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Composition operators between model and Hardy spaces

Evgueni Doubtsov

Abstract

Let $n\ge 1$ and $\varphi: \mathbb{D}^n\to\mathbb{D}$ be a holomorphic function, where $\mathbb{D}$ denotes the open unit disk of $\mathbb{C}$. Let $Θ: \mathbb{D} \to \mathbb{D}$ be an inner function and $K^p_Θ$, $p>0$, denote the corresponding model space. We obtain characterizations of the compact composition operators $C_\varphi: K^p_Θ\to H^p(\mathbb{D}^n)$, $1<p<\infty$, where $H^p(\mathbb{D}^n)$ denotes the Hardy space.

Composition operators between model and Hardy spaces

Abstract

Let and be a holomorphic function, where denotes the open unit disk of . Let be an inner function and , , denote the corresponding model space. We obtain characterizations of the compact composition operators , , where denotes the Hardy space.

Paper Structure

This paper contains 15 sections, 11 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. Composition operators
  3. Model spaces
  4. Composition operators on model spaces
  5. Auxiliary results
  6. Littlewood--Paley identity and its generalizations
  7. Subharmonic property for the Nevanlinna counting function
  8. Reproducing kernels for $K_\Theta$
  9. Compact composition operators on model spaces
  10. Compact composition operators on $K_\Theta$
  11. Compact composition operators on $K_\Theta^p$, $p>1$
  12. One-component inner functions and Clark measures
  13. Spectrum of a one-component inner function
  14. Clark measures
  15. Compact composition operators and Clark measures

Key Result

Theorem 2.1

Let $\phi: \mathbb D\to\mathbb D$ be a holomorphic function. Then $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (17)

  • Definition 1.1
  • Theorem 2.1: ShJoel87
  • Corollary 2.2
  • proof
  • Proposition 2.3: ShJoel87
  • Corollary 2.4
  • proof
  • Lemma 2.5: LM13
  • Theorem 3.1
  • proof : About the proof
  • ...and 7 more