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In Linear Dynamics, the most studied class of linear operators is certainly that of weighted shifts, on the separable Banach spaces c0and ℓp, 1≤p<∞. Over the last decades, the intensive study of such operators has produced an incredible number of versatile, deep and beautiful results that are applicable in various areas of Mathematics; and the relationships between various important notions, especially concerning chaos and hyperbolic properties, as well as spectrum of weighted shifts, have been investigated. In this paper, we investigate the point spectrum of weighted shifts and, under some regularity hypotheses on the weight sequence, we deduce the spectrum.

In this note, we investigate the two notions of expansivity and strong structural stability for composition operators on Lpspaces, 1≤p<∞. Necessary and sufficient conditions for such operators to be expansive are provided, both in the general and the dissipative case. We also show that, in the dissipative setting, the shadowing property implies the strong structural stability and we prove that these two notions are equivalent under the extra hypothesis of positive expansivity.