Invariance of chaos from backward shift on the Köthe sequence space

Abstract

In this paper it is proved that the backward shift operator on the Köthe sequence space admits a pair which is not asymptotic, if and only if it has an uncountable invariant -scrambled set for some > 0, if and only if it has an -scrambled subspace for some > 0, if and only if it has an invariant scrambled linear manifold. An analogous result for distributional chaos of type 2 is also obtained.

Some remarks on distributional chaos for bounded linear operators

“…Wu et al discussed an invariant DC2-scrambled linear manifold for backward shift on Köthe sequence space in [19]. To describe distributional chaos more precisely, we will give a necessary condition for DC2.…”

Reiterative $m_{n}$-distributional chaos of type $s$ in Fr\' echet spaces

“…In particular, any m n -distributionally chaotic sequence is distributionally chaotic. (iii) It is worth noting that the distributional chaos of type s ∈ {1, 2, 3} for backward shift operators in Köthe sequence spaces has been analyzed for the first time by X. Wu et al [33], under certain assumptions other from ours.…”

“…Finally, we will use the notion of Li-Yorke chaos below. Li-Yorke chaos in Fréchet spaces has been recently investigated by N. C. Bernardes Jr et al [8] and M. Kostić [24] (see also [5], [9] and [33]- [35]): Definition 2.5. We say that the sequence (T j ) j∈N is X-Li-Yorke chaotic iff there exists an uncountable set S ⊆ j∈N D(T j ) X such that for every pair (x, y) ∈ S × S of distinct points, we have lim inf j→∞ d Y T j x, T j y = 0 and lim sup j→∞ d Y T j x, T j y > 0.…”

Distributional Chaoticity of C0-Semigroup on a Frechet Space

“…Wu and Zhu [15] further proved that the principal measure of the annihilation operator studied in Reference [14] is 1. Since then, distributional chaos for linear operators has been studied by many authors, see for instance References [16][17][18][19][20][21].…”