# Supercyclicity and weighted shifts

Studia Mathematica (1999)

• Volume: 135, Issue: 1, page 55-74
• ISSN: 0039-3223

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## Abstract

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An operator (linear and continuous) in a Fréchet space is hypercyclic if there exists a vector whose orbit under the operator is dense. If the scalar multiples of the elements in the orbit are dense, the operator is supercyclic. We give, for Fréchet space operators, a Supercyclicity Criterion reminiscent of the Hypercyclicity Criterion. We characterize the supercyclic bilateral weighted shifts in terms of their weight sequences. As a consequence, we show that a bilateral weighted shift is supercyclic if and only if it satisfies the Supercyclicity Criterion. We exhibit two supercyclic, irreducible Hilbert space operators which are C*-isomorphic, but one is hypercyclic and the other is not. We prove that a Banach space operator which satisfies a version of the Supercyclicity Criterion, and has zero in its left essential spectrum, has an infinite-dimensional closed subspace whose nonzero vectors are supercyclic.

## How to cite

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Salas, Héctor. "Supercyclicity and weighted shifts." Studia Mathematica 135.1 (1999): 55-74. <http://eudml.org/doc/216643>.

@article{Salas1999,
abstract = {An operator (linear and continuous) in a Fréchet space is hypercyclic if there exists a vector whose orbit under the operator is dense. If the scalar multiples of the elements in the orbit are dense, the operator is supercyclic. We give, for Fréchet space operators, a Supercyclicity Criterion reminiscent of the Hypercyclicity Criterion. We characterize the supercyclic bilateral weighted shifts in terms of their weight sequences. As a consequence, we show that a bilateral weighted shift is supercyclic if and only if it satisfies the Supercyclicity Criterion. We exhibit two supercyclic, irreducible Hilbert space operators which are C*-isomorphic, but one is hypercyclic and the other is not. We prove that a Banach space operator which satisfies a version of the Supercyclicity Criterion, and has zero in its left essential spectrum, has an infinite-dimensional closed subspace whose nonzero vectors are supercyclic.},
author = {Salas, Héctor},
journal = {Studia Mathematica},
keywords = {hypercyclic and supercyclic vectors; Hypercyclicity Criterion; Supercyclicity Criterion; weighted shifts; semi-Fredholm operators in Banach spaces; left essential spectrum; basic sequences.; cyclicity; hypercyclicity; supercyclicity; Fréchet spaces; supercyclic bilateral shifts; weight sequences; -isomorphisms},
language = {eng},
number = {1},
pages = {55-74},
title = {Supercyclicity and weighted shifts},
url = {http://eudml.org/doc/216643},
volume = {135},
year = {1999},
}

TY - JOUR
AU - Salas, Héctor
TI - Supercyclicity and weighted shifts
JO - Studia Mathematica
PY - 1999
VL - 135
IS - 1
SP - 55
EP - 74
AB - An operator (linear and continuous) in a Fréchet space is hypercyclic if there exists a vector whose orbit under the operator is dense. If the scalar multiples of the elements in the orbit are dense, the operator is supercyclic. We give, for Fréchet space operators, a Supercyclicity Criterion reminiscent of the Hypercyclicity Criterion. We characterize the supercyclic bilateral weighted shifts in terms of their weight sequences. As a consequence, we show that a bilateral weighted shift is supercyclic if and only if it satisfies the Supercyclicity Criterion. We exhibit two supercyclic, irreducible Hilbert space operators which are C*-isomorphic, but one is hypercyclic and the other is not. We prove that a Banach space operator which satisfies a version of the Supercyclicity Criterion, and has zero in its left essential spectrum, has an infinite-dimensional closed subspace whose nonzero vectors are supercyclic.
LA - eng
KW - hypercyclic and supercyclic vectors; Hypercyclicity Criterion; Supercyclicity Criterion; weighted shifts; semi-Fredholm operators in Banach spaces; left essential spectrum; basic sequences.; cyclicity; hypercyclicity; supercyclicity; Fréchet spaces; supercyclic bilateral shifts; weight sequences; -isomorphisms
UR - http://eudml.org/doc/216643
ER -

## References

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1. [1] S. I. Ansari, Existence of hypercyclic operators on topological vector spaces, J. Funct. Anal. 148 (1997), 384-390. Zbl0898.47019
2. [2] S. I. Ansari and P. S. Bourdon, Some properties of cyclic operators, Acta Sci. Math. (Szeged) 63 (1997), 195-207. Zbl0892.47004
3. [3] B. Beauzamy, Un opérateur sans sous-espace invariant: Simplification de l'exemple de P. Enflo, J. Integral Equations Operator Theory 8 (1985), 314-384. Zbl0571.47002
4. [4] B. Beauzamy, Introduction to Operator Theory and Invariant Subspaces, North-Holland Math. Library 42, North-Holland, 1988.
5. [5] L. Bernal-González and A. Montes-Rodríguez, Non-finite dimensional closed vector spaces of universal functions for composition operators, J. Approx. Theory 82 (1995), 375-391. Zbl0831.30024
6. [6] J. P. Bès, Hereditary hypercyclic operators and the hypercyclicity criterion, preprint.
7. [7] G. D. Birkhoff, Démonstration d'un théorème élémentaire sur les fonctions entières, C. R. Acad. Sci. Paris 189 (1929), 473-475. Zbl55.0192.07
8. [8] P. S. Bourdon, Invariant manifolds of hypercyclic operators, Proc. Amer. Math. Soc. 118 (1993), 1577-1581. Zbl0853.54036
9. [9] P. S. Bourdon, Orbits of hyponormal operators, Michigan Math. J. 44 (1997), 345-353. Zbl0896.47020
10. [10] K. C. Chan and J. H. Shapiro, The cyclic behavior of translation operators on Hilbert spaces of entire functions, Indiana Univ. Math. J. 40 (1991), 1421-1449. Zbl0771.47015
11. [11] J. B. Conway, A Course in Functional Analysis, 2nd ed., Springer, New York, 1990. Zbl0706.46003
12. [12] R. M. Gethner and J. H. Shapiro, Universal vectors for operators on spaces of holomorphic functions, Proc. Amer. Math. Soc. 100 (1987), 281-288. Zbl0618.30031
13. [13] G. Godefroy and J. H. Shapiro, Operators with dense, invariant, cyclic vector manifolds, J. Funct. Anal. 98 (1991), 239-269. Zbl0732.47016
14. [14] K.-G. Große-Erdmann, Holomorphe Monster und universelle Funktionen, Mitt. Math. Sem. Giessen 176 (1987), 1-84.
15. [15] D. A. Herrero, Limits of hypercyclic and supercyclic operators, J. Funct. Anal. 99 (1991), 179-190. Zbl0758.47016
16. [16] D. A. Herrero, Hypercyclic operators and chaos, J. Operator Theory 28 (1992), 93-103. Zbl0806.47020
17. [17] D. A. Herrero and C. Kitai, On invertible hypercyclic operators, Proc. Amer. Math. Soc. 116 (1992), 873-875. Zbl0780.47006
18. [18] D. A. Herrero and Z. Y. Wang, Compact perturbations of hypercyclic and supercyclic operators, Indiana Univ. Math. J. 39 (1990), 819-829. Zbl0724.47009
19. [19] G. Herzog, On linear operators having supercyclic vectors, Studia Math. 103 (1992), 295-298. Zbl0811.47018
20. [20] H. M. Hilden and L. J. Wallen, Some cyclic and non-cyclic vectors of certain operators, Indiana Univ. Math. J. 24 (1974), 557-565. Zbl0274.47004
21. [21] C. Kitai, Invariant closed sets for linear operators, thesis, Univ. of Toronto, 1982.
22. [22] F. León-Saavedra and A. Montes-Rodríguez, Linear structure of hypercyclic vectors, J. Funct. Anal. 148 (1997), 524-545. Zbl0999.47009
23. [23] F. León-Saavedra and A. Montes-Rodríguez, Spectral theory and hypercyclic subspaces, Trans. Amer. Math. Soc., to appear. Zbl0961.47003
24. [24] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I, Springer, Berlin, 1977. Zbl0362.46013
25. [25] V. G. Miller, Remarks on finitely hypercyclic and finitely supercyclic operators, Integral Equations Operator Theory 29 (1997), 110-115. Zbl0902.47018
26. [26] A. Montes-Rodríguez, Banach spaces of hypercyclic vectors, Michigan Math. J. 43 (1996), 419-436. Zbl0907.47023
27. [27] A. Montes-Rodríguez and H. N. Salas, Supercyclic operators, in preparation.
28. [28] D. P. O'Donovan, Weighted shifts and covariance algebras, Trans. Amer. Math. Soc. 208 (1975), 1-25. Zbl0308.46050
29. [29] C. J. Read, The invariant subspace problem for a class of Banach spaces, 2: Hypercyclic operators, Israel J. Math. 63 (1988), 1-40. Zbl0782.47002
30. [30] S. Rolewicz, On orbits of elements, Studia Math. 32 (1969), 17-22. Zbl0174.44203
31. [31] H. N. Salas, Semigroup of isometries with commuting range projections, J. Operator Theory 14 (1985), 331-346. Zbl0613.47041
32. [32] H. N. Salas, A hypercyclic operator whose adjoint is also hypercyclic, Proc. Amer. Math. Soc. 112 (1991), 765-770. Zbl0748.47023
33. [33] H. N. Salas, Hypercyclic weighted shifts, Trans. Amer. Math. Soc. 347 (1995), 993-1004. Zbl0822.47030
34. [34] J. H. Shapiro, Composition Operators and Classical Function Theory, Springer, New York, 1993. Zbl0791.30033
35. [35] A. L. Shields, Weighted shift operators and analytic function theory, in: Topics in Operator Theory, Math. Surveys 13, Amer. Math. Soc., Providence, R.I., 2nd printing, 1979, 49-128.
36. [36] J. Zemánek, The semi-Fredholm radius of a linear operator, Bull. Polish Acad. Sci. Math. 32 (1984), 67-76. Zbl0583.47016

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