Illinois Journal of Mathematics

Common bounded universal functions for composition operators

Abstract

Let $\mathcal{A}$ be the set of automorphisms of the unit disk with $1$ as attractive fixed point. We prove that there exists a single Blaschke product that is universal for every composition operator $C_\phi$, $\phi\in\mathcal{A}$, acting on the unit ball of $H^\infty(\mathbb{D})$.

Article information

Source
Illinois J. Math., Volume 52, Number 3 (2008), 995-1006.

Dates
First available in Project Euclid: 1 October 2009

https://projecteuclid.org/euclid.ijm/1254403727

Digital Object Identifier
doi:10.1215/ijm/1254403727

Mathematical Reviews number (MathSciNet)
MR2546020

Zentralblatt MATH identifier
1181.47019

Citation

Bayart, Frédéric; Grivaux, Sophie; Mortini, Raymond. Common bounded universal functions for composition operators. Illinois J. Math. 52 (2008), no. 3, 995--1006. doi:10.1215/ijm/1254403727. https://projecteuclid.org/euclid.ijm/1254403727

References

• F. Bayart, Common hypercyclic vectors for composition operators, J. Operator Theory 52 (2004), 353–370.
• F. Bayart and P. Gorkin, How to get universal inner functions, Math. Ann. 337 (2007), 875–886.
• F. Bayart, P. Gorkin, S. Grivaux and R. Mortini, Bounded universal functions for sequences of holomorphic self-maps of the disk, to appear in Ark. Mat.
• F. Bayart and S. Grivaux, Hypercyclicity and unimodular point spectrum, J. Funct. Anal. 226 (2005), 281–300.
• F. Bayart and E. Matheron, How to get common universal vectors, Indiana Univ. Math. J. 56 (2007), 553–580.
• G. Costakis and M. Sambarino, Genericity of wild holomorphic functions and common hypercyclic vectors, Adv. Math. 182 (2004), 278–306.
• J. B. Garnett, Bounded analytic functions, Academic Press, New York–London, 1981.
• P. Gorkin and R. Mortini, Universal Blaschke products, Math. Proc. Cambridge Philos. Soc. 136 (2004), 175–184.
• P. Gorkin and R. Mortini, Radial limits of interpolating Blaschke products, Math. Ann. 331 (2005), 417–444.
• M. Heins, A universal Blaschke product, Arch. Math. 6 (1955), 41–44.
• R. Mortini, Infinite dimensional universal subspaces generated by Blaschke products, Proc. Amer. Math. Soc. 135 (2007), 1795–1801.