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

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 $H^{\infty }(\mathbb {D} ) .$