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Supercyclicity and weighted shifts

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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.
Opis fizyczny
  • Departamento de Matemáticas, Universidad de Puerto Rico, Mayagüez, Puerto Rico 00681
  • [1] S. I. Ansari, Existence of hypercyclic operators on topological vector spaces, J. Funct. Anal. 148 (1997), 384-390.
  • [2] S. I. Ansari and P. S. Bourdon, Some properties of cyclic operators, Acta Sci. Math. (Szeged) 63 (1997), 195-207.
  • [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.
  • [4] B. Beauzamy, Introduction to Operator Theory and Invariant Subspaces, North-Holland Math. Library 42, North-Holland, 1988.
  • [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.
  • [6] J. P. Bès, Hereditary hypercyclic operators and the hypercyclicity criterion, preprint.
  • [7] G. D. Birkhoff, Démonstration d'un théorème élémentaire sur les fonctions entières, C. R. Acad. Sci. Paris 189 (1929), 473-475.
  • [8] P. S. Bourdon, Invariant manifolds of hypercyclic operators, Proc. Amer. Math. Soc. 118 (1993), 1577-1581.
  • [9] P. S. Bourdon, Orbits of hyponormal operators, Michigan Math. J. 44 (1997), 345-353.
  • [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.
  • [11] J. B. Conway, A Course in Functional Analysis, 2nd ed., Springer, New York, 1990.
  • [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.
  • [13] G. Godefroy and J. H. Shapiro, Operators with dense, invariant, cyclic vector manifolds, J. Funct. Anal. 98 (1991), 239-269.
  • [14] K.-G. Große-Erdmann, Holomorphe Monster und universelle Funktionen, Mitt. Math. Sem. Giessen 176 (1987), 1-84.
  • [15] D. A. Herrero, Limits of hypercyclic and supercyclic operators, J. Funct. Anal. 99 (1991), 179-190.
  • [16] D. A. Herrero, Hypercyclic operators and chaos, J. Operator Theory 28 (1992), 93-103.
  • [17] D. A. Herrero and C. Kitai, On invertible hypercyclic operators, Proc. Amer. Math. Soc. 116 (1992), 873-875.
  • [18] D. A. Herrero and Z. Y. Wang, Compact perturbations of hypercyclic and supercyclic operators, Indiana Univ. Math. J. 39 (1990), 819-829.
  • [19] G. Herzog, On linear operators having supercyclic vectors, Studia Math. 103 (1992), 295-298.
  • [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.
  • [21] C. Kitai, Invariant closed sets for linear operators, thesis, Univ. of Toronto, 1982.
  • [22] F. León-Saavedra and A. Montes-Rodríguez, Linear structure of hypercyclic vectors, J. Funct. Anal. 148 (1997), 524-545.
  • [23] F. León-Saavedra and A. Montes-Rodríguez, Spectral theory and hypercyclic subspaces, Trans. Amer. Math. Soc., to appear.
  • [24] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I, Springer, Berlin, 1977.
  • [25] V. G. Miller, Remarks on finitely hypercyclic and finitely supercyclic operators, Integral Equations Operator Theory 29 (1997), 110-115.
  • [26] A. Montes-Rodríguez, Banach spaces of hypercyclic vectors, Michigan Math. J. 43 (1996), 419-436.
  • [27] A. Montes-Rodríguez and H. N. Salas, Supercyclic operators, in preparation.
  • [28] D. P. O'Donovan, Weighted shifts and covariance algebras, Trans. Amer. Math. Soc. 208 (1975), 1-25.
  • [29] C. J. Read, The invariant subspace problem for a class of Banach spaces, 2: Hypercyclic operators, Israel J. Math. 63 (1988), 1-40.
  • [30] S. Rolewicz, On orbits of elements, Studia Math. 32 (1969), 17-22.
  • [31] H. N. Salas, Semigroup of isometries with commuting range projections, J. Operator Theory 14 (1985), 331-346.
  • [32] H. N. Salas, A hypercyclic operator whose adjoint is also hypercyclic, Proc. Amer. Math. Soc. 112 (1991), 765-770.
  • [33] H. N. Salas, Hypercyclic weighted shifts, Trans. Amer. Math. Soc. 347 (1995), 993-1004.
  • [34] J. H. Shapiro, Composition Operators and Classical Function Theory, Springer, New York, 1993.
  • [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] J. Zemánek, The semi-Fredholm radius of a linear operator, Bull. Polish Acad. Sci. Math. 32 (1984), 67-76.
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