Weak generators of the algebra $l^\infty$ and the commutant of a~model operator

M. F. Gamal'

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 24, Tome 232 (1996), pp. 73-85

Résumé

Let $B$ be a Blaschke product with simple zeros in the unit disc, let $\Lambda$ be the set of its zeros, and let $\varphi\in H^\infty$. It is known that a necessary condition (which is also sufficient in the case where $В$ satisfies the Carleson condition (C)) for $\varphi+BH^\infty$ to be a weak$^*$ generator of the algebra $H^\infty/BH^\infty$ is that $\varphi(\Lambda)$ be a weak$^*$ generator of the algebra $l^\infty$. We show that for any Blaschke product $В$ with simple zeros not satisfying condition (C) and having representation $B=B_1\cdot\ldots\cdot B_N$, where $B_1,\dots,B_N$ are Blaschke products satisfying condition (C), there exists a function $\varphi\in H^\infty$ such that $\varphi(\Lambda)$ is a weak$^*$ generator of $l^\infty$ and $\varphi+BH^\infty$ is not a weak$^*$ generator $H^\infty/BH^\infty$. Bibl. 12 titles.