The forward and backward shift on the Hardy space of a tree

In this paper we initiate the study of the forward and backward shifts on the discrete generalized Hardy space of a tree and the discrete generalized little Hardy space of a tree. In particular, we investigate when these shifts are bounded, find the norm of the shifts if they are bounded, characterize the trees in which they are an isometry, compute the spectrum in some concrete examples, and completely determine when they are hypercyclic.