LAUSR

We study several notions of chaos for hyperspace dynamics associated to continuous linear operators. More precisely, we consider a continuous linear operator $$T : X \rightarrow X$$T:X→X on a topological vector space X, and the natural hyperspace extensions $$\overline{T}$$T¯ and $$\widetilde{T}$$T~ of T to the spaces $$\mathcal {K}(X)$$K(X) of compact subsets of X and $$\mathcal {C}(X)$$C(X) of convex compact subsets of X, respectively, endowed with the Vietoris topology. We show that, when X is a complete locally convex space (respectively, a locally convex space), then Devaney chaos (respectively, topological ergodicity) is equivalent for the maps T, $$\overline{T}$$T¯ and $$\widetilde{T}$$T~. Also, under very general conditions, we obtain analogous equivalences for Li-Yorke chaos. Finally, some remarks concerning the topological transitivity and weak mixing properties are included, extending results in Banks (Chaos Solitons Fractals 25(3):681–685, 2005) and Peris (Chaos Solitons Fractals 26(1):19–23, 2005).