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On hypercyclic operators on Banach spaces

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

[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

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Additional Information:

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

DOI: 10.1090/S0002-9939-99-04657-2
PII: S 0002-9939(99)04657-2
Keywords: Hypercyclic vector, linear operator, infinite-dimensional 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 Andaluciá.
Communicated by: David R. Larson
Copyright of article: Copyright 1999, American Mathematical Society

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