Dynamics of perturbations of the identity operator by multiples of the backward shift on l^∞(ℕ)

2013/02/07 by George Costakis, Costakis, George, Antonios Manoussos +3
Mathematics · #Advanced Banach Space Theory #Holomorphic and Operator Theory #advanced mathematical theories #math.FA #msc:47A16

paper · pdf · doi:10.48550/arxiv.1302.1736

12 pages

arxiv created 2013/02/07 · arxiv updated 2013/02/08

Abstract

Let B, I be the unweighted backward shift and the identity operator respectively on l^∞(ℕ), the space of bounded sequences over the complex numbers endowed with the supremum norm. We prove that I+λB is locally topologically transitive if and only if |λ|>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(ℕ), 1≤ p<+∞, fails to hold for the notion of local topological transitivity on l^∞(ℕ). We also obtain further results which complement certain results from \citeCosMa.

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