Li-Yorke chaos translation set for linear operators

TL;DR

This paper introduces the Li-Yorke chaos translation set for bounded linear operators on Banach spaces, analyzing its properties for various operator classes and demonstrating that for the Kalisch operator on a Hilbert space, it is a singleton set.

Contribution

It defines the Li-Yorke chaos translation set for linear operators and investigates its structure for different classes, including a specific example with the Kalisch operator.

Findings

Li-Yorke chaos translation set is a singleton for Kalisch operator

The set is characterized for normal and compact operators

The set's structure varies across different operator classes

Abstract

In order to study Li-Yorke chaos by the scalar perturbation for a given bounded linear operator $T$ on Banach spaces $X$, we introduce the Li-Yorke chaos translation set of $T$, which is defined by $S_{LY}(T)=\{\lambda\in\mathbb{C};\lambda+T \text{ is Li-Yorke chaotic}\}$. In this paper, some operator classes are considered, such as normal operator, compact operator, shift and so on. In particular, we show that the Li-Yorke chaos translation set of Kalisch operator on Hilbert space $\mathcal{L}^2[0,2\pi]$ is a simple point set $\{0\}$.