Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On hypercyclic operators on Banach spaces

Author: Luis Bernal-González
Journal: Proc. Amer. Math. Soc. 127 (1999), 1003-1010
MSC (1991): Primary 47A65; Secondary 47B37, 47B99
MathSciNet review: 1476119
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We provide in this paper a direct and constructive proof of the following fact: for a Banach space $X$ there are bounded linear operators having hypercyclic vectors if and only if $X$ is separable and dim$\, X = \infty $. This is a special case of a recent result, which in turn solves a problem proposed by S. Rolewicz.

References [Enhancements On Off] (What's this?)

  • [An1] S. I. Ansari, Hypercyclic and cyclic vectors, J. Funct. Anal. 128 (1995), 374-383. MR 96h:47002
  • [An2] S. I. Ansari, Hypercyclic operators on topological vectors spaces, J. Funct. Anal., to appear. CMP 98:01
  • [Be1] B. Beauzamy, Un opérateur sur l'espace de Hilbert, dont tous les polynômes sont hypercyclic, C. R. Acad. Sci. Paris, Sér. I Math. 303 (1986), 923-927. MR 88g:47010
  • [Be2] B. Beauzamy, An operator in a separable Hilbert space with many hupercyclic vectors, Studia Math. 87 (1987), 71-78. MR 89j:47004
  • [Be3] B. Beauzamy, An operator on a separable Hilbert space with all polynomial hypercyclic, Studia Math. 96 (1990)), 81-90. MR 91d:47004
  • [Bi] G. D. Birkhoff, Démonstration d'un théorème elémentaire sur les fonctions entières, C. R. Acad. Sci. Paris 189 (1929), 473-475.
  • [Bo] P. S. Bourdon, Invariant manifolds of hypercyclic vectors, Proc. Amer. Math. Soc. 118 (1993), 845-847. MR 93i:47002
  • [GeS] R. M. Gethner and J. H. Shapiro, Universal vectors for operators on spaces of holomorphic functions, Proc. Amer. Math. Soc. 100 (1987), 281-288. MR 88g:47060
  • [GoS] G. Godefroy and J. H. Shapiro, Operators with dense, invariant, cyclic vector manifolds, J. Funct. Anal. 98 (1991), 229-269. MR 92d:47029
  • [Gr1] K. G. Grosse-Erdmann, Holomorphe Monster und universelle Funktionen, Mitt. Math. Sem. Giessen 176, 1987. MR 88i:30060
  • [Gr2] K. G. Grosse-Erdmann, On the universal functions of G. R. MacLane, Complex Variables 15 (1990), 193-196. MR 91i:30021
  • [HaKR] I. Halperin, C. Kitai and P. Rosenthal, On orbits of linear operators, J. London Math. Soc. (2) 31 (1985), 561-565. MR 87e:47025
  • [He] D. A. Herrero, Hypercyclic operators and chaos, J. Operator Theory 28 (1992), 93-103. MR 95g:47031
  • [HeK] D. A. Herrero and C. Kitai, On invertible hypercyclic operators, Proc. Amer. Math. Soc. 116 (1992), 873-875. MR 93a:47023
  • [Hz] G. Herzog, On linear operators having supercyclic vectors, Studia Math. 103 (3) (1992), 295-298. MR 93k:47033
  • [Ki] C. Kitai, Invariant closed sets for linear operators, Dissertation, University of Toronto, 1982.
  • [Ma] G. R. MacLane, Sequences of derivatives and normal families, J. Analyse Math. 2 (1952), 72-87. MR 14:741d
  • [OP] R. I. Ovsepian and A. Pelczynski, The existence in every separable Banach space of a fundamental total and bounded biorthogonal sequence and related constructions of uniformly bounded orthogonal systems in $L^{2}$, Studia Math. 54 (1975), 149-155. MR 52:14941
  • [Re] C. Read, The invariant subspace problem for a class of Banach spaces, 2: Hypercyclic operators, Israel J. Math. 63 (1988), 1-40. MR 90b:47013
  • [Ro] S. Rolewicz, On orbits of elements, Studia Math. 32 (1969), 17-22. MR 39:3292
  • [Sa] H. N. Salas, Hypercyclic weighted shifts, Trans. Amer. Math. Soc. 347 (1995), 993-1003. MR 95e:47042

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 47A65, 47B37, 47B99

Retrieve articles in all journals with MSC (1991): 47A65, 47B37, 47B99

Additional Information

Luis Bernal-González
Affiliation: Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Sevilla, Apdo. 1160, Sevilla 41080, Spain

Keywords: Hypercyclic vector, linear operator, infinite-dimensio\-nal separable Banach space, biorthogonal system, backward weighted shift
Received by editor(s): May 29, 1997
Received by editor(s) in revised form: July 6, 1997
Additional Notes: The author’s research was supported in part by DGES grant #PB93–0926 and the Junta de Andalucıá.
Communicated by: David R. Larson
Article copyright: © Copyright 1999 American Mathematical Society