Disjoint distributional chaos in Fréchet spaces

Abstract

We introduce several different notions of disjoint distributional chaos for sequences of multivalued linear operators in Fréchet spaces. Any of these notions seems to be new and not considered elsewhere even for linear continuous operators in Banach spaces. We focus special attention to the analysis of some specific classes of linear continuous operators having a certain disjoint distributionally chaotic behaviour, providing also a great number of illustrative examples and applications of our abstract theoretical results.

Disjoint Li-Yorke chaos in Fréchet spaces

“…The statements (i) and (ii) are trivial consequences of [25,Proposition 3.9], which slightly extend one of the main results of article [33] by V. Müller and J. Vršovský.…”

“…Arguing as in [25,Example 5.3], we can prove that there exist two distributionally chaotic unilateral backward weighted shifts on the space X := c 0 (N) which cannot be (d, X, n,…”

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

“…A series of elaborate and very plain counterexamples shows that the conclusions established in our previous research study [10] are no longer true for general sequences of linear continuous operators, even on finite-dimensional spaces (for more details about this problematic, see [26]). This also holds for Xmn -reiterative distributional chaos of types 0, 1 + and 2−, which are introduced as follows: Definition 2.3.…”

“…To verify this, we will prove the following proper extension of [7,Theorem 25]; cf. also [5,Remark 21], [26] and Theorem 2.10:…”

“…x n n∈N ∈ X, (4.7) about which we assume that it is not necessarily continuous. Albeit it is without scope of this paper to consider m n -distributionally chaotic properties of unbounded bilateral weighted shift operators (see [26], where we have recently initiated the study of disjoint distributionally chaotic properties of such operators), we will state here only one result regarding this question, which can be deduced with the help of Corollary 4.5(iii) and the proof of [26,Theorem 4.11]: Theorem 4.7. Let X := l p for some 1 ≤ p < ∞ or X := c 0 , and let (m n ) ∈ R. Suppose that there exists a bounded sequence (a n ) n∈N of positive reals such that for each k ∈ N we have…”