On hypercyclic operators on Banach spaces
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- by Luis Bernal-González PDF
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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
- Shamim I. Ansari, Hypercyclic and cyclic vectors, J. Funct. Anal. 128 (1995), no. 2, 374–383. MR 1319961, DOI 10.1006/jfan.1995.1036
- S. I. Ansari, Hypercyclic operators on topological vectors spaces, J. Funct. Anal., to appear.
- Bernard Beauzamy, Un opérateur, sur l’espace de Hilbert, dont tous les polynômes sont hypercycliques, C. R. Acad. Sci. Paris Sér. I Math. 303 (1986), no. 18, 923–925 (French, with English summary). MR 873395
- Bernard Beauzamy, An operator on a separable Hilbert space with many hypercyclic vectors, Studia Math. 87 (1987), no. 1, 71–78. MR 924762, DOI 10.4064/sm-87-1-71-78
- Bernard Beauzamy, An operator on a separable Hilbert space with all polynomials hypercyclic, Studia Math. 96 (1990), no. 1, 81–90. MR 1055079, DOI 10.4064/sm-96-1-81-90
- G. D. Birkhoff, Démonstration d’un théorème elémentaire sur les fonctions entières, C. R. Acad. Sci. Paris 189 (1929), 473–475.
- Paul S. Bourdon, Invariant manifolds of hypercyclic vectors, Proc. Amer. Math. Soc. 118 (1993), no. 3, 845–847. MR 1148021, DOI 10.1090/S0002-9939-1993-1148021-4
- Robert M. Gethner and Joel H. Shapiro, Universal vectors for operators on spaces of holomorphic functions, Proc. Amer. Math. Soc. 100 (1987), no. 2, 281–288. MR 884467, DOI 10.1090/S0002-9939-1987-0884467-4
- Gilles Godefroy and Joel H. Shapiro, Operators with dense, invariant, cyclic vector manifolds, J. Funct. Anal. 98 (1991), no. 2, 229–269. MR 1111569, DOI 10.1016/0022-1236(91)90078-J
- Karl-Goswin Große-Erdmann, Holomorphe Monster und universelle Funktionen, Mitt. Math. Sem. Giessen 176 (1987), iv+84 (German). Dissertation, University of Trier, Trier, 1987. MR 877464
- Karl-Goswin Grosse-Erdmann, On the universal functions of G. R. MacLane, Complex Variables Theory Appl. 15 (1990), no. 3, 193–196. MR 1074061, DOI 10.1080/17476939008814450
- Israel Halperin, Carol Kitai, and Peter Rosenthal, On orbits of linear operators, J. London Math. Soc. (2) 31 (1985), no. 3, 561–565. MR 812786, DOI 10.1112/jlms/s2-31.3.561
- Domingo A. Herrero, Hypercyclic operators and chaos, J. Operator Theory 28 (1992), no. 1, 93–103. MR 1259918
- Domingo A. Herrero and Carol Kitai, On invertible hypercyclic operators, Proc. Amer. Math. Soc. 116 (1992), no. 3, 873–875. MR 1123653, DOI 10.1090/S0002-9939-1992-1123653-7
- Gerd Herzog, On linear operators having supercyclic vectors, Studia Math. 103 (1992), no. 3, 295–298. MR 1202014, DOI 10.4064/sm-103-3-295-298
- C. Kitai, Invariant closed sets for linear operators, Dissertation, University of Toronto, 1982.
- P. Hebroni, Sur les inverses des éléments dérivables dans un anneau abstrait, C. R. Acad. Sci. Paris 209 (1939), 285–287 (French). MR 14
- R. I. Ovsepian and A. Pełczyński, The existence in every separable Banach space of a fundamental total and bounded biorthogonal sequence and related constructions of uniformly bounded orthonormal systems in $L^{2}$, Séminaire Maurey-Schwartz (1973–1974), Espaces Lp, applications radonifiantes et géométrie des espaces de Banach, Exp. No. 20, Centre de Math., École Polytech., Paris, 1974, pp. 16. MR 0394136
- C. J. Read, The invariant subspace problem for a class of Banach spaces. II. Hypercyclic operators, Israel J. Math. 63 (1988), no. 1, 1–40. MR 959046, DOI 10.1007/BF02765019
- S. Rolewicz, On orbits of elements, Studia Math. 32 (1969), 17–22. MR 241956, DOI 10.4064/sm-32-1-17-22
- Héctor N. Salas, Hypercyclic weighted shifts, Trans. Amer. Math. Soc. 347 (1995), no. 3, 993–1004. MR 1249890, DOI 10.1090/S0002-9947-1995-1249890-6
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
- 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
- © Copyright 1999 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 127 (1999), 1003-1010
- MSC (1991): Primary 47A65; Secondary 47B37, 47B99
- DOI: https://doi.org/10.1090/S0002-9939-99-04657-2
- MathSciNet review: 1476119