On a generalization of the Corona problem (Q1819324)

Let \(H^{\infty}\) denote the algebra of bounded analytic functions in the open unit disk and let \(f,f_ 1,...,f_ N\) be functions in \(H^{\infty}\) satisfying \[ (1)\quad | f| \leq \sum^{N}_{i=1}| f_ i|. \] As a generalization of the Corona problem, T. H. Wolff proved that (1) implies the existence of functions \(g_ 1,...,g_ N\in H^{\infty}\) such that \[ (2)\quad f^ 3=\sum^{N}_{i=1}g_ if_ i. \] As was noted earlier by R. Rao, the conclusion does not hold (in general) if the exponent 3 is replaced by 1. The question of whether (2) holds for \(f^ 2\) is still a challenging open problem. The authors of the present paper now give conditions on the functions \(f_ 1,...,f_ N\) in order that (1) implies that f resp. \(f^ 2\) belongs to the ideal generated by \(f_ 1,...,f_ N.\) Reviewers remark. Apparently the authors are not aware of the work of M. v. Renteln and V. A. Tolokonnikov. For instance, Theorem 6 has been proved a long time ago by \textit{M. v. Renteln} [Collect. Math. 26, 115-126 (1975; Zbl 0315.46044)]. It is today a standard fact. Even far more reaching results have been obtained by \textit{V. A. Tolokonnikov} [Sov. J. Math. 27, 2549-2553 (1984; Zbl 0518.46046)]. On the whole the paper under review does not yield any substantial progress on the generalized corona problem. Finally let us point out an error on p. 786. The ideal \(I_ H\) should be replaced by \(I_ H\cap H^{\infty}\).