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 T 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 T is sensitive dependence iff sup { ‖ T n ‖ : n ∈ N } = ∞ . Finally, the following results are obtained: (1) T is chaotic iff T n is chaotic for each n ∈ N . (2) The product operator T n ∗ = ∏ i = 1 n T i is chaotic iff T k is chaotic for some k ∈ { 1 , 2 , … , n } .