Cyclicity of Vectors with Orbital Limit Points for Backward Shifts |
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Authors: | Kit Chan Irina Seceleanu |
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Institution: | 1. Department of Mathematics and Statistics, Bowling Green State University, Bowling Green, OH, 43403, USA 2. Department of Mathematics, Bridgewater State University, Bridgewater, MA, 02325, USA
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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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