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

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- by Hermann Render
- Proc. Amer. Math. Soc.
**127**(1999), 1409-1411 - DOI: https://doi.org/10.1090/S0002-9939-99-04697-3
- Published electronically: January 29, 1999
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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 $H^{\infty }(\mathbb {D} ) .$## References

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## Bibliographic Information

**Hermann Render**- Affiliation: Universität Duisburg, Fachbereich Mathematik, Lotharstr. 65, D-47057 Duisburg, Federal Republic of Germany
- MR Author ID: 268351
- Email: render@math.uni-duisburg.de
- 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
- © Copyright 1999 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**127**(1999), 1409-1411 - MSC (1991): Primary 46J15; Secondary 30B10
- DOI: https://doi.org/10.1090/S0002-9939-99-04697-3
- MathSciNet review: 1476388