Abstract and Applied Analysis

An Extension of Hypercyclicity for N -Linear Operators

Juan Bès and J. Alberto Conejero

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Grosse-Erdmann and Kim recently introduced the notion of bihypercyclicity for studying the existence of dense orbits under bilinear operators. We propose an alternative notion of orbit for N -linear operators that is inspired by difference equations. Under this new notion, every separable infinite dimensional Fréchet space supports supercyclic N -linear operators, for each N 2 . Indeed, the nonnormable spaces of entire functions and the countable product of lines support N -linear operators with residual sets of hypercyclic vectors, for N = 2 .

Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2014), Article ID 609873, 11 pages.

Dates
First available in Project Euclid: 2 October 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1412278776

Digital Object Identifier
doi:10.1155/2014/609873

Mathematical Reviews number (MathSciNet)
MR3212435

Zentralblatt MATH identifier
07022716

Citation

Bès, Juan; Conejero, J. Alberto. An Extension of Hypercyclicity for $N$ -Linear Operators. Abstr. Appl. Anal. 2014, Special Issue (2014), Article ID 609873, 11 pages. doi:10.1155/2014/609873. https://projecteuclid.org/euclid.aaa/1412278776


Export citation

References

  • G. D. Birkhoff, “Demonstration d'un theoreme elementaire sur les fonctions entieres,” Comptes Rendus de l'Académie des Sciences, vol. 189, no. 2, pp. 473–475, 1929.
  • G. R. MacLane, “Sequences of derivatives and normal families,” Journal d'Analyse Mathématique, vol. 2, pp. 72–87, 1952.
  • S. Rolewicz, “On orbits of elements,” Polska Akademia Nauk. Instytut Matematyczny. Studia Mathematica, vol. 32, pp. 17–22, 1969.
  • C. Kitai, Invariant closed sets for linear operators [Ph.D. thesis], University of Toronto, 1982.
  • B. Beauzamy, “An operator on a separable Hilbert space with many hypercyclic vectors,” Polska Akademia Nauk. Instytut Matematyczny. Studia Mathematica, vol. 87, no. 1, pp. 71–78, 1987.
  • R. M. Gethner and J. H. Shapiro, “Universal vectors for operators on spaces of holomorphic functions,” Proceedings of the American Mathematical Society, vol. 100, no. 2, pp. 281–288, 1987.
  • G. Godefroy and J. H. Shapiro, “Operators with dense, invariant, cyclic vector manifolds,” Journal of Functional Analysis, vol. 98, no. 2, pp. 229–269, 1991.
  • K.-G. Grosse-Erdmann, “Universal families and hypercyclic operators,” Bulletin of the American Mathematical Society, vol. 36, no. 3, pp. 345–381, 1999.
  • K.-G. Grosse-Erdmann, “Recent developments in hypercyclicity,” Revista de la Real Academia de Ciencias Exactas, Físicas Y Naturales A: Matemáticas, vol. 97, no. 2, pp. 273–286, 2003.
  • J. Bonet, F. Martínez-Giménez, and A. Peris, “Linear chaos on Fréchet spaces,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 13, no. 7, pp. 1649–1655, 2003.
  • F. Bayart and E. Matheron, Dynamics of linear operators, vol. 179 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, UK, 1 edition, 2009.
  • K.-G.s Grosse-Erdmann and A. Peris-Manguillot, Linear chaos, Universitext, Springer, London, UK, 2011.
  • R. G. Douglas, H. S. Shapiro, and A. L. Shields, “Cyclic vectors and invariant subspaces for the backward shift operator,” Annales de l'Institut Fourier, vol. 20, no. 1, pp. 37–76, 1970.
  • H. M. Hilden and L. J. Wallen, “Some cyclic and non-cyclic vectors of certain operators,” Indiana University Mathematics Journal, vol. 23, pp. 557–565, 1973/74.
  • D. A. Herrero, “Hypercyclic operators and chaos,” Journal of Operator Theory, vol. 28, no. 1, pp. 93–103, 1992.
  • J. Bès and A. Peris, “Hereditarily hypercyclic operators,” Journal of Functional Analysis, vol. 167, no. 1, pp. 94–112, 1999.
  • F. Bayart and S. Grivaux, “Frequently hypercyclic operators,” Transactions of the American Mathematical Society, vol. 358, no. 11, pp. 5083–5117, 2006.
  • L. Bernal-González, “Disjoint hypercyclic operators,” Studia Mathematica, vol. 182, no. 2, pp. 113–131, 2007.
  • J. Bès and A. Peris, “Disjointness in hypercyclicity,” Journal of Mathematical Analysis and Applications, vol. 336, no. 1, pp. 297–315, 2007.
  • F. Martínez-Giménez, P. Oprocha, and A. Peris, “Distributional chaos for backward shifts,” Journal of Mathematical Analysis and Applications, vol. 351, no. 2, pp. 607–615, 2009.
  • S. Bartoll, F. Martínez-Giménez, and A. Peris, “The specification property for backward shifts,” Journal of Difference Equations and Applications, vol. 18, no. 4, pp. 599–605, 2012.
  • G. T. Prǎjiturǎ, “Irregular vectors of Hilbert space operators,” Journal of Mathematical Analysis and Applications, vol. 354, no. 2, pp. 689–697, 2009.
  • T. Bermúdez, A. Bonilla, F. Martínez-Giménez, and A. Peris, “Li-Yorke and distributionally chaotic operators,” Journal of Mathematical Analysis and Applications, vol. 373, no. 1, pp. 83–93, 2011.
  • K.-G. Grosse-Erdmann and S. G. Kim, “Bihypercyclic bilinear mappings,” Journal of Mathematical Analysis and Applications, vol. 399, no. 2, pp. 701–708, 2013.
  • J. Bonet and A. Peris, “Hypercyclic operators on non-normable Fréchet spaces,” Journal of Functional Analysis, vol. 159, no. 2, pp. 587–595, 1998.
  • R. I. Ovsepian and A. Pełczyński, “On the existence of a fundamental total and bounded biorthogonal sequence in every separable Banach space, and related constructions of uniformly bounded orthonormal systems in L$^{2}$ ,” Polska Akademia Nauk. Instytut Matematyczny. Studia Mathematica, vol. 54, no. 2, pp. 149–159, 1975.
  • G. Herzog, “On linear operators having supercyclic vectors,” Studia Mathematica, vol. 103, no. 3, pp. 295–298, 1992.
  • H. N. Salas, “Supercyclicity and weighted shifts,” Studia Mathematica, vol. 135, no. 1, pp. 55–74, 1999.
  • T. Bermúdez, A. Bonilla, and A. Peris, “On hypercyclicity and supercyclicity criteria,” Bulletin of the Australian Mathematical Society, vol. 70, no. 1, pp. 45–54, 2004.
  • K.-G. Grosse-Erdmann, Holomorphe Monster und Universelle Funktionen, vol. 176 of Mitteilungen aus dem Mathematischen Seminar Giessen, University of Trier, Trier, Germany, 1987.
  • J. Bès and J. A. Conejero, “Hypercyclic subspaces in omega,” Journal of Mathematical Analysis and Applications, vol. 316, no. 1, pp. 16–23, 2006.
  • H. Petersson, “Hypercyclicity in omega,” Proceedings of the American Mathematical Society, vol. 135, no. 4, pp. 1145–1149, 2007.
  • H. N. Salas, “Eigenvalues and hypercyclicity in omega,” Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matematicas. RACSAM, vol. 105, no. 2, pp. 379–388, 2011.
  • R. Aron and D. Markose, “On universal functions,” Journal of the Korean Mathematical Society, vol. 41, no. 1, pp. 65–76, 2004.
  • R. L. Devaney, An Introduction to Chaotic Dynamical Systems, Addison-Wesley, Redwood City, Calif, USA, 2nd edition, 1989. \endinput