Some dynamical properties for linear operators

TL;DR

This paper explores the properties of a new dynamical concept called norm-unimodality for linear operators, its invariance, spectral characteristics, and implications for chaos, including applications in nest algebras and perturbations.

Contribution

It introduces and analyzes the invariance and spectral properties of norm-unimodality, and demonstrates the existence of distributionally chaotic operators in various classes.

Findings

Norm-unimodality is similar invariant.

Spectra of norm-unimodal operators are characterized.

Distributional chaos exists in nest algebras and under small perturbations.

Abstract

In our another recent article, we introduce a new dynamical property for linear operators called norm-unimodality which implies distributional chaos. In the present paper, we'll give a further discussion of norm-unimodality. It is showed that norm-unimodality is similar invariant and the spectra of norm-unimodal operator is referred to. As an application, in each nest algebra there exist distributional chaotic operators. Moreover, normal operators and compact operators with regard to norm-unimodality and Li-Yorke chaos are also be considered. Specially, a small compact perturbation of the unit operator could be distributionally chaotic.