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Infinite dimensional universal subspaces generated by Blaschke products

Author: Raymond Mortini
Journal: Proc. Amer. Math. Soc. 135 (2007), 1795-1801
MSC (2000): Primary 30D50; Secondary 47B33, 46J15, 30H05
Published electronically: December 28, 2006
MathSciNet review: 2286090
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Abstract: Let $ H^\infty$ be the Banach algebra of all bounded analytic functions in the unit disk $ \mathbb{D}$. A function $ f\in H^\infty$ is said to be universal with respect to the sequence $ (\frac{z+z_n}{1+\overline{z}_nz})_n$ of noneuclidian translates, if the set $ \{f(\frac{z+z_n}{1+\overline {z}_nz}):n\in\mathbb{N}\}$ is locally uniformly dense in the set of all holomorphic functions bounded by $ \vert\vert f\vert\vert _\infty$. We show that for any sequence of points $ (z_n)$ in $ \mathbb{D}$ tending to the boundary there exists a closed subspace of $ H^\infty$, topologically generated by Blaschke products, and linear isometric to $ \ell^1$, such that all of its elements $ f$ are universal with respect to noneuclidian translates. The proof is based on certain interpolation problems in the corona of $ H^\infty$. Results on cyclicity of composition operators in $ H^2$ are deduced.

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Additional Information

Raymond Mortini
Affiliation: Département de Mathématiques, Université Paul Verlaine, Ile du Saulcy F-57045 Metz, France

Keywords: Universal Blaschke products, interpolation in the corona, composition operators on Hardy spaces, joint cyclic vectors
Received by editor(s): September 6, 2005
Received by editor(s) in revised form: February 5, 2006
Published electronically: December 28, 2006
Additional Notes: The author thanks the referee for his/her comments improving the exposition of this work
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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