Shadowing phenomenon for composition operators on the Hardy space $H^2(\mathbb{D})$

TL;DR

This paper investigates the shadowing phenomenon for composition operators on the Hardy space, specifically characterizing those induced by linear fractional self-maps that exhibit positive shadowing.

Contribution

It provides a complete characterization of linear fractional self-maps of the disk that induce composition operators with the positive shadowing property.

Findings

Characterization of linear fractional self-maps with positive shadowing

Analysis of shadowing phenomenon for composition operators

Insights into the structure of operators with shadowing property

Abstract

Let $\phi$ be a holomorphic self-map of the open unit disk $\mathbb{D}.$ In this article, we study the shadowing phenomenon for composition operators $C_{\phi}f=f\circ \phi$ on the Hardy space $H^2(\mathbb{D}).$ We mainly characterize all the composition operators induced by linear fractional self-maps of $\mathbb{D}$ that have the positive shadowing property.