Narrow Results

Let 1 ≤ p < ∞. We give the sufficient and necessary condition for cosine operator functions, generated by bilateral weighted shifts on ℓ^p(ℤ), to be chaotic. Moreover, such a cosine operator function is chaotic if, and only if, its weighted shift is chaotic.

This paper investigates the dynamics of bilateral operator-weighted shifts on $L^2({𝒦},\mathbb{Z})$ with a weight sequence of positive diagonal operators on a Hilbert space ${𝒦}$. Necessary and sufficient conditions for the bilateral weighted shifts to be hypercyclic (subspace-hypercyclic, frequently hypercyclic, Devaney chaotic, respectively) are provided. As a consequence, it is shown that for any $G_{\delta }$-set $G$ of positive numbers which is bounded and bounded away from zero, there exists an invertible bilateral operator-weighted shift $T$ such that $G=\{a>0:aT\mathrm{\text{ is recurrent}}\}$. Furthermore, the (hereditary) Cesàro-hypercyclicity of the bilateral weighted shifts is characterized.