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

TL;DR

This paper investigates the local topological transitivity of perturbations of the identity operator by the backward shift on the space of bounded sequences, revealing conditions under which these operators exhibit this property.

Contribution

It establishes a precise criterion for local topological transitivity of I + λB on l^∞(N), extending and contrasting classical results on hypercyclicity.

Findings

I + λB is locally topologically transitive if and only if |λ| > 2

Classical hypercyclicity results do not extend to local topological transitivity on l^∞(N)

Provides new insights into perturbations of the identity operator on bounded sequence spaces

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}.