Remarks on multiples of distributionally chaotic operators
Volume 243 / 2018
Abstract
This paper investigates distributional chaos and the existence of common distributionally irregular vectors for multiples of linear operators on Banach spaces. We focus on the topological property of the set $DC_T : = \{\lambda \gt 0: \lambda T$ is distributionally chaotic$\}$ for a given operator $T$. For any open set $U \subset (0, \infty )$ which is bounded away from zero, we prove that there is a bounded operator $T$ on $l^{p} (1\leq p \lt \infty )$ such that $U= DC_T$. As a consequence, there exists an operator $T_1$ such that $T_1$ and $3 T_1$ are distributionally chaotic but $2 T_1$ is not. We also construct an invertible operator $T$ such that $DC_T$ is a singleton. Furthermore, sufficient conditions for the existence of common distributionally irregular vectors for the family of operators $\{\lambda T : \lambda \in DC_T\}$ are provided.
