Distributional chaos for operators on Banach spaces

Four notions of distributional chaos, namely DC1, DC2, DC212 and DC3, are studied within the framework of operators on Banach spaces. It is known that, for general dynamical systems, DC1 ⊂ DC2 ⊂ DC212 ⊂ DC3. We show that DC1 and DC2 coincide in our context, which answers a natural question. In contrast, there exist DC212 operators which are not DC2. Under the condition that there exists a dense set X0⊂X such that Tnx→0 for any x∈X0, DC3 operators are shown to be DC1. Moreover, we prove that any upper-frequently hypercyclic operator is DC212. Finally, several examples are provided to distinguish between different notions of distributional chaos, Li-Yorke chaos and irregularity.