Weak* Hypercyclicity and Supercyclicity of Shifts on ℓ∞ |
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Authors: | Juan Bès Kit C Chan Rebecca Sanders |
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Institution: | (1) Department of Mathematics and Statistics, Bowling Green State University, Bowling Green, Ohio 43403, USA;(2) Department of Mathematics, Statistics and Computer Sciences, Marquette University, Milwaukee, Wisconsin 53201, USA |
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Abstract: | We study hypercyclicity and supercyclicity of weighted shifts on ℓ∞, with respect to the weak * topology. We show that there exist bilateral shifts that are weak * hypercyclic but fail to be
weak * sequentially hypercyclic. In the unilateral case, a shift T is weak * hypercyclic if and only if it is weak * sequentially hypercyclic, and this is equivalent to T being either norm, weak, or weak-sequentially hypercyclic on c0 or ℓp (1 ≤ p < ∞). We also show that the set of weak * hypercyclic vectors of any unilateral or bilateral shift on ℓ∞ is norm nowhere dense. Finally, we show that ℓ∞ supports an isometry that is weak * sequentially supercyclic. |
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Keywords: | Primary 47A16 46A45 Secondary 46A03 46B26 |
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