Abstract
We consider generalized convolution operators generated by operators of Gel’fond-Leont’ev generalized differentiation. In this paper, we prove that any such operator not coinciding with the operator of multiplication by a constant is hypercyclic and chaotic on the space of all entire functions. This result generalizes earlier results belonging to the classical convolution operators as well as to the generalized convolution operators constructed by using Dunkl operators.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. O. Gel’fond and A. F. Leont’ev, “On a generalization of Fourier series. Mat. Sb. 29(3), 477–500 (1951).
A. F. Leont’ev, Generalization of Series of Exponentials (Nauka, Moscow, 1981) [in Russian].
B. Beauzamy, “Un opérateur, sur l’espace de Hilbert, dont tous les polynômes sont hypercycliques,” C. R. Acad. Sci. Paris Ser. I Math. 303(18), 923–925 (1986).
K.-G. Grosse-Erdmann, “Universal families and hypercyclic operators,” Bull. Amer. Math. Soc. (N. S.) 36(3), 345–381 (1999).
R. L. Devaney, An Introduction to Chaotic Dynamical Systems, in Addison-Wesley Stud. Nonlinearity (Addison-Wesley Publ., Redwood City, CA, 1989).
G. D. Birkhoff, “Démonstration d’un théorèmeélémentaire sur les fonctions entières,” C. R. Acad. Sci. Paris 189, 473–475 (1929).
G. R. MacLane, “Sequences of derivatives and normal families,” J. Analyse Math. 2(1), 72–87 (1952).
G. Godefroy and J. H. Shapiro, “Operators with dense, invariant, cyclic vector manifolds,” J. Funct. Anal. 98(2), 229–269 (1991).
J. J. Betancor, M. Sifi, and K. Triméche, “Hypercyclic and chaotic convolution operators associated with the Dunkl operators on ℂ,” Acta Math. Hungar. 106(1–2), 101–116 (2005).
C. F. Dunkl, “Differential-difference operators associated with reflections groups,” Trans. Amer. Math. Soc. 311(1), 167–183 (1989).
M. Rösler, “Dunkl operators: theory and applications,” in Lecture Notes in Math., Vol. 1817: Orthogonal Polynomials and Special Functions (Leuven, 2002) (Springer-Verlag, Berlin, 2003), pp. 93–135.
A. F. Leont’ev, Entire Functions. Series of Exponentials (Nauka, Moscow, 1983) [in Russian].
R. M. Gethner and J. H. Shapiro, “Universal vectors for operators on spaces of holomorphic functions,” Proc. Amer. Math. Soc. 100(2), 281–288 (1987).
J. Banks, J. Brooks, G. Cairns, G. Davis, and P. Stacey, “On Devaney’s definition of chaos,” Amer. Math. Monthly 99(4), 332–334 (1992).
R. M. Kronover, Fractals and Chaos in Dynamical Systems: Fundamentals of the Theory (Postmarket, Moscow, 2000) [in Russian].
J. J. Betancor, J. D. Betancor, and J. M. Méndez, “Hypercyclic and chaotic convolution operators in Chébli-Trimèche hypergroups,” Rocky Mountain J. Math. 34(4), 1207–1237 (2004).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © V. É. Kim, 2009, published in Matematicheskie Zametki, 2009, Vol. 85, No. 6, pp. 849–856.
Rights and permissions
About this article
Cite this article
Kim, V.É. Hypercyclicity and chaotic character of generalized convolution operators generated by Gel’fond-Leont’ev operators. Math Notes 85, 807–813 (2009). https://doi.org/10.1134/S000143460905023X
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S000143460905023X