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

Costakis G, Manoussos A, Nasseri AB. Dynamics of perturbations of the identity operator by multiples of the backward shift on $l^{\infty}(\mathbb{N})$. 2013.Let $B$, $I$ be the unweighted backward shift and the identity operatorrespectively on $l^{\infty}(\mathbb{N})$, the space of bounded sequences overthe complex numbers endowed with the supremum norm. We prove that $I+\lambda B$is locally topologically transitive if and only if $|\lambda |>2$. This, showsthat a classical result of Salas, which says that backward shift perturbationsof the identity operator are always hypercyclic, or equivalently topologicallytransitive, on $l^p(\mathbb{N})$, $1\leq p<+\infty$, fails to hold for thenotion of local topological transitivity on $l^{\infty}(\mathbb{N})$. We alsoobtain further results which complement certain results from \cite{CosMa}