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BANACH SPACES WITH SEPARABLE DUALS SUPPORT DUAL HYPERCYCLIC OPERATORS

Published online by Cambridge University Press:  09 August 2007

HÉCTOR N. SALAS*
Affiliation:
Department of Mathematics, University of Puerto Rico, Mayagüez, Puerto Rico 00681 e-mail: salas@math.uprm.edu
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Abstract

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Let E be a Banach space such that its dual E* is separable. We show that there exists a hypercyclic bounded operator T on E such that its adjoint T* is also hypercyclic on E*. We also exhibit a new kind of dual hypercyclic operator. Thus answers affirmatively two of the questions raised by Henrik Petersson in a recent paper.

Type
Research Article
Copyright
Copyright © Glasgow Mathematical Journal Trust 2007

References

REFERENCES

1. Abakumov, E. and Gordon, J., Common hypercyclic vectors for multiples of backward shift, J. Funct. Anal. 200 (2003), no. 2, 494504.Google Scholar
2. Ansari, S. I., Existence of hypercyclic operators on topological vector spaces, J. Funct. Anal. 148 (1997), no. 2, 384390.Google Scholar
3. Aron, R., Bès, J., León, F. and Peris, A., Operators with common hypercyclic subspaces, J. Operator Theory 54 (2005), no. 2, 251260.Google Scholar
4. Bayart, F., Common hypercyclic subspaces, Integral Equations and Operator Theory 53 (2005), no. 4, 467476.Google Scholar
5. Bernal-González, L., On hypercyclic operators on Banach spaces, Proc. Amer. Math. Soc. 127 (1999), no. 4, 10031010.CrossRefGoogle Scholar
6. Bès, J. P. and Peris, A., Hereditarily hypercyclic operators, J. Funct. Anal. 167 (1999), no. 1, 94112.Google Scholar
7. Bonet, J. and Peris, A., Hypercyclic operators on non-normable Fréchet spaces, J. Funct. Anal. 159 (1998), no. 2, 587595.Google Scholar
8. Godefroy, G. and Shapiro, J. H., Operators with dense, invariant, cyclic vector manifolds J. Funct. Anal. 98 (1991), 229269.Google Scholar
9. González, M., León-Saavedra, F. and Montes-Rodríguez, A., Semi Fredholm theory, hypercyclic and supercyclic subspaces, Proc. London Math. Soc. 80 (2000), no 1, 169189.Google Scholar
10. Grivaux, S., Hypercyclic operators, mixing operators, and the bounded steps problem, J. Operator Theory 54 (2005), no. 1, 147168.Google Scholar
11. Grosse-Erdman, K. G., Universal families and hypercyclic operators, Bull. Amer. Math. Soc. (N.S.) 36 (1999), no. 3, 345381.Google Scholar
12. León-Saavedra, F. and Montes-Rodríguez, A., Linear structure of hypercyclic vectors, J. Funct. Anal. 148 (1997), no. 2, 524545.Google Scholar
13. Lindenstrauss, J. and Tzafriri, L., Classical Banach spaces. I. Sequence spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 92 (Springer-Verlag, 1977).Google Scholar
14. Herrero, D. A., Limits of hypercyclic and supercyclic operators, J. Funct. Anal. 99 (1991), 179190.Google Scholar
15. Petersson, H., Spaces that admit hypercyclic operators with hypercyclic adjoints, Proc. Amer. Math. Soc. 134 (2005), 16711676.Google Scholar
16. Rolewicz, S., On orbits of elements, Studia Math. 32 (1969), 1722.Google Scholar
17. Salas, H. N., A hypercyclic operator whose adjoint is also hypercyclic, Proc. Amer. Math. Soc. 112 (1991), no. 3, 765770.Google Scholar
18. Salas, H. N., Hypercyclic weighted shifts, Trans. Amer. Math. Soc. 347 (1995), no. 3, 9931004.Google Scholar
19. Shields, A. L., Weighted shift operators and analytic function theory, Math. Survey Monographs, Vol 12 Amer. Math. Soc. Providence, RI 1974; Second Printing 1979), 49–128.Google Scholar