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

TL;DR

This paper characterizes when certain composition operators on the little Lipschitz space over rooted trees are frequently hypercyclic, using an isomorphism to classical sequence spaces to simplify analysis.

Contribution

It provides a characterization of frequently hypercyclic composition operators on Lipschitz spaces over rooted trees, leveraging an isomorphism to classical sequence spaces.

Findings

Characterization of strictly increasing symbols for frequent hypercyclicity

Establishment of an isomorphism between Lipschitz spaces over trees and classical sequence spaces

Simplification and improvement of previous results using the isomorphism

Abstract

We characterize the strictly increasing symbols $\varphi:\mathbb{N}_0\longrightarrow\mathbb{N}_0$ whose composition operators $C_{\varphi}$ satisfy the Frequent Hypercyclicity Criterion on the little Lipschitz space $\mathcal{L}_0(\mathbb{N}_0)$. With this result we continue the recent research about this kind of spaces and operators, but our approach relies on establishing a natural isomorphism between the Lipschitz-type spaces over rooted trees and the classical spaces $\ell^{\infty}$ and $c_0$. Such isomorphism provides an alternative framework that simplifies and allows to improve many previous results about these spaces and the operators defined there.