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

Antoni López-Martínez

TL;DR

The paper addresses the problem of identifying when composition operators $C_{\\varphi}$ on the little Lipschitz space $\\mathcal{L}_0(\\mathbb{N}_0)$ are frequently hypercyclic. The core approach is to establish an isometric isomorphism $D$ between the Lipschitz-type spaces on rooted trees and classical sequence spaces $\\ell^{\\infty}(T)$ and $c_0(T)$, translating operators into their $\\ell^{\\infty}$ counterparts. This framework yields sharp norm formulas and a conjugacy for composition, multiplication, and backward-shift operators, enabling precise dynamical analysis. The main result characterizes strictly increasing symbols $\\varphi:\\mathbb{N}_0\\to\\mathbb{N}_0$ for which $C_{\\varphi}$ satisfies the Frequent Hypercyclicity Criterion on $\\mathcal{L}_0(\\mathbb{N}_0)$, showing equivalence with hypercyclicity and specific distance-growth conditions. Overall, the isomorphism-based method simplifies existing results, offers a complete dynamical picture in the studied setting, and suggests broader extensions to other rooted trees and dynamical notions.

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.

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

TL;DR

The paper addresses the problem of identifying when composition operators on the little Lipschitz space are frequently hypercyclic. The core approach is to establish an isometric isomorphism between the Lipschitz-type spaces on rooted trees and classical sequence spaces and , translating operators into their counterparts. This framework yields sharp norm formulas and a conjugacy for composition, multiplication, and backward-shift operators, enabling precise dynamical analysis. The main result characterizes strictly increasing symbols for which satisfies the Frequent Hypercyclicity Criterion on , showing equivalence with hypercyclicity and specific distance-growth conditions. Overall, the isomorphism-based method simplifies existing results, offers a complete dynamical picture in the studied setting, and suggests broader extensions to other rooted trees and dynamical notions.

Abstract

We characterize the strictly increasing symbols whose composition operators~ satisfy the Frequent Hypercyclicity Criterion on the little Lipschitz space . 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 and . Such isomorphism provides an alternative framework that simplifies and allows to improve many previous results about these spaces and the operators defined there.
Paper Structure (6 sections, 4 theorems, 65 equations)

This paper contains 6 sections, 4 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Notation, a useful isomorphism, and some consequences
  3. Composition operators
  4. Multiplication and the backward shift operators
  5. Frequently hypercyclic composition operators on $\mathcal{L}_0(T)$
  6. Conclusions and future lines of research

Key Result

Corollary 2.3

Let $\varphi$ be a self-map of a rooted tree $T$. If the composition operator $C_{\varphi}$ is bounded in the space $\mathcal{L}(T)$, then its operator norm fulfills the equality $\|C_{\varphi}\|_{\mathcal{L}} = \max\{ 1+|\varphi(\mathfrak{o})| , L_{\varphi} \}$, where $L_{\varphi}$ is the Lipschitz

Theorems & Definitions (17)

  • Remark 2.1: A useful isomorphism
  • Remark 2.2: A useful conjugacy
  • Corollary 2.3
  • proof
  • Example 2.4
  • Remark 2.5: Multiplication operators
  • Corollary 2.6
  • proof
  • Remark 2.7: The backward shift
  • Corollary 2.8
  • ...and 7 more