Asymptotic behaviour and cyclic properties of weighted shifts on directed trees

In this paper we investigate a new class of bounded operators called weighted shifts on directed trees introduced recently in [11]. This class is a natural generalization of the so called weighted bilateral, unilateral and backward shift operators. In the first part of the paper we calculate the asymptotic limit and the isometric asymptote of a contractive weighted shift on a directed tree and that of the adjoint. Then we use the asymptotic behaviour and similarity properties in order to obtain cyclicity results. We also show that a weighted backward shift operator is cyclic if and only if there is at most one zero weight.