Bilateral Upper-Left Shifts on Double Sequence Spaces

Abstract

In this paper, we study the bilateral upper-left shifts $\B$ on the weighted double sequence spaces $\mathcal{L}^{p}(\Z, v)$ and characterize the hypercyclicity and supercyclicity of $\B: \mathcal{L}^{p}(\Z, v)\rightarrow\mathcal{L}^{p}(\Z, v)$
based on Salas's previous results about the bilateral weighted backward shifts acting on $l^{2}({\Z}$). Furthermore, we construct a special weight sequence such that $\B$ and the weighted bilateral upper-left shifts $\B_{w}$ are conjugate, and we generalize the results to $\B_{w}:\mathcal{L}^{p (\Z)\rightarrow\mathcal{L}^{p}(\Z)$ via this conjugacy. Finally, we investigate the chaoticity of the bilateral upper-left shifts.