The Forward and Backward Shift on the Lipschitz Space of a Tree

Abstract

We initiate the study of the forward and backward shifts on the Lipschitz space of an undirected tree, $$\mathcal {L}$$, and on the little Lipschitz space of an undirected tree, $$\mathcal {L}_0$$. We determine that the forward shift is bounded both on $$\mathcal {L}$$ and on $$\mathcal {L}_0$$ and, when the tree is leafless, it is an isometry; we also calculate its spectrum. For the backward shift, we determine when it is bounded on $$\mathcal {L}$$ and on $$\mathcal {L}_0$$, we find the norm when the tree is homogeneous, we calculate the spectrum for the case when the tree is homogeneous, and we determine, for a general tree, when it is hypercyclic.