Skip to main content Accesibility Help
×
×
Home

Epsilon-hypercyclic operators

  • CATALIN BADEA (a1), SOPHIE GRIVAUX (a1) and VLADIMIR MÜLLER (a2)
Abstract

Let X be a separable infinite-dimensional Banach space, and T a bounded linear operator on X; T is hypercyclic if there is a vector x in X with dense orbit under the action of T. For a fixed ε∈(0,1), we say that T is ε-hypercyclic if there exists a vector x in X such that for every non-zero vector yX there exists an integer n with . The main result of this paper is a construction of a bounded linear operator T on the Banach space 1 which is ε-hypercyclic without being hypercyclic. This answers a question from V. Müller [Three problems, Mini-Workshop: Hypercyclicity and linear chaos, organized by T. Bermudez, G. Godefroy, K.-G. Grosse-Erdmann and A. Peris. Oberwolfach Rep.3 (2006), 2227–2276].

Copyright
References
Hide All
[1]Bayart, F. and Grivaux, S.. Frequently hypercylic operators. Trans. Amer. Math. Soc. 358 (2006), 50835117.
[2]Bayart, F. and Matheron, É.. Dynamics of Linear Operators (Cambridge Tracts in Mathematics, 179). Cambridge University Press, Cambridge, 2009.
[3]Bayart, F. and Matheron, É.. (Non)-weakly mixing operators and hypercyclicity sets. Ann. Inst. Fourier to appear.
[4]Bourdon, P. and Feldman, N.. Somewhere dense orbits are everywhere dense. Indiana Univ. Math. J. 52 (2003), 811819.
[5]Chan, K. and Sanders, R.. A weakly hypercyclic operator that is not norm hypercyclic. J. Operator Theory 52 (2004), 3959.
[6]Costakis, G.. On a conjecture of D. Herrero concerning hypercyclic operators. C. R. Math. Acad. Sci. Paris 330 (2000), 179182.
[7]Feldman, N.. Perturbations of hypercyclic vectors. J. Math. Anal. Appl. 273 (2002), 6774.
[8]Grosse-Erdmann, K.-G. and Peris, A.. Frequently dense orbits. C. R. Math. Acad. Sci. Paris 341 (2005), 123128.
[9]Müller, V.. Three problems, Mini-Workshop: Hypercyclicity and linear chaos, organized by T. Bermudez, G. Godefroy, K.-G. Grosse-Erdmann and A. Peris. Oberwolfach Rep. 3 (2006), 2227–2276.
[10]Peris, A.. Multi-hypercyclic operators are hypercyclic. Math. Z. 236 (2001), 779786.
[11]Rolewicz, S.. On orbits of elements. Studia Math. 32 (1969), 1722.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
  • URL: /core/journals/ergodic-theory-and-dynamical-systems
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed