Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

A hypercyclic operator whose adjoint is also hypercyclic


Author: Héctor Salas
Journal: Proc. Amer. Math. Soc. 112 (1991), 765-770
MSC: Primary 47A65; Secondary 47B37
DOI: https://doi.org/10.1090/S0002-9939-1991-1049848-8
MathSciNet review: 1049848
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: An operator $ T$ acting on a Hilbert space $ H$ is hypercyclic if, for some vector $ x$ in $ H$, the orbit $ \{ {T^n}x:n \geq 0\} $ is dense in $ H$. We show the existence of a hypercyclic operator--in fact, a bilateral weighted shift--whose adjoint is also hypercyclic. This answers positively a question of D. A. Herrero.


References [Enhancements On Off] (What's this?)

  • [1] C. Apostol, L. A. Fialkow, D. A. Herrero, and D. Voiculescu, Approximation of Hilbert space operators, Vol. II, Research Notes in Math. 102, Pitman, Boston, 1984. MR 735080 (85m:47002)
  • [2] 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 884467 (88g:47060)
  • [3] G. Godefroy and J. H. Shapiro, Operators with dense, invariant cyclic vector manifolds, J. Funct. Anal. (to appear). MR 1111569 (92d:47029)
  • [4] I. Halperin, C. Kitai, and P. Rosenthal, On orbits of linear operators, J. London Math. Soc. 31 (1985), 561-565. MR 812786 (87e:47025)
  • [5] D. A. Herrero, The diagonal entries in the formula 'Quasitriangular - compact = triangular' and restrictions of quasitriangularity, Trans. Amer. Math. Soc. 289 (1987), 1-42. MR 857432 (88c:47022)
  • [6] -, Spectral pictures of operators in the Cowen-Douglas class $ {B_n}(\Omega )$ and its closure, J. Operator Theory 18 (1988), 213-222. MR 915506 (89b:47032)
  • [7] -, Limits of hypercyclic and supercyclic operators, preprint. MR 1670822 (99m:47018)
  • [8] D. A. Herrero and W. R. Wogen, On the multiplicity of $ T \oplus T \oplus \cdots \oplus T$, Rocky Mountain J. Math. (to appear). MR 1065843 (91f:47027)
  • [9] D. A. Herrero and Z. Y. Wang, Compact perturbations of hypercyclic and supercyclic operators, Indiana Univ. Math. J. (to appear). MR 1078739 (91k:47042)
  • [10] C. Kitai, Invariant closed sets for linear operators, Thesis, Univ. of Toronto, 1982.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 47A65, 47B37

Retrieve articles in all journals with MSC: 47A65, 47B37


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1991-1049848-8
Keywords: Cyclic vectors, hypercyclic vectors and operators, weighted shifts
Article copyright: © Copyright 1991 American Mathematical Society

American Mathematical Society