: Trans. Amer. Math. Soc. 306 (1988), 697-714
MSC: Primary 32A35; Secondary 32E25, 46J15
MathSciNet review: 933313
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Abstract: Let $ f$ be a function in the bidisc algebra $ A({{\mathbf{D}}^2})$ whose zero set $ Z(f)$ is contained in $ \{ 1\} \times \overline {\mathbf{D}} $. We show that the closure of the ideal generated by $ f$ coincides with the ideal of functions vanishing on $ Z(f)$ if and only if $ f( \cdot ,\,\alpha )$ is an outer function for all $ \alpha \in \overline {\mathbf{D}} $, and $ f(1,\, \cdot )$ either vanishes identically or is an outer function. Similar results are obtained for a few other function algebras on $ {{\mathbf{D}}^2}$ as well.

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