Abstract
Recently a lot of research has been done on hypercyclicity of a bounded linear operator on a Banach space, based on the hypercyclicity criterion obtained by Kitai in 1982, and independently by Gethner and Shapiro in 1987. By combining this criterion with one extra condition, Montes-Rodríguez obtained in 1996 a sufficient condition for the operator to have a closed infinite dimensional hypercyclic subspace, with a very technical proof. Since then, this result has been used extensively to generate new results on hypercyclic subspaces. In the present paper, we give a simple proof of the result of Montes-Rodríguez, by first establishing a few elementary results about the algebra of operators on a Banach space.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
B. Beauzamy,Un opérator sur l'espace de Hilbert, dont tous les polynômes sont hypercyclic, C. R. Acad. Sci Paris Sér. I Math.303, (1986), 923–927.
B. Beauzamy,An operator on a separable Hilbert space, with many hypercyclic vectors, Studia Math.87 (1988), 71–78.
B. Beauzamy,An operator on a separable Hilbert space with all polynomials hypercyclic, Studia Math.96, (1990), 81–90.
L. Bernal-González & A. Montes-Rodríguez,Non-finite dimensional closed vector spaces of universal functions for composition operators, J. Approx. Theory82, (1995), 375–391.
J. Bès,Invariant manifolds of hypercyclic vectors for the real scalar case, Proc. Amer. Math. Soc.127 (1999), 1801–1804.
G. D. Birkhoff,Démonstration d'un Théoreme elementaire sur les fonctions entiéres, C. R. Acad. Sci. Paris189 (1929), 473–475.
P. S. Bourdon,Invariant manifolds of hypercyclic vectors, Proc. Amer. Math. Soc.118 (1993), 845–847.
K. C. Chan,Hypercyclicity of the operator algebra for a separable Hilbert space, J. Operator Theory42, (1999), 231–244.
R. M. Gethner & J. H. Shapiro,Universal vectors for operators on spaces of holomorphic functions, Proc. Amer. Math. Soc.100 (1987), 281–288.
G. Godefroy & J. H. Shapiro,Operators with dense, invariant, cyclic vector manifolds, J. Funct. Anal.98 (1991), 229–269.
D. A. Herrero,Limits of hypercyclic and supercyclic operators, J. Funct. Anal.99 (1991), 179–190.
C. Kitai,Invariant Closed Sets for Linear Operators, Ph. D. Thesis, Univ. of Toronto, 1982.
F. León-Saavedra & A. Montes-Rodríguez,Linear structure of hypercyclic vectors, J. Functional Analysis148, (1997), 524–545.
F. León-Saavedra & A. Montes-Rodríguez,Spectral theory and hypercyclic subspaces, Transactions of the Amer. Math. Soc, to appear.
G. R. MacLane,Sequences of derivatives and normal families, J. Analyse Math.2 (1952), 72–87.
A. Montes-Rodríguez,Banach spaces of hypercyclic vectors, Michigan Math. J.43 (1996) 419–436.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Chan, K.C., Taylor, R.D. Hypercyclic subspaces of a Banach space. Integr equ oper theory 41, 381–388 (2001). https://doi.org/10.1007/BF01202099
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01202099