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Li–Yorke chaos for composition operators on Orlicz spaces

  • Yousef Estaremi EMAIL logo and Shah Muhammad
Published/Copyright: May 15, 2024

Abstract

In this paper we characterize Li–Yorke chaotic composition operators on Orlicz spaces. Indeed, some necessary and sufficient conditions are provided for Li–Yorke chaotic composition operator C φ on the Orlicz space L Φ ( μ ) . In some cases we have equivalent conditions for composition operators on Orlicz spaces to be Li–Yorke chaotic. The results of this paper extend similar results in L p -spaces.

MSC 2020: 47A16; 47B33

Communicated by Siegfried Echterhoff


Funding statement: This research is funded by “Researchers Supporting Project number (RSPD2024R733), King Saud University, Riyadh, Saudi Arabia”.

References

[1] F. Bayart, U. B. Darji and B. Pires, Topological transitivity and mixing of composition operators, J. Math. Anal. Appl. 465 (2018), no. 1, 125–139. 10.1016/j.jmaa.2018.04.063Search in Google Scholar

[2] F. Bayart and E. Matheron, Dynamics of Linear Operators, Cambridge Tracts in Math. 179, Cambridge University, Cambridge, 2009. 10.1017/CBO9780511581113Search in Google Scholar

[3] T. Bermúdez, A. Bonilla, F. Martínez-Giménez and A. Peris, Li–Yorke and distributionally chaotic operators, J. Math. Anal. Appl. 373 (2011), no. 1, 83–93. 10.1016/j.jmaa.2010.06.011Search in Google Scholar

[4] N. C. Bernardes, Jr., A. Bonilla, V. Müller and A. Peris, Distributional chaos for linear operators, J. Funct. Anal. 265 (2013), no. 9, 2143–2163. 10.1016/j.jfa.2013.06.019Search in Google Scholar

[5] N. C. Bernardes, Jr., A. Bonilla, V. Müller and A. Peris, Li–Yorke chaos in linear dynamics, Ergodic Theory Dynam. Systems 35 (2015), no. 6, 1723–1745. 10.1017/etds.2014.20Search in Google Scholar

[6] N. C. Bernardes, Jr., P. R. Cirilo, U. B. Darji, A. Messaoudi and E. R. Pujals, Expansivity and shadowing in linear dynamics, J. Math. Anal. Appl. 461 (2018), no. 1, 796–816. 10.1016/j.jmaa.2017.11.059Search in Google Scholar

[7] N. C. Bernardes, Jr., U. B. Darji and B. Pires, Li–Yorke chaos for composition operators on L p -spaces, Monatsh. Math. 191 (2020), no. 1, 13–35. 10.1007/s00605-019-01341-2Search in Google Scholar

[8] N. C. Bernardes, Jr. and A. Messaoudi, Shadowing and structural stability for operators, Ergodic Theory Dynam. Systems 41 (2021), no. 4, 961–980. 10.1017/etds.2019.107Search in Google Scholar

[9] J. Bès, Q. Menet, A. Peris and Y. Puig, Recurrence properties of hypercyclic operators, Math. Ann. 366 (2016), no. 1–2, 545–572. 10.1007/s00208-015-1336-3Search in Google Scholar

[10] Y. Cui, H. Hudzik, R. Kumar and L. Maligranda, Composition operators in Orlicz spaces, J. Aust. Math. Soc. 76 (2004), no. 2, 189–206. 10.1017/S1446788700008892Search in Google Scholar

[11] S. Grivaux, E. Matheron and Q. Menet, Linear dynamical systems on Hilbert spaces: Typical properties and explicit examples, Mem. Amer. Math. Soc. 269 (2021), no. 1315, 1–147. 10.1090/memo/1315Search in Google Scholar

[12] K.-G. Grosse-Erdmann and A. Peris Manguillot, Linear Chaos, Universitext, Springer, London, 2011. 10.1007/978-1-4471-2170-1Search in Google Scholar

[13] M. A. Krasnosel’skii and Y. B. Rutickii, Convex Functions and Orlicz Spaces, Noordhoff, Netherlands, 1961. Search in Google Scholar

[14] A. Kufner, O. John and S. Fučík, Function Spaces, Noordhoff International, Leiden, 1977. Search in Google Scholar

[15] T. Y. Li and J. A. Yorke, Period three implies chaos, Amer. Math. Monthly 82 (1975), no. 10, 985–992. 10.1080/00029890.1975.11994008Search in Google Scholar

[16] Q. Menet, Linear chaos and frequent hypercyclicity, Trans. Amer. Math. Soc. 369 (2017), no. 7, 4977–4994. 10.1090/tran/6808Search in Google Scholar

[17] M. M. Rao and Z. D. Ren, Theory of Orlicz Spaces, Monogr. Textb. Pure Appl. Math. 146, Marcel Dekker, New York, 1991. Search in Google Scholar

Received: 2022-12-13
Revised: 2024-04-20
Published Online: 2024-05-15
Published in Print: 2024-07-01

© 2024 Walter de Gruyter GmbH, Berlin/Boston

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