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FREQUENTLY HYPERCYCLIC BILATERAL SHIFTS

Part of: Special classes of linear operators General theory of linear operators

Published online by Cambridge University Press:  20 June 2018

KARL-G. GROSSE-ERDMANN Open the ORCID record for KARL-G. GROSSE-ERDMANN [Opens in a new window]
KARL-G. GROSSE-ERDMANN*
Affiliation:
Département de Mathématique, Institut Complexys, Université de Mons, 20 Place du Parc, 7000 Mons, Belgium e-mail: kg.grosse-erdmann@umons.ac.be
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Abstract

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It is not known whether the inverse of a frequently hypercyclic bilateral weighted shift on c0(ℤ) is again frequently hypercyclic. We show that the corresponding problem for upper frequent hypercyclicity has a positive answer. We characterise, more generally, when bilateral weighted shifts on Banach sequence spaces are (upper) frequently hypercyclic.

MSC classification

Primary: 47A16: Cyclic vectors, hypercyclic and chaotic operators
Secondary: 47B37: Operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 61 , Issue 2 , May 2019 , pp. 271 - 286
Copyright
Copyright © Glasgow Mathematical Journal Trust 2018 

References

REFERENCES

1. Bayart, F. and Grivaux, S., Frequently hypercyclic operators, Trans. Amer. Math. Soc. 358 (2006), 5083–5117.Google Scholar
2. Bayart, F. and Grivaux, S., Invariant Gaussian measures for operators on Banach spaces and linear dynamics, Proc. Lond. Math. Soc. (3) 94 (2007), 181–210.Google Scholar
3. Bayart, F. and Matheron, É., Dynamics of linear operators (Cambridge University Press, Cambridge, 2009).Google Scholar
4. Bayart, F. and Ruzsa, I. Z., Difference sets and frequently hypercyclic weighted shifts, Ergodic Theory Dyn. Syst. 35 (2015), 691–709.Google Scholar
5. Bellenot, S. F., Somewhat quasireflexive Banach spaces, Ark. Mat. 22 (1984), 175–183.Google Scholar
6. Bès, J., Menet, Q., A. Peris and Y. Puig, Recurrence properties of hypercyclic operators, Math. Ann. 366 (2016), 545–572.Google Scholar
7. Bonilla, A. and Grosse-Erdmann, K.-G., Upper frequent hypercyclicity and related notions, Rev. Mat. Complut. (to appear).Google Scholar
8. Grosse-Erdmann, K.-G., Holomorphe Monster und universelle Funktionen, Mitt. Math. Sem. Giessen 176 (1987).Google Scholar
9. Grosse-Erdmann, K.-G. and Manguillot, A. Peris, Linear chaos (Springer, London, 2011).Google Scholar
10. Guirao, A. J., Montesinos, V. and Zizler, V., Open problems in the geometry and analysis of Banach spaces (Springer, Cham, 2016).Google Scholar
11. Herzog, G., On zero-free universal entire functions, Arch. Math. (Basel) 63 (1994), 329–332.Google Scholar
12. Madore, B. F. and Martínez-Avendaño, R. A., Subspace hypercyclicity, J. Math. Anal. Appl. 373 (2011), 502–511.Google Scholar
13. Singer, I., Bases in Banach spaces. I (Springer, New York/Berlin, 1970).Google Scholar