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

2019/06/06 by Rivera-Guasco, Emmanuel, Martínez-Avendaño, Rubén A. · 1 citation
#05C05 #05C63 #47A16 #47B37 #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1906.02372

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 L₀. We determine that the forward shift is bounded both on \mathcal L and on \mathcal L₀ 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 L₀, 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.

Cited by

• Frequently hypercyclic composition operators on the little Lipschitz space of a rooted tree

Related

• The Forward and Backward Shift on the Hardy Space of a Tree
• Spectral rigidity for primitive elements of F_N
• Composition Operators on the Little Lipschitz space of a rooted tree
• Tree metrics and their Lipschitz-free spaces
• Lipschitz Functions on Quasiconformal Trees