Hostname: page-component-cb9f654ff-rkzlw Total loading time: 0 Render date: 2025-07-31T09:29:43.921Z Has data issue: false hasContentIssue false
  • English
  • Français

Disjoint Hypercyclicity and Weighted Translations on Discrete Groups

Published online by Cambridge University Press:  20 November 2018

Chung-Chuan Chen
Chung-Chuan Chen*
Affiliation:
Department of Mathematics Education, National Taichung University of Education, Taiwan e-mail: chungchuan@mail.ntcu.edu.tw
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let $1\le p<\infty $, and let $G$ be a discrete group. We give a sufficient and necessary condition for weighted translation operators on the Lebesgue space ${{\ell }^{p}}(G)$ to be densely disjoint hypercyclic. The characterization for the dual of a weighted translation to be densely disjoint hypercyclic is also obtained.

Keywords

47A1647B3843A15disjoint hypercyclicitytopological transitivityweighted translationℓp-space

Information

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 60 , Issue 4 , 01 December 2017 , pp. 712 - 720
Copyright
Copyright © Canadian Mathematical Society 2017

References

[1] Bayart, F. and E. Matheron, Dynamics of linear operators. Cambridge Tracts in Mathematics, 179, Cambridge University Press, Cambridge, 2009. http://dx.doi.org/10.1017/CBO9780511581113 Google Scholar
[2] Bernal-Gonzalez, L., Disjoint hypercydic operators. Studia Math. 182(2007), no. 2,113-131. http://dx.doi.org/10.4064/sm182-2-2 Google Scholar
[3] Bes, J. and O. Martin, Compositional disjoint hypercyclicity equals disjoint supercyclicity. Houston J. Math. 38(2012), no. 4, 1149–1163. Google Scholar
[4] Bes, J., O. Martin, and A. Peris, Disjoint hypercyclic linear fractional composition operators. J. Math. Anal. Appl. 381(2011), no. 2, 843–856. http://dx.doi.org/10.101 6/j.jmaa.2O11.03.072 Google Scholar
[5] Bes, J., O. Martin, A. Peris, and S. Shkarin, Disjoint mixing operators. J. Funct. Anal. 263(2012), no. 5, 1283–1322. http://dx.doi.org/10.1016/j.jfa.2012.05.018 Google Scholar
[6] Bes, J., O. Martin, and R. Sanders, Weighted shifts and disjoint hypercyclicity. J. Operator Theory 72(2014), no. 1, 15–40. http://dx.doi.org/10.7900/jot.2012aug20.1970 Google Scholar
[7] Bes, J. and A. Peris, Disjointness in hypercyclicity. J. Math. Anal. Appl. 336(2007), no. 1, 297–315. http://dx.doi.org/10.1016/j.jmaa.2007.02.043 Google Scholar
[8] Chen, C.-C., Chaotic weighted translations on groups. Arch. Math. 97(2011), no. 1, 61–68. http://dx.doi.org/10.1007/s00013–011-0262-1Google Scholar
[9] Chen, C.-C., Supercyclic and Cesdro hypercyclic weighted translations on groups. Taiwanese J. Math. 16(2012), no. 5, 1815–1827.Google Scholar
[10] Chen, C.-C., Hypercyclic weighted translations generated by non-torsion elements. Arch. Math. 101(2013), no. 2, 135–141. http://dx.doi.org/10.1007/s00013-013-0538-8 Google Scholar
[11] Chen, C.-C. and C.-H. Chu, Hypercyclicity of weighted convolution operators on homogeneous spaces. Proc. Amer. Math. Soc. 137(2009), no. 8, 2709–2718. http://dx.doi.org/10.1090/S0002-9939-09-09889-X Google Scholar
[12] Chen, C.-C., Hypercyclic weighted translations on groups. Proc. Amer. Math. Soc. 139(2011), no. 8, 2839–2846. http://dx.doi.org/10.1090/S0002-9939-2011-10718-4 Google Scholar
[13] Costakis, G. and M. Sambarino, Topologically mixing hypercyclic operators. Proc. Amer. Math. Soc. 132(2004), no. 2, 385–389. http://dx.doi.org/10.1090/S0002-9939-03-07016-3 Google Scholar
[14] Grosse-Erdmann, K.-G., Hypercyclic and chaotic weighted shifts. Studia Math. 139(2000), no. 1, 47–68.Google Scholar
[15] K.-G. Grosse-Erdmann and A. Peris, Linear chaos. Universitext, Springer, London, 2011. http://dx.doi.org/10.1007/978-1-4471-2170-1 Google Scholar
[16] Han, S.-A. and Y.-X. Liang, Disjoint hypercyclic weighted translations generated by aperiodic elements. Collect. Math. 67(2016), no. 3, 347-356. http://dx.doi.org/10.1007/s13348-015-0136-0 Google Scholar
[17] Leon-Saavedra, F., Operators with hypercyclic Cesdro means. Studia Math. 152(2002), no. 3, 201-215. http://dx.doi.org/10.4064/sm152-3-1 Google Scholar
[18] Martin, O., Disjoint hypercyclic and supercyclic composition operators. PhD thesis, Bowling Green State University, 2010.Google Scholar
[19] Rolewicz, S., On orbits of elements. Studia Math. 32(1969) 17–22. Google Scholar
[20] Salas, H., A hypercyclic operator whose adjoint is also hypercyclic. Proc. Amer. Math. Soc. 112(1991), no. 3, 765–770. http://dx.doi.org/10.1090/S0002-9939-1991-1049848-8 Google Scholar
[21] Salas, H., Hypercyclic weighted shifts. Trans. Amer. Math. Soc. 347(1995), no. 3, 993–1004. http://dx.doi.org/10.1090/S0002-9947-1995-1249890-6 Google Scholar
[22] Salas, H., Dual disjoint hypercyclic operators. J. Math. Anal. Appl. 374(2011), 106–117. http://dx.doi.org/10.101 6/j.jmaa.2O10.09.003 Google Scholar
[23] Salas, H., The strong disjoint blow-up/collapse property. J. Funct. Spaces Appl. 2013, Art. ID 146517, 6pp. Google Scholar
[24] Shkarin, S., A short proof of existence of disjoint hypercyclic operators. J. Math. Anal. Appl. 367(2010), no. 2, 713–715. http://dx.doi.org/10.101 6/j.jmaa.2O10.01.005 Google Scholar
[25] Shkarin, S. and R. Sanders, Existence of disjoint weekly mixing operators that fail to satisfy the disjoint hypercyclicity criterion. J. Math. Anal. Appl. 417(2014), 834–855. http://dx.doi.org/! 0.101 6/j.jmaa.2O14.03.063 Google Scholar