Frequently supercyclic operators and frequently supercyclic C0-semigroups
DOI:
https://doi.org/10.5269/bspm.67430Abstract
In this paper, the concept of frequent supercyclicity for operators and for C0-semigroups is defined. It is proved that if an operator T is frequently supercyclic, then T^n and {\lambda}T are frequently supercyclic for any natural number n and any non-zero scalar {\lambda}. Also, it is established that frequent supercyclicity of a C0-semigroup implies frequent supercyclicity of any of their operators. Moreover, by using discretization and autonomous discretization of a C0-semigroup, some equivalent conditions for frequent supercyclicity are stated.
References
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2. Bayart, F., Matheron, E., Dynamics of Linear Operators, Cambridge, Cambridge University Press, 2009.
3. Bayart, F., Grivaux, S., Frequently hypercyclic operators, Trans. Amer. Math. Soc. 358, 5083-5117, (2006).
4. Bermudez, T., Bonilla, A., Conejero, J. A., Peris, A., Hypercyclic, Topologically Mixing and chaotic semigroups on Banach spaces, Studia Math. 131, 57-75, (2005).
5. Bermudez, T., Bonilla, A., Martinon, A., On the Existence of Chaotic and Hypercyclic Semigroups on Banach Spaces, Proc. Amer. Math. Soc. 170, 2435-2441, (2003).
6. Bernal-Gonzalez, L., Grosse-Erdmann, K. G., Existence and Nonexistence of Hypercyclic Semigroups, Proc. Amer. Math. Soc. 135, 755-766, (2007).
7. Bonilla, A., Grosse-Erdmann, K. G., Frequently hypercyclic operators and vectors, Ergod. Theory Dyn. Syst. 27, 383-404, (2007).
8. Chaouch, B., Kosti, M., Pilipovi, S., Velinov, D., f-Frequently hypercyclic C0-semigroups on complex sectors, Banach J. Math. Anal. 14, 1080-1110, (2020).
9. Conejero, J. A., Muller, V., Peris, A., Hypercyclic behaviour of operators in a hypercyclic C0-semigroup, J. Funct. Anal. 244, 342-348, (2007).
10. Desh, W., Schappacher, W., Webb, G. F., Hypercyclic and chaotic semigroups of linear operators, Ergod. Theory Dyn. Syst. 17, 793–819, (1997).
11. Grosse-Erdmann, K. G., Peris Manguillot, A., Linear Chaos. London: Springer-Verlag, 2011.
12. Grosse-Erdmann, K. G., Peris, A., Frequently dense orbits, C. R. Acad. Sci. Paris 341, 123-128, (2005).
13. Grosse-Erdmann, K. G., Recent developments in hypercyclicity, RACSAM. Rev. R. Acad. Cien. Serie. A Mat. 97, 273-286, (2003).
14. Guo, Z., Liu, L., Shu, Y., Frequent hypercyclicity and chaoticity of Toeplitz operators and their tensor products, Proc. Indian Acad. Sci. 131, 49, (2021).
15. Kumar, A., Srivastava, S., Supercyclicity criteria for C0-semigroups, Adv. Oper. Theory 5, 1646-1666, (2020).
16. Kumar, A., Srivastava, S., Supercyclic C0-semigroups, stability and somewhere dense orbits, J. Math. Anal. Appl. 476, 539-548, (2019).
17. Mangino, E., Peris, A., Frequently hypercyclic semigroups, Studia Math. 202, 227-242, (2011).
18. Mangino, E. M., Murillo-Arcila, M., Frequently hypercyclic translation semigroups, Studia Math. 227, 219-238 (2015).
19. Moosapoor, M., Supermixing and hypermixing of strongly continuous semigroups and their direct sum, J. Taibah Univ. Sci. 15, 953-959, (2021).
20. Moosapoor, M., Shahriari, M., About subspace-frequently hypercyclic operators, Sahand Commun. Math. Anal. 17, 107-116 (2020).
21. Shkarin, S., On supercyclicity of operators from a supercyclic semigroup, J. Math. Anal. Appl. 382, 516-522, (2011).
22. Shkarin, S., On the spectrum of frequently hypercyclic operators, Proc. Amer. Math. Soc. 137, 123-134, (2009).
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2025-02-14
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