Hostname: page-component-745bb68f8f-hvd4g Total loading time: 0 Render date: 2025-01-23T14:35:23.605Z Has data issue: false hasContentIssue false
  • English
  • Français

On hypercyclicity and supercyclicity criteria

Published online by Cambridge University Press:  17 April 2009

Teresa Bermúdez ,
Antonio Bonilla  and
Alfredo Peris
Teresa Bermúdez
Affiliation:
Departamento de Análisis Matemático, Universidad de La Laguna, 38271 La Laguna (Tenerife), Spain, e-mail: [email protected]
Antonio Bonilla
Affiliation:
Departamento de Análisis Matemático, Universidad de La Laguna, 38271 La Laguna (Tenerife), Spain, e-mail: [email protected]
Alfredo Peris
Affiliation:
E.T.S. Arquitectura, Departament de Matemàtica Aplicada, Universitat Politècnica de València, E-46022 València, Spain, e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We show that the Hypercyclicity Criterion coincides with other existing hypercyclicity criteria and prove that a wide class of hypercyclic operators satisfy the Criterion. The results obtained extend or improve earlier work of several authors. We also unify the different versions of the Supercyclicity Criterion and show that operators with dense generalised kernel and dense range are supercyclic.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 70 , Issue 1 , August 2004 , pp. 45 - 54
Copyright
Copyright © Australian Mathematical Society 2004

References

[1]Bermúdez, T. and Miller, V., ‘On operators T such that f(T) is hypercyclic’, Integral Equation Operator Theory 37 (2000), 332–340.CrossRefGoogle Scholar
[2]Bernal, L., ‘Hypercyclic sequences of differential and antidifferential operators’, J. Approx. Theory 96 (2000), 323–337.CrossRefGoogle Scholar
[3]Bernal, L., ‘Densely hereditarily hypercyclic sequences and large hypercyclic manifolds’, Proc. Amer. Math. Soc. 127 (1999), 3279–3285.CrossRefGoogle Scholar
[4]Bernel, L., Prado-Tendero, J.A., ‘Sequences of differential operators: exponentials, hypercyclicity and equicontinuity’, Ann. Polon. Math. 77 (2001), 169–187.Google Scholar
[5]Bernel, L. and Grosse-Erdmann, K.G., ‘The hypercyclicity criterion for sequences of operators’, Studia Math. 157 (2003), 17–32.Google Scholar
[6]Bès, J., Three problems on hypercyclic operators, Ph.D. thesis (Kent State University, 1998).Google Scholar
[7]Bès, J. and Peris, A., ‘Hereditarily hypercyclic operators’, J. Funct. Anal. 167 (1999), 94–112.CrossRefGoogle Scholar
[8]Birkhoff, G.D., ‘Démonstration d'un théorème élémentaire sur les fonctions entières’, C. R. Acad. Sci. Paris 189 (1929), 473–475.Google Scholar
[9]Bonet, J., Domański, P. and Lindström, M., ‘Pointwise multiplication operators on weighted Banach spaces of analytic functions’, Studia Math. 137 (1999), 177–194.Google Scholar
[10]Devaney, R.L., An introduction to chaotic dynamical systems, (2nd ed.) (Addison-Wesley, Reading, MA, 1989).Google Scholar
[11]Feldman, N., Miller, V. and Miller, L., ‘Hypercyclic and supercyclic cohyponormal operators’, Acta Sci. Math. 68 (2002), 303–328.Google Scholar
[12]Gethner, R.M. and Shapiro, J., ‘Universal vectors for operators on space of holomorphic functions’, Proc. Amer. Math. Soc. 100 (1987), 281–288.CrossRefGoogle Scholar
[13]Godefroy, G. and Shapiro, J., ‘Operators with dense invariant cyclic vectors manifolds’, J. Funct. Anal. 98 (1991), 229–269.CrossRefGoogle Scholar
[14]Grosse-Erdmann, K.-G., ‘Universal families and hypercyclic operators’, Bull. Amer. Math Soc. 36 (1999), 345–381.CrossRefGoogle Scholar
[15]Herzog, G., ‘On linear having supercyclic operators’, Studia Math. 103 (1992), 295–298.CrossRefGoogle Scholar
[16]Herzog, G. and Schmoeger, C., ‘On operators T such that f(T) is hypercyclic’, Studia Math. 108 (1994), 209–216.Google Scholar
[17]Hilden, H. and Wallen, L., ‘Some cyclic and non-cyclic vectors of certain operators’, Indiana Univ. Math. J. 23 (1974), 557–565.CrossRefGoogle Scholar
[18]Kitai, C., Invariant closed sets for linear operators, Thesis (Univ. Toronto, 1982).Google Scholar
[19]Laursen, K. B. and Neumann, M. M., Introduction to local spectral theory, London Math. Soc. Monographs New Series 20 (Oxford University Press, New York, 2000).CrossRefGoogle Scholar
[20]MacLane, G.R., ‘Sequences of derivatives and normal families’, J. Analyse Math. 2 (1952), 72–87.CrossRefGoogle Scholar
[21]Montes-Rodríguez, A. and Salas, H., ‘Supercyclic subspaces: spectral theory and weighted shifts’, Adv. Math. 163 (2001), 74–134.CrossRefGoogle Scholar
[22]Pál, J., ‘Zwei Kleine Bemerkungen’, Tôhoku Math. J. 6 (1914/1915), 42–43.Google Scholar
[23]Peris, A., ‘Hypercyclicity criteria and the Mittag-Leffler theorem’, Bull. Soc. Roy. Sci. Liège 70 (2001), 365–371.Google Scholar
[24]Rolewicz, S., ‘On orbits of elements’, Stud. Math. 33 (1969), 17–22.CrossRefGoogle Scholar
[25]Salas, H., ‘Hypercyclic weighted shifts’, Trans. Amer. Math. Soc. 347 (993–1004).CrossRefGoogle Scholar
[26]Salas, H., ‘Supercyclicity and weighted shifts’, Studia Math. 135 (1999), 55–74.CrossRefGoogle Scholar