# The forward and backward shift on the Lipschitz space of a tree

## TL;DR

This paper studies the properties of forward and backward shift operators on Lipschitz spaces of trees, establishing boundedness, isometry conditions, spectra, and hypercyclicity criteria for various types of trees.

## Contribution

It introduces the analysis of shift operators on Lipschitz spaces of trees, providing boundedness, spectral, and hypercyclicity results, which are new for this setting.

## Key findings

- Forward shift is bounded and an isometry on leafless trees.
- Spectrum of the forward shift is computed for leafless trees.
- Backward shift's boundedness, norm, spectrum, and hypercyclicity are characterized.

## Abstract

We initiate the study of the forward and backward shifts on the Lipschitz space of a tree, $\mathcal L$, and on the little Lipshitz space of a 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.