Characteristics of (α,β)-Mean Li–Yorke Chaos of Linear Operators on Banach Spaces

Abstract

In this paper, we introduce a new concept, [Formula: see text]-mean Li–Yorke chaotic operator, which includes the standard mean Li–Yorke chaotic operators as special cases. We show that when [Formula: see text] or [Formula: see text], [Formula: see text]-mean Li–Yorke chaotic dynamics is strictly stronger than the ones that appeared in mean Li–Yorke chaos. When [Formula: see text] or [Formula: see text], it has completely different characteristics from the mean Li–Yorke chaos. We prove that no finite-dimensional Banach space can support [Formula: see text]-mean Li–Yorke chaotic operators. Moreover, we show that an operator is [Formula: see text]-mean Li–Yorke chaos if and only if there exists an [Formula: see text]-mean semi-irregular vector for the underlying operator, and if and only if there exists an [Formula: see text]-mean irregular vector when [Formula: see text], which generalizes the recent results by Bernardes et al. given in 2018. When [Formula: see text], we construct a counterexample in which it is an [Formula: see text]-mean Li–Yorke chaotic operator but does not admit an [Formula: see text]-mean irregular vector. In addition, we show that an operator with dense generalized kernel is [Formula: see text]-mean Li–Yorke chaotic if and only if there exists a residual set of [Formula: see text]-mean irregular vectors, and if and only if there exists an [Formula: see text]-mean unbounded orbit.