Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Common hypercyclic vectors for families of operators

Authors: Eva A. Gallardo-Gutierrez and Jonathan R. Partington
Journal: Proc. Amer. Math. Soc. 136 (2008), 119-126
MSC (2000): Primary 47A16; Secondary 47B33, 47B37
Published electronically: September 25, 2007
MathSciNet review: 2350396
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We provide a criterion for the existence of a residual set of common hypercyclic vectors for an uncountable family of hypercyclic operators which is based on a previous one given by Costakis and Sambarino. As an application, we get common hypercyclic vectors for a particular family of hypercyclic scalar multiples of the adjoint of a multiplier in the Hardy space, generalizing recent results by Abakumov and Gordon and also Bayart. The criterion is applied to other specific families of operators.

References [Enhancements On Off] (What's this?)

  • 1. E. Abakumov and J. Gordon, Common hypercyclic vectors for multiples of backward shift, J. Funct. Anal., 200, (2003), no. 2, 494-504. MR 1979020 (2004g:47012)
  • 2. F. Bayart, Common hypercyclic vectors for composition operators, J. Operator Theory, 52, (2004), no. 2, 353-370. MR 2119275 (2006a:47014)
  • 3. F. Bayart and S. Grivaux, Hypercyclicité: le r$ \hat{o}$le du spectre ponctuel unimodulaire C. R. Math. Acad. Sci. Paris, 338, (2004), no. 9, 703-708. MR 2065378 (2005c:47009)
  • 4. F. Bayart and S. Grivaux, Hypercyclicity and unimodular point spectrum, J. Funct. Anal., 226, (2005), no. 2, 281-300. MR 2159459 (2006i:47014)
  • 5. P. S. Bourdon and J. H. Shapiro, Spectral synthesis and common cyclic vectors, Michigan Math. J., 37, (1990), no. 1, 71-90. MR 1042515 (91m:47039)
  • 6. I. Chalendar and J. R. Partington, On the structure of invariant subspaces for isometric composition operators on $ H\sp 2(\mathbb{D})$ and $ H\sp 2(\mathbb{C}\sb +)$, Arch. Math. (Basel), 81, (2003), no. 2, 193-207. MR 2009562 (2004g:47030)
  • 7. G. Costakis and M. Sambarino, Genericity of wild holomorphic functions and common hypercyclic vectors, Adv. Math., 182, (2004), no. 2, 278-306. MR 2032030 (2004k:47009)
  • 8. C. C. Cowen and B. D. MacCluer, Composition Operators on Spaces of Analytic Functions, CRC Press, 1995. MR 1397026 (97i:47056)
  • 9. G. Godefroy and J. H. Shapiro, Operators with dense, invariant, cyclic vector manifolds, J. Funct. Anal., 98, (1991), no. 2, 229-269. MR 1111569 (92d:47029)
  • 10. K. G. Grosse-Erdmann, Universal families and hypercyclic operators, Bull. Amer. Math. Soc (NS), 36, (1999), 345-381. MR 1685272 (2000c:47001)
  • 11. 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. MR 2068180 (2005c:47010)
  • 12. J. E. Littlewood, On inequalities in the theory of functions, Proc. London Math. Soc., 23, (1925) 481-519.
  • 13. J.R. Partington, Linear operators and linear systems, London Mathematical Society Student Texts, 60. Cambridge University Press, Cambridge, 2004. MR 2158502 (2006d:93001)
  • 14. S. Rolewicz, On orbits of elements, Studia Math., 32, (1969), 17-22. MR 0241956 (39:3292)
  • 15. H. Salas, Supercyclicity and weighted shifts, Studia Math., 135, (1999), no. 1, 55-74. MR 1686371 (2000b:47020)
  • 16. J. H. Shapiro, Composition Operators and Classical Function Theory, Springer-Verlag, Berlin, 1993. MR 1237406 (94k:47049)
  • 17. K.H. Zhu, Operator theory in function spaces, Marcel Dekker, Inc., New York, 1990. MR 1074007 (92c:47031)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 47A16, 47B33, 47B37

Retrieve articles in all journals with MSC (2000): 47A16, 47B33, 47B37

Additional Information

Eva A. Gallardo-Gutierrez
Affiliation: Departamento de Matemáticas, Universidad de Zaragoza e IUMA, Plaza San Francisco s/n, 50009 Zaragoza, Spain

Jonathan R. Partington
Affiliation: School of Mathematics, University of Leeds, Leeds LS2 9JT, United Kingdom

Received by editor(s): August 15, 2006
Published electronically: September 25, 2007
Additional Notes: This work was partially supported by Plan Nacional I+D grant no. MTM2006-06431, Gobierno de Aragón research group Análisis Matemático y Aplicaciones, ref. DGA E-64 and a Scheme 4 grant from the London Mathematical Society
Communicated by: Joseph Ball
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society