Linear operators with wild dynamics

Abstract

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