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

 

The maximal ideal space of $ H^{\infty }(\mathbb{D} )$
with respect to the Hadamard product


Author: Hermann Render
Journal: Proc. Amer. Math. Soc. 127 (1999), 1409-1411
MSC (1991): Primary 46J15; Secondary 30B10
Published electronically: January 29, 1999
MathSciNet review: 1476388
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Abstract: It is shown that the space of all regular maximal ideals in the Banach algebra $H^{\infty }(\mathbb{D} ) $ with respect to the Hadamard product is isomorphic to $ \mathbb{N} _{0}. $ The multiplicative functionals are exactly the evaluations at the $n$-th Taylor coefficient. It is a consequence that for a given function $ f(z) =\sum _{n=0}^{\infty }a_{n} z^{n} $ in $H^{\infty }(\mathbb{D} ) $ and for a function $ F(z) $ holomorphic in a neighborhood $U$ of $ 0 $ with $ F(0) =0 $ and $ a_{n} \in U $ for all $ n \in \mathbb{N}_{0} $ the function $ g(z) =\sum _{n=0}^{\infty }F(a_{n} ) z^{n} $ is in $H^{\infty }(\mathbb{D} ) . $


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Additional Information

Hermann Render
Affiliation: Universität Duisburg, Fachbereich Mathematik, Lotharstr. 65, D-47057 Duisburg, Federal Republic of Germany
Email: render@math.uni-duisburg.de

DOI: http://dx.doi.org/10.1090/S0002-9939-99-04697-3
PII: S 0002-9939(99)04697-3
Keywords: Hadamard product, bounded analytic functions
Received by editor(s): March 27, 1997
Received by editor(s) in revised form: August 19, 1997
Published electronically: January 29, 1999
Communicated by: Albert Baernstein II
Article copyright: © Copyright 1999 American Mathematical Society