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Zapiski Nauchnykh Seminarov POMI, 1996, Volume 232, Pages 73–85
(Mi znsl3676)
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Weak generators of the algebra $l^\infty$ and the commutant of a model operator
M. F. Gamal' St. Petersburg Department of V. A. Steklov Institute of Mathematics, Russian Academy of Sciences
Abstract:
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.
Received: 20.11.1995
Citation:
M. F. Gamal', “Weak generators of the algebra $l^\infty$ and the commutant of a model operator”, Investigations on linear operators and function theory. Part 24, Zap. Nauchn. Sem. POMI, 232, POMI, St. Petersburg, 1996, 73–85; J. Math. Sci. (New York), 92:1 (1998), 3589–3596
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https://www.mathnet.ru/eng/znsl3676 https://www.mathnet.ru/eng/znsl/v232/p73
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Abstract page: | 113 | Full-text PDF : | 55 |
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