Supercyclicity and weighted shifts

Salas, Héctor

Studia Mathematica, Tome 133 (1999), p. 55-74 / Harvested from The Polish Digital Mathematics Library

Résumé

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.

Détail

Publié le : 1999-01-01

Zbl 0940.47005

EUDML-ID : urn:eudml:doc:216643

Bibliographie

[00000] [1] S. I. Ansari, Existence of hypercyclic operators on topological vector spaces, J. Funct. Anal. 148 (1997), 384-390. | Zbl 0898.47019

[00001] [2] S. I. Ansari and P. S. Bourdon, Some properties of cyclic operators, Acta Sci. Math. (Szeged) 63 (1997), 195-207. | Zbl 0892.47004

[00002] [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. | Zbl 0571.47002

[00003] [4] B. Beauzamy, Introduction to Operator Theory and Invariant Subspaces, North-Holland Math. Library 42, North-Holland, 1988.

[00004] [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. | Zbl 0831.30024

[00005] [6] J. P. Bès, Hereditary hypercyclic operators and the hypercyclicity criterion, preprint.

[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] P. S. Bourdon, Invariant manifolds of hypercyclic operators, Proc. Amer. Math. Soc. 118 (1993), 1577-1581. | Zbl 0853.54036

[00008] [9] P. S. Bourdon, Orbits of hyponormal operators, Michigan Math. J. 44 (1997), 345-353. | Zbl 0896.47020

[00009] [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. | Zbl 0771.47015

[00010] [11] J. B. Conway, A Course in Functional Analysis, 2nd ed., Springer, New York, 1990. | Zbl 0706.46003

[00011] [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. | Zbl 0618.30031

[00012] [13] G. Godefroy and J. H. Shapiro, Operators with dense, invariant, cyclic vector manifolds, J. Funct. Anal. 98 (1991), 239-269. | Zbl 0732.47016

[00013] [14] K.-G. Große-Erdmann, Holomorphe Monster und universelle Funktionen, Mitt. Math. Sem. Giessen 176 (1987), 1-84.

[00014] [15] D. A. Herrero, Limits of hypercyclic and supercyclic operators, J. Funct. Anal. 99 (1991), 179-190. | Zbl 0758.47016

[00015] [16] D. A. Herrero, Hypercyclic operators and chaos, J. Operator Theory 28 (1992), 93-103. | Zbl 0806.47020

[00016] [17] D. A. Herrero and C. Kitai, On invertible hypercyclic operators, Proc. Amer. Math. Soc. 116 (1992), 873-875. | Zbl 0780.47006

[00017] [18] 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

[00018] [19] G. Herzog, On linear operators having supercyclic vectors, Studia Math. 103 (1992), 295-298. | Zbl 0811.47018

[00019] [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. | Zbl 0274.47004

[00020] [21] C. Kitai, Invariant closed sets for linear operators, thesis, Univ. of Toronto, 1982.

[00021] [22] F. León-Saavedra and A. Montes-Rodríguez, Linear structure of hypercyclic vectors, J. Funct. Anal. 148 (1997), 524-545. | Zbl 0999.47009

[00022] [23] F. León-Saavedra and A. Montes-Rodríguez, Spectral theory and hypercyclic subspaces, Trans. Amer. Math. Soc., to appear. | Zbl 0961.47003

[00023] [24] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I, Springer, Berlin, 1977. | Zbl 0362.46013

[00024] [25] V. G. Miller, Remarks on finitely hypercyclic and finitely supercyclic operators, Integral Equations Operator Theory 29 (1997), 110-115. | Zbl 0902.47018

[00025] [26] A. Montes-Rodríguez, Banach spaces of hypercyclic vectors, Michigan Math. J. 43 (1996), 419-436. | Zbl 0907.47023

[00026] [27] A. Montes-Rodríguez and H. N. Salas, Supercyclic operators, in preparation.

[00027] [28] D. P. O'Donovan, Weighted shifts and covariance algebras, Trans. Amer. Math. Soc. 208 (1975), 1-25. | Zbl 0308.46050

[00028] [29] C. J. Read, The invariant subspace problem for a class of Banach spaces, 2: Hypercyclic operators, Israel J. Math. 63 (1988), 1-40. | Zbl 0782.47002

[00029] [30] S. Rolewicz, On orbits of elements, Studia Math. 32 (1969), 17-22. | Zbl 0174.44203

[00030] [31] H. N. Salas, Semigroup of isometries with commuting range projections, J. Operator Theory 14 (1985), 331-346. | Zbl 0613.47041

[00031] [32] H. N. Salas, A hypercyclic operator whose adjoint is also hypercyclic, Proc. Amer. Math. Soc. 112 (1991), 765-770. | Zbl 0748.47023

[00032] [33] H. N. Salas, Hypercyclic weighted shifts, Trans. Amer. Math. Soc. 347 (1995), 993-1004. | Zbl 0822.47030

[00033] [34] J. H. Shapiro, Composition Operators and Classical Function Theory, Springer, New York, 1993. | Zbl 0791.30033

[00034] [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.

[00035] [36] J. Zemánek, The semi-Fredholm radius of a linear operator, Bull. Polish Acad. Sci. Math. 32 (1984), 67-76. | Zbl 0583.47016