Infinite dimensional universal subspaces generated by Blaschke products
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- Proc. Amer. Math. Soc. 135 (2007), 1795-1801 Request permission
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 $||f||_\infty$. 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.References
- R. Aron, P. Gorkin: An infinite dimensional vector space of universal functions for $H^\infty$ on the ball, to appear in Canad. Math. Bull.
- Paul S. Bourdon and Joel H. Shapiro, Cyclic phenomena for composition operators, Mem. Amer. Math. Soc. 125 (1997), no. 596, x+105. MR 1396955, DOI 10.1090/memo/0596
- John B. Garnett, Bounded analytic functions, Pure and Applied Mathematics, vol. 96, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. MR 628971
- Pamela Gorkin and Raymond Mortini, Universal Blaschke products, Math. Proc. Cambridge Philos. Soc. 136 (2004), no. 1, 175–184. MR 2034021, DOI 10.1017/S0305004103007023
- Karl-Goswin Grosse-Erdmann, Universal families and hypercyclic operators, Bull. Amer. Math. Soc. (N.S.) 36 (1999), no. 3, 345–381. MR 1685272, DOI 10.1090/S0273-0979-99-00788-0
- K.-G. Grosse-Erdmann, Recent developments in hypercyclicity, RACSAM. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 97 (2003), no. 2, 273–286 (English, with English and Spanish summaries). MR 2068180
- Maurice Heins, A universal Blaschke product, Arch. Math. (Basel) 6 (1954), 41–44. MR 65644, DOI 10.1007/BF01899211
- Kenneth Hoffman, Bounded analytic functions and Gleason parts, Ann. of Math. (2) 86 (1967), 74–111. MR 215102, DOI 10.2307/1970361
Additional Information
- Raymond Mortini
- Affiliation: Département de Mathématiques, Université Paul Verlaine, Ile du Saulcy F-57045 Metz, France
- Email: mortini@math.univ-metz.fr
- 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
- © Copyright 2006
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 135 (2007), 1795-1801
- MSC (2000): Primary 30D50; Secondary 47B33, 46J15, 30H05
- DOI: https://doi.org/10.1090/S0002-9939-06-08669-2
- MathSciNet review: 2286090