Topological transitivity for cosine operator functions on groups

Let G be a locally compact group and let 1≤p<∞. We give a necessary and sufficient condition for a cosine operator function induced by a weighted translation operator on the Lebesgue space Lp(G) to be topologically transitive. By the same argument, the characterization for mixing cosine operator functions is given. We also determine when the direct sum of a sequence of cosine operator functions is transitive.