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

 

An extreme point in $ H\sp{\infty }(U\sp{2})$


Author: Nathaniel R. Riesenberg
Journal: Proc. Amer. Math. Soc. 76 (1979), 129-130
MSC: Primary 32A35; Secondary 30D55, 46J15
MathSciNet review: 534402
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Abstract: In this paper an example of a function $ f \in {H^\infty }({U^2})$ with $ {\left\Vert f \right\Vert _\infty } = 1$ and

$\displaystyle \int_{{T^2}} {\log (1 - \vert{f^ \ast }(z)\vert)dm > - \infty ,\quad z \in {T^2}} ,$

yet f is an extreme point in the unit ball of $ {H^\infty }$, is given. For functions $ f \in {H^\infty }({U^1})$ that

$\displaystyle \int_T {\log (1 - \vert{f^ \ast }(z)\vert)dm = - \infty ,\quad z \in T,} $

is both necessary and sufficient for f to be an extreme point in $ {H^\infty }$.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1979-0534402-9
PII: S 0002-9939(1979)0534402-9
Keywords: Extreme point, polydisc algebra, sup norm, integral log condition finite
Article copyright: © Copyright 1979 American Mathematical Society