Translation operators on weighted spaces of entire functions

Tien, Pham

doi:10.1090/proc/13254

Translation operators on weighted spaces of entire functions
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Proc. Amer. Math. Soc. 145 (2017), 805-815 Request permission

Abstract:

We study the dynamical properties of translation operators on both weighted Hilbert and Banach spaces of entire functions. We show that the translation operator on these weighted spaces is always mixing when it is continuous and give necessary and sufficient conditions in terms of weights for the chaos of this operator. We also prove that translation operators can arise as compact perturbations of the identity on weighted Banach spaces.

References

Alexander V. Abanin and Pham Trong Tien, The differentiation and integration operators on weighted Banach spaces of holomorphic functions. arXiv: 1505.04350v2 (2016).
Frédéric Bayart and Étienne Matheron, Dynamics of linear operators, Cambridge Tracts in Mathematics, vol. 179, Cambridge University Press, Cambridge, 2009. MR 2533318, DOI 10.1017/CBO9780511581113
María J. Beltrán, Dynamics of differentiation and integration operators on weighted spaces of entire functions, Studia Math. 221 (2014), no. 1, 35–60. MR 3194061, DOI 10.4064/sm221-1-3
María J. Beltrán, José Bonet, and Carmen Fernández, Classical operators on weighted Banach spaces of entire functions, Proc. Amer. Math. Soc. 141 (2013), no. 12, 4293–4303. MR 3105871, DOI 10.1090/S0002-9939-2013-11685-0
Klaus D. Bierstedt, José Bonet, and Jari Taskinen, Associated weights and spaces of holomorphic functions, Studia Math. 127 (1998), no. 2, 137–168. MR 1488148, DOI 10.4064/sm-127-2-137-168
G. D. Birkhoff, Démonstration d’un théoreme elementaire sur les fonctions entiéres, C. R. Acad. Sci. Paris 189 (1929), 473–475.
José Bonet, Hypercyclic and chaotic convolution operators, J. London Math. Soc. (2) 62 (2000), no. 1, 253–262. MR 1772185, DOI 10.1112/S0024610700001174
José Bonet, Dynamics of the differentiation operator on weighted spaces of entire functions, Math. Z. 261 (2009), no. 3, 649–657. MR 2471093, DOI 10.1007/s00209-008-0347-0
José Bonet and Antonio Bonilla, Chaos of the differentiation operator on weighted Banach spaces of entire functions, Complex Anal. Oper. Theory 7 (2013), no. 1, 33–42. MR 3010787, DOI 10.1007/s11785-011-0134-5
Kit C. Chan and Joel H. Shapiro, The cyclic behavior of translation operators on Hilbert spaces of entire functions, Indiana Univ. Math. J. 40 (1991), no. 4, 1421–1449. MR 1142722, DOI 10.1512/iumj.1991.40.40064
Robert L. Devaney, An introduction to chaotic dynamical systems, 2nd ed., Addison-Wesley Studies in Nonlinearity, Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, CA, 1989. MR 1046376
S. M. Duyos-Ruiz, On the existence of universal functions, Soviet Math. Dokl. 27 (1983), 9–13.
Gilles Godefroy and Joel H. Shapiro, Operators with dense, invariant, cyclic vector manifolds, J. Funct. Anal. 98 (1991), no. 2, 229–269. MR 1111569, DOI 10.1016/0022-1236(91)90078-J
Karl-G. Grosse-Erdmann and Alfredo Peris Manguillot, Linear chaos, Universitext, Springer, London, 2011. MR 2919812, DOI 10.1007/978-1-4471-2170-1
Joel H. Shapiro, Composition operators and classical function theory, Universitext: Tracts in Mathematics, Springer-Verlag, New York, 1993. MR 1237406, DOI 10.1007/978-1-4612-0887-7