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

Artur Blois, Ben-Hur Eidt, Paulo Lupatini, Osmar R. Severiano

Abstract

Let $φ$ be a holomorphic self-map of the open unit disk $\mathbb{D}.$ In this article, we study the shadowing phenomenon for composition operators $C_φf=f\circ φ$ 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.

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

Abstract

Let be a holomorphic self-map of the open unit disk In this article, we study the shadowing phenomenon for composition operators on the Hardy space We mainly characterize all the composition operators induced by linear fractional self-maps of that have the positive shadowing property.
Paper Structure (10 sections, 13 theorems, 46 equations, 1 table)

This paper contains 10 sections, 13 theorems, 46 equations, 1 table.

Table of Contents

  1. Introduction
  2. notations and background
  3. Shadowing in linear dynamics
  4. Linear fractional composition operators and canonical forms
  5. Shadowing property on
  6. EA, HNA II and LOX cases
  7. PA and PNA cases
  8. HA case
  9. HNA I case
  10. Comments on the shadowing property on

Key Result

Proposition 2.3

If $T_1\in \mathcal{B}(X_1)$ and $T_2\in \mathcal{B}(X_2)$ are similar, then $T_1$ has positive shadowing property if and only if so does $T_2.$

Theorems & Definitions (26)

  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Definition 2.4
  • Remark 2.5
  • Theorem 2.6
  • Proposition 3.1
  • proof
  • Corollary 3.2
  • ...and 16 more