Linear operators with wild dynamics

TL;DR

This paper constructs a bounded linear operator on an infinite-dimensional Banach space with wild dynamics, showing the set of points with unbounded orbits can have non-empty interior and not be dense, answering recent open questions.

Contribution

It provides a novel example of an operator with specific dynamical properties, combining identity plus compact operator form and non-dense unbounded orbit set.

Findings

Constructed an operator with non-dense unbounded orbit set

Demonstrated the set of points with unbounded orbits can have non-empty interior

Answered open questions by He1jek and Smith

Abstract

If $X$ is a separable infinite dimensional Banach space, we construct a bounded and linear operator $R$ on $X$ such that $$ A_R=\{x \in X, \|R^tx\| \rightarrow \infty\} $$ is not dense and has non empty interior with the additional property that $R$ can be written $I+K$, where $I$ is the identity and $K$ is a compact operator. This answers two recent questions of H\'ajek and Smith.