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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Bès, J., Chan, K.C. & Sanders, R. Weak* Hypercyclicity and Supercyclicity of Shifts on ℓ∞. Integr. equ. oper. theory 55, 363–376 (2006). https://doi.org/10.1007/s00020-005-1394-0
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DOI: https://doi.org/10.1007/s00020-005-1394-0