Abstract
In this paper, various notions of chaos for continuous linear operators on Fréchet spaces are investigated. It is shown that an operator is Li–Yorke chaotic if and only if it is mean Li–Yorke chaotic in a sequence whose upper density equals one; that an operator is mean Li–Yorke chaotic if and only if it admits a mean Li–Yorke pair, if and only if it is distributionally chaotic of type 2, if and only if it has an absolutely mean irregular vector. As a consequence, mean Li–Yorke chaos is not conjugacy invariant for continuous self-maps acting on complete metric spaces. Moreover, the existence of invariant scrambled sets (with respect to certain Furstenberg families) of a class of weighted shift operators is proved.
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