Abstract
On a separable, infinite dimensional Banach space X, a bounded linear operator T : X → X is said to be hypercyclic, if there exists a vector x in X such that its orbit Orb(T, x) = {x, Tx, T 2 x, …} is dense in X. In a recent paper (Chan and Seceleanu in J Oper Theory 67:257–277, 2012), it was shown that if a unilateral weighted backward shift has an orbit with a single non-zero limit point, then it possesses a dense orbit, and hence the shift is hypercyclic. However, the orbit with the non-zero limit point may not be dense, and so the vector x inducing the orbit need not be hypercyclic. Motivated by this result, we provide conditions for x to be a cyclic vector.
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References
Bourdon P.S., Feldman N.S.: Somewhere dense orbits are everywhere dense. Indiana Univ. Math. J. 52, 811–819 (2003)
Chan K.C., Seceleanu I.: Hypercyclicity of shifts as a zero-one law of orbital limit points. J. Oper. Theory 67, 257–277 (2012)
Chan K.C., Seceleanu I.: Orbital limit points and hypercyclicity of operators on analytic function spaces. Math. Proc. R. Ir. Acad. 110, 99–109 (2010)
Salas H.: Hypercyclic weighted shifts. Trans. Am. Math. Soc. 347(3), 993–1004 (1995)
Shields A.: Weighted Shift Operators and Analytic Function Theory. In: Topics in Operator Theory, pp. 149–128. Mathematical Surveys Number 13, American Mathematical Society (1974)
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Chan, K., Seceleanu, I. Cyclicity of Vectors with Orbital Limit Points for Backward Shifts. Integr. Equ. Oper. Theory 78, 225–232 (2014). https://doi.org/10.1007/s00020-013-2100-2
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DOI: https://doi.org/10.1007/s00020-013-2100-2