On the existence of chaotic and hypercyclic semigroups on Banach spaces

Bermúdez, Teresa; Bonilla, Antonio; Martinón, Antonio

doi:10.1090/S0002-9939-02-06762-X

On the existence of chaotic and hypercyclic semigroups on Banach spaces
HTML articles powered by AMS MathViewer

by Teresa Bermúdez, Antonio Bonilla and Antonio Martinón PDF

Proc. Amer. Math. Soc. 131 (2003), 2435-2441 Request permission

Abstract:

We prove that every separable infinite dimensional complex Banach space admits a hypercyclic uniformly continuous semigroup. We also prove that there exist Banach spaces admitting no chaotic strongly continuous semigroups.

References

Shamim I. Ansari, Existence of hypercyclic operators on topological vector spaces, J. Funct. Anal. 148 (1997), no. 2, 384–390. MR 1469346, DOI 10.1006/jfan.1996.3093
S. A. Argyros and V. Felouzis, Interpolating hereditarily indecomposable Banach spaces, J. Amer. Math. Soc. 13 (2000), no. 2, 243–294. MR 1750954, DOI 10.1090/S0894-0347-00-00325-8
Á. Császár and J. Deák, Simultaneous extensions of proximities, semi-uniformities, contiguities and merotopies. III, Math. Pannon. 2 (1991), no. 2, 3–23. MR 1149946
Luis Bernal-González, On hypercyclic operators on Banach spaces, Proc. Amer. Math. Soc. 127 (1999), no. 4, 1003–1010. MR 1476119, DOI 10.1090/S0002-9939-99-04657-2
José Bonet, Félix Martínez-Giménez, and Alfredo Peris, A Banach space which admits no chaotic operator, Bull. London Math. Soc. 33 (2001), no. 2, 196–198. MR 1815423, DOI 10.1112/blms/33.2.196
José Bonet and Alfredo Peris, Hypercyclic operators on non-normable Fréchet spaces, J. Funct. Anal. 159 (1998), no. 2, 587–595. MR 1658096, DOI 10.1006/jfan.1998.3315
R. DeLaubenfels, H. Emamirad, Chaos for functions of discrete and continuous weighted shift operators, Ergodic Theory Dynam. Systems 21 (2001), 1411-1427.
Wolfgang Desch, Wilhelm Schappacher, and Glenn F. Webb, Hypercyclic and chaotic semigroups of linear operators, Ergodic Theory Dynam. Systems 17 (1997), no. 4, 793–819. MR 1468101, DOI 10.1017/S0143385797084976
Robert L. Devaney, An introduction to chaotic dynamical systems, 2nd ed., Addison-Wesley Studies in Nonlinearity, Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, CA, 1989. MR 1046376
H. Emamirad, Hypercyclicity in the scattering theory for linear transport equation, Trans. Amer. Math. Soc. 350 (1998), no. 9, 3707–3716. MR 1451598, DOI 10.1090/S0002-9947-98-02062-5
V. Ferenczi, Quotient hereditarily indecomposable Banach spaces, Canad. J. Math. 51 (1999), no. 3, 566–584. MR 1701326, DOI 10.4153/CJM-1999-026-4
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
W. T. Gowers and B. Maurey, The unconditional basic sequence problem, J. Amer. Math. Soc. 6 (1993), no. 4, 851–874. MR 1201238, DOI 10.1090/S0894-0347-1993-1201238-0
Karl-Goswin Grosse-Erdmann, Universal families and hypercyclic operators, Bull. Amer. Math. Soc. (N.S.) 36 (1999), no. 3, 345–381. MR 1685272, DOI 10.1090/S0273-0979-99-00788-0
Domingo A. Herrero, Limits of hypercyclic and supercyclic operators, J. Funct. Anal. 99 (1991), no. 1, 179–190. MR 1120920, DOI 10.1016/0022-1236(91)90058-D
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
F. Martínez-Giménez, A. Peris, Chaos for backward shift operators, Int. J. of Bifurcation and Chaos 12 (2002), 1703–1715.
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 $L^{2}$, Studia Math. 54 (1975), no. 2, 149–159. MR 394137, DOI 10.4064/sm-54-2-149-159
Sergio Sispanov, Generalización del teorema de Laguerre, Bol. Mat. 12 (1939), 113–117 (Spanish). MR 3
A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. MR 710486, DOI 10.1007/978-1-4612-5561-1
A. Peris, Personal communication.
F. Räbiger and W. J. Ricker, $C_0$-groups and $C_0$-semigroups of linear operators on hereditarily indecomposable Banach spaces, Arch. Math. (Basel) 66 (1996), no. 1, 60–70. MR 1363778, DOI 10.1007/BF01323983
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
Angus Ellis Taylor and David C. Lay, Introduction to functional analysis, 2nd ed., John Wiley & Sons, New York-Chichester-Brisbane, 1980. MR 564653