Orbits of the backward shifts with limit points

Abstract

We show that the bilateral backward shift on ℓp(Z,ω) that has a projective orbit with a non-zero limit point is supercyclic. This phenomenon holds also for Γ-supercyclicity, which extends a result obtained for the first time by Chan and Seceleanu. Moreover, we show that if K is a compact subset of ℓp(N,ω) such that its orbit under the unilateral backward shift B on ℓp(N,ω) has a non-zero weak limit point, then B is hypercyclic. Similar results for translation semigroups on weighted Lebesgue spaces are obtained.