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 a tree, L, and on the little Lipshitz space of a tree, L 0 . We determine that the forward shift is bounded both on L and on 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 L and on 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.

Dynamics of Weighted Backward Shifts on the Little Lipschitzspace

“…In [10,Theorem 7.7], the authors obtained that bounded backward shift on L 0 is hypercyclic if and only if T has no free ends. Therefore, we studied the hypercyclicity of weighted shifts on L 0 where T has no free ends.…”

“…At present, there are also some scholars studying shift operators on an infinite tree. For instance, in [10] and [12], the authors studied the norm of the backward shift on the Lipschitz space of a tree. In particular, in [10], the authors also discussed the spectral properties of the backward shift on the Lipschitz space of a tree and the hypercyclity of the backward shift on the little Lipschitz space of a tree, which is a separable subspace of the Lipschitz space.…”

“…For instance, in [10] and [12], the authors studied the norm of the backward shift on the Lipschitz space of a tree. In particular, in [10], the authors also discussed the spectral properties of the backward shift on the Lipschitz space of a tree and the hypercyclity of the backward shift on the little Lipschitz space of a tree, which is a separable subspace of the Lipschitz space. For the hypercyclicity of shift operators on weighted L p spaces of directed tree we can refer to [9].…”

“…Moreover, in [13], Grosse-Erdmann and Papathanasiou investigated the dynamical (hypercyclic, weaking mixing, mixing) properties of weighted shifts on sequence spaces of a directed tree. Combining [10] and [13], a natural thought is: What are the dynamical properties of weighted shift operators on the little Lipschitz space of a tree? Motivated by the above thought, we study the dynamical behaviour of weighted backward shifts defined on the little Lipschitz space.…”