On the equivalence of four chaotic operators

In this paper, we study chaos for bounded operators on Banach spaces. First, it is proved that, for a bounded operator TT defined on a Banach space, Li–Yorke chaos, Li–Yorke sensitivity, spatio-temporal chaos, and distributional chaos in a sequence are equivalent, and they are all strictly stronger than sensitivity. Next, we show that TT is sensitive dependence iff sup{‖Tn‖:n∈N}=∞sup{‖Tn‖:n∈N}=∞. Finally, the following results are obtained: (1) TT is chaotic iff TnTn is chaotic for each n∈Nn∈N. (2) The product operator Tn∗=∏i=1nTi is chaotic iff TkTk is chaotic for some k∈{1,2,…,n}k∈{1,2,…,n}.