Inner functions and division in Douglas algebras
In this interesting paper the authors continue the study of division in Douglas algebras begun by \textit{C. Guillory} and \textit{D. Sarason} [Mich. Math. J. 28, 173-181 (1981; Zbl 0417.30024)]. In that paper the problem was posed of finding the best constant \(N\in {\mathbb{N}}\) such that the following holds. If I is an inner function and if \(f\in H^{\infty}+C\) such that \(| f| \leq | I|\) on the maximal ideal space of \(H^{\infty}+C\), then \(f^ N\bar I\in H^{\infty}+C\). It was known that N cannot be one. The paper under review now settles this problem; it tells us that one may take \(N=2\). Quite unexpectedly, the techniques of the proof are essentially the same as those of Guillory and Sarason, who had shown the existence of such an N. Several related questions are also answered. The paper concludes with a nice factorization theorem for Blaschke products. Remark. Meanwhile, the reviewer has obtained from the authors a manuscript where the result mentioned above is generalized to arbitrary Douglas algebras.
factorization theorem for Blaschke products
scientific article; zbMATH DE number 3994303