Dynamics of perturbations of the identity operator by multiples of the backward shift on $l^{\infty}(\mathbb{N})$

Abstract

Let $B$, $I$ be the unweighted backward shift and the identity operator respectively on $l^{\infty}(\mathbb{N})$, the space of bounded sequences over the complex numbers endowed with the supremum norm. We prove that $I+\lambda B$ is locally topologically transitive if and only if $|\lambda |>2$. This, shows that a classical result of Salas, which says that backward shift perturbations of the identity operator are always hypercyclic, or equivalently topologically transitive, on $l^p(\mathbb{N})$, $1\leq p<+\infty$, fails to hold for the notion of local topological transitivity on $l^{\infty}(\mathbb{N})$. We also obtain further results which complement certain results from \cite{CosMa}.