On finitely generated closed ideals of \(H^ \infty\). (Chinese) Zbl 0810.46058

Wang, Jian

Chin. Ann. Math., Ser. A 12, No. 2, 210-217 (1991).

Let \(\Omega\) be a Denjoy domain and \(g, f_ 1,\dots, f_ n\in H^ \infty(\Omega)\). Let \(\alpha(t)\) be a nonnegative function on \(\mathbb{R}^ +\) with \(\alpha(t)/t\to 0\). If the following conditions are satisfied:
(1) \(| g(z)|\leq \alpha\left(\sum^ n_{j=1} | f_ j(z)|\right)\), \(z\in \Omega\);
(2) \(g(\bar z)= \overline{g(z)}\), \(z\in \Omega\);
(3) there exists \(\delta>0\) such that \(| g(z)|\geq \delta\), \(x\in \Omega\cap \mathbb{R}\);
then \(g\in \overline{I(f_ 1,\dots, f_ n)}\).
Here \(I(f_ 1,\dots, f_ n)\) is the ideal generated by \(\{f_ 1,\dots, f_ n\}\) and \(\overline{I(f_ 1,\dots, f_ n)}\) is the closure of \(I(f_ 1,\dots, f_ n)\).

Reviewer: Ren Fu-Yao (Shanghai)

MSC:

46J15

Banach algebras of differentiable or analytic functions, \(H^p\)-spaces

Keywords:

Denjoy domain