Dense Lineable Criterion for Linear Dynamics

TL;DR

This paper introduces a new dense lineable criterion for linear dynamics, demonstrating the existence and properties of certain operator sequences in infinite-dimensional spaces, and addressing a question in the field.

Contribution

It develops the D-phenomenon to unify properties like recurrence and chaos, and constructs specific operator sequences with unique lineability characteristics.

Findings

Existence of operator sequences with dense irregular vectors but no dense irregular manifold

Recurrent operator with non-dense-lineable recurrent vectors

Negative resolution to a question by Grivaux et al.

Abstract

We study Li-Yorke chaos for sequences of continuous linear operators from an \(F\)-space to a normed space. We introduce the \emph{D-phenomenon} to establish a common dense lineable criterion that encompasses properties such as recurrence, universality, and Li-Yorke chaos. We show that in every infinite-dimensional separable complex Banach space, there exists a sequence of operators with a dense set of irregular vectors but without a dense irregular manifold, and we exhibit a recurrent operator whose set of recurrent vectors is not dense-lineable. This resolves in the negative a question posed by Grivaux et al.