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Common hypercyclic functions for multiples of convolution and non-convolution operators


Author: Luis Bernal-González
Journal: Proc. Amer. Math. Soc. 137 (2009), 3787-3795
MSC (2000): Primary 47A16; Secondary 30E10, 47B33
DOI: https://doi.org/10.1090/S0002-9939-09-09943-2
Published electronically: June 5, 2009
MathSciNet review: 2529888
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Abstract: We prove the existence of a residual set of entire functions, all of whose members are hypercyclic for every non-zero scalar multiple of $ T$, where $ T$ is the differential operator associated to an entire function of order less than $ 1/2$. The same result holds if $ T$ is a finite-order linear differential operator with non-constant coefficients.


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  • 1. E. Abakumov and J. Gordon, Common hypercyclic vectors for multiples of backward shift, J. Funct. Anal. 200 (2003), 494-504. MR 1979020 (2004g:47012)
  • 2. L.V. Ahlfors, Complex analysis, 3rd ed., MacGraw-Hill, London, 1978. MR 510197 (80c:30001)
  • 3. S.I. Ansari, Hypercyclic and cyclic vectors, J. Funct. Anal. 128 (1995), 374-383. MR 1319961 (96h:47002)
  • 4. R. Aron, J. Bès, F. León and A. Peris, Operators with common hypercyclic subspaces, J. Operator Theory 54 (2005), 251-260. MR 2186352 (2006h:47010)
  • 5. F. Bayart, Common hypercyclic vectors for composition operators, J. Operator Theory 52 (2004), 353-370. MR 2119275 (2006a:47014)
  • 6. F. Bayart, Common hypercyclic subspaces, Integr. Equ. Oper. Theory 53 (2005), 467-476. MR 2187432 (2006k:47016)
  • 7. F. Bayart, Topological and algebraic genericity of divergence and universality, Studia Math. 167 (2005), 161-181. MR 2134382 (2006b:46024)
  • 8. F. Bayart and É. Matheron, How to get common universal vectors, Indiana Univ. Math. J. 56 (2007), 553-580. MR 2317538 (2008f:47008)
  • 9. C.A. Berenstein and R. Gay, Complex analysis and special topics in harmonic analysis, Springer, New York, 1995. MR 1344448 (96j:30001)
  • 10. G.D. Birkhoff, Démonstration d'un théorème élémentaire sur les fonctions entières, C. R. Acad. Sci. Paris 189 (1929), 473-475.
  • 11. R.P. Boas, Entire functions, Academic Press, New York, 1954. MR 0068627 (16:914f)
  • 12. K. Chan and R. Sanders, Common supercyclic vectors for a path of operators, J. Math. Anal. Appl. 337 (2008), 646-658. MR 2356099 (2008k:47019)
  • 13. K. Chan and R. Sanders, Two criteria for a path of operators to have common hypercyclic vectors, J. Operator Theory, 61 (2009), 191-223.
  • 14. G. Costakis, Common Cesàro hypercyclic vectors, preprint.
  • 15. G. Costakis and P. Mavroudis, Common hypercyclic entire functions for multiples of differential operators, Colloq. Math. 111 (2008), 199-203. MR 2365797 (2008m:47009)
  • 16. G. Costakis and M. Sambarino, Genericity of wild holomorphic functions and common hypercyclic vectors, Adv. Math. 182 (2004), 278-306. MR 2032030 (2004k:47009)
  • 17. J. Delsarte and J.L. Lions, Transmutations d'opérateurs différentiels dans le domaine complexe, Comment. Math. Helv. 32 (1957), 113-128. MR 0091386 (19:959c)
  • 18. L. Ehrenpreis, Mean periodic functions. I, Amer. J. Math. 77 (1955), 293-328. MR 0070047 (16:1122d)
  • 19. D. Gaier, Lectures on complex approximation, Birkhäuser, Boston, 1987. MR 894920 (88i:30059b)
  • 20. E.A. Gallardo-Gutiérrez and J.R. Partington, Common hypercyclic vectors for families of operators, Proc. Amer. Math. Soc. 136 (2008), 119-126. MR 2350396 (2008i:47021)
  • 21. G. Godefroy and J.H. Shapiro, Operators with dense, invariant, cyclic vector manifolds, J. Funct. Anal. 98 (1991), 229-269. MR 1111569 (92d:47029)
  • 22. K.-G. Grosse-Erdmann, Universal families and hypercyclic operators, Bull. Amer. Math. Soc. (N.S.) 36 (1999), 345-381. MR 1685272 (2000c:47001)
  • 23. K.-G. Grosse-Erdmann, Recent developments in hypercyclicity, RACSAM Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. 97 (2003), 273-286. MR 2068180 (2005c:47010)
  • 24. F. León-Saavedra and V. Müller, Rotations of hypercyclic and supercyclic operators, Integr. Equ. Oper. Theory 50 (2004), 385-391. MR 2104261 (2005g:47009)
  • 25. G.R. MacLane, Sequences of derivatives and normal families, J. Anal. Math. 2 (1952), 72-87. MR 0053231 (14:741d)
  • 26. B. Malgrange, Existence et approximation des solutions des équations aux dérivées partielles et des équations de convolution, Ann. Inst. Fourier (Grenoble) 6 (1955/1956), 271-355. MR 0086990 (19:280a)
  • 27. A. Peris, Common hypercyclic vectors for backward shifts, Operator theory seminar, Michigan State University, 2000-2001.
  • 28. J. Wengenroth, Hypercyclic operators on non-locally convex spaces, Proc. Amer. Math. Soc. 131 (2003), 1759-1761. MR 1955262 (2003j:47007)

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Additional Information

Luis Bernal-González
Affiliation: Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Sevilla, Apdo. 1160, Avda. Reina Mercedes, Sevilla-41080, Spain
Email: lbernal@us.es

DOI: https://doi.org/10.1090/S0002-9939-09-09943-2
Keywords: Hypercyclic operators, common hypercyclic vectors, entire functions, linear differential operators, Borel transform
Received by editor(s): July 7, 2008
Received by editor(s) in revised form: February 23, 2009
Published electronically: June 5, 2009
Additional Notes: The author has been partially supported by the Plan Andaluz de Investigación de la Junta de Andalucía FQM-127, by MEC Grant MTM2006-13997-C02-01 and by MEC Acción Especial MTM2006-26627-E
Dedicated: Dedicated to the memory of Professor Antonio Aizpuru, who died in March 2008
Communicated by: Nigel J. Kalton
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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