Extending previous results of H. Salas we obtain a characterisation of hypercyclic weighted shifts on an arbitrary F-sequence space in which the canonical unit vectors form a Schauder basis. If the basis is unconditional we give a characterisation of those hypercyclic weighted shifts that are even chaotic.
@article{bwmeta1.element.bwnjournal-article-smv139i1p47bwm,
author = {K.-G. Grosse-Erdmann},
title = {Hypercyclic and chaotic weighted shifts},
journal = {Studia Mathematica},
volume = {141},
year = {2000},
pages = {47-68},
zbl = {0991.47013},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv139i1p47bwm}
}
Grosse-Erdmann, K.-G. Hypercyclic and chaotic weighted shifts. Studia Mathematica, Tome 141 (2000) pp. 47-68. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv139i1p47bwm/
[00000] [1] J. Banks, J. Brooks, G. Cairns, G. Davis and P. Stacey, On Devaney's definition of chaos, Amer. Math. Monthly 99 (1992), 332-334. | Zbl 0758.58019
[00001] [2] B. Beauzamy, Introduction to Operator Theory and Invariant Subspaces, North-Holland, Amsterdam, 1988. | Zbl 0663.47002
[00002] [3] L. Bernal-González, Derivative and antiderivative operators and the size of complex domains, Ann. Polon. Math. 59 (1994), 267-274. | Zbl 0843.47019
[00003] [4] L. Bernal-González, Universal functions for Taylor shifts, Complex Variables Theory Appl. 31 (1996), 121-129.
[00004] [5] L. Bernal-González and A. Montes-Rodríguez, Universal functions for composition operators, ibid. 27 (1995), 47-56. | Zbl 0838.30032
[00005] [6] J. Bès and A. Peris, Hereditarily hypercyclic operators, J. Funct. Anal. 167 (1999), 94-112.
[00006] [7] G. D. Birkhoff, Démonstration d'un théorème élémentaire sur les fonctions entières, C. R. Acad. Sci. Paris 189 (1929), 473-475. | Zbl 55.0192.07
[00007] [8] R. L. Devaney, An Introduction to Chaotic Dynamical Systems, Benjamin/Cummings, Menlo Park, CA, 1986. | Zbl 0632.58005
[00008] [9] P. L. Duren, Theory of Spaces, Academic Press, New York, 1970. | Zbl 0215.20203
[00009] [10] R. M. Gethner and J. H. Shapiro, Universal vectors for operators on spaces of holomorphic functions, Proc. Amer. Math. Soc. 100 (1987), 281-288. | Zbl 0618.30031
[00010] [11] G. Godefroy and J. H. Shapiro, Operators with dense invariant cyclic vector manifolds, J. Funct. Anal. 98 (1991), 229-269. | Zbl 0732.47016
[00011] [12] K.-G. Grosse-Erdmann, Holomorphe Monster und universelle Funktionen, Mitt. Math. Sem. Giessen 176 (1987).
[00012] [13] K.-G. Grosse-Erdmann, On the universal functions of G. R. MacLane, Complex Variables Theory Appl. 15 (1990), 193-196. | Zbl 0682.30021
[00013] [14] K.-G. Grosse-Erdmann, Universal families and hypercyclic operators, Bull. Amer. Math. Soc. (N.S.) 36 (1999), 345-381. | Zbl 0933.47003
[00014] [15] A. Gulisashvili and C. R. MacCluer, Linear chaos in the unforced quantum harmonic oscillator, J. Dynam. Systems Measurement Control 118 (1996), 337-338. | Zbl 0870.58057
[00015] [16] D. A. Herrero and Z.-Y. Wang, Compact perturbations of hypercyclic and supercyclic operators, Indiana Univ. Math. J. 39 (1990), 819-829. | Zbl 0724.47009
[00016] [17] N. J. Kalton, N. T. Peck and J. W. Roberts, An F-space Sampler, Cambridge Univ. Press, Cambridge, 1984.
[00017] [18] P. K. Kamthan and M. Gupta, Sequence Spaces and Series, Marcel Dekker, New York, 1981.
[00018] [19] C. Kitai, Invariant closed sets for linear operators, thesis, Univ. of Toronto, Toronto, 1982.
[00019] [20] R. deLaubenfels and H. Emamirad, Chaos for functions of discrete and continuous weighted shift operators, preprint. | Zbl 0997.47027
[00020] [21] F. León-Saavedra and A. Montes-Rodríguez, Linear structure of hypercyclic vectors, J. Funct. Anal. 148 (1997), 524-545. | Zbl 0999.47009
[00021] [22] F. León-Saavedra and A. Montes-Rodríguez, Spectral theory and hypercyclic subspaces, Trans. Amer. Math. Soc., to appear. | Zbl 0961.47003
[00022] [23] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces. I, Springer, Berlin, 1977. | Zbl 0362.46013
[00023] [24] G. R. MacLane, Sequences of derivatives and normal families, J. Anal. Math. 2 (1952/53), 72-87. | Zbl 0049.05603
[00024] [25] F. Martínez-Giménez and A. Peris, Hypercyclic and chaotic backward shift operators on Köthe echelon spaces, preprint. | Zbl 1070.47024
[00025] [26] V. Mathew, A note on hypercyclic operators on the space of entire sequences, Indian J. Pure Appl. Math. 25 (1994), 1181-1184. | Zbl 0821.47019
[00026] [27] R. I. Ovsepian and A. Pełczyński, On the existence of a fundamental total and bounded biorthogonal sequence in every separable Banach space, and related constructions of uniformly bounded orthonormal systems in , Studia Math. 54 (1975), 149-159. | Zbl 0317.46019
[00027] [28] A. Peris, Chaotic polynomials on Fréchet spaces, Proc. Amer. Math. Soc. 127 (1999), 3601-3603. | Zbl 0942.46028
[00028] [29] S. Rolewicz, On orbits of elements, Studia Math. 32 (1969), 17-22. | Zbl 0174.44203
[00029] [30] S. Rolewicz, Metric Linear Spaces, second ed., D. Reidel, Dordrecht, 1985. | Zbl 0573.46001
[00030] [31] H. Salas, A hypercyclic operator whose adjoint is also hypercyclic, Proc. Amer. Math. Soc. 112 (1991), 765-770. | Zbl 0748.47023
[00031] [32] H. Salas, Hypercyclic weighted shifts, Trans. Amer. Math. Soc. 347 (1995), 993-1004. | Zbl 0822.47030
[00032] [33] I. Singer, Bases in Banach Spaces. I, Springer, Berlin, 1970.
[00033] [34] I. Singer, Bases in Banach Spaces. II, Springer, Berlin, 1981.
[00034] [35] R. L. Wheeden and A. Zygmund, Measure and Integral, Marcel Dekker, New York, 1977.
[00035] [36] A. Wilansky, Summability through Functional Analysis, North-Holland, Amsterdam, 1984. | Zbl 0531.40008