A generalized Wolff's Ideal Theorem on certain subalgebras of \(H^{\infty}(\mathbb{D})\) (Q902272)

Let \(H^\infty(\mathbb{D})\) denote the space of bounded analytic functions on the unit disc \(\mathbb{D}\). Let \(B\) denote a Blaschke product. In this paper the authors study the ideal question for the algebra \(\mathbb{C}+BH^\infty(\mathbb{D})\). Namely they provide strong sufficient conditions on the collection of generators \(\{f_j\}\subset \mathbb{C}+BH^\infty(\mathbb{D})\) and a function \(h\) so that \(h\) belongs to the ideal generated by the \(\{f_j\}\). This can be seen as an extension of the work of Treil for the case of \(H^\infty(\mathbb{D})\). These results are further generalized to two other algebras of analytic functions on the unit disc of a similar type. The method of proof is to use the result of Treil for \(H^\infty(\mathbb{D})\) to find a solution in \(H^\infty(\mathbb{D})\). This is then corrected to belong to the algebra \(\mathbb{C}+BH^\infty(\mathbb{D})\) through a suitable projection onto the kernels of certain multiplication operators (motivated by the Koszul complex).