Composition operators between model and Hardy spaces

TL;DR

This paper characterizes when composition operators between model spaces and Hardy spaces on polydisks are compact, extending understanding of operator behavior in complex function spaces.

Contribution

It provides new characterizations of compact composition operators from model spaces to Hardy spaces in multiple complex variables.

Findings

Characterizations of compactness for composition operators between specific function spaces.

Extension of known results to higher-dimensional polydisks.

Insights into the structure of model spaces and their operators.

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 $\Theta: \mathbb{D} \to \mathbb{D}$ be an inner function and $K^p_\Theta$, $p>0$, denote the corresponding model space. We obtain characterizations of the compact composition operators $C_\varphi: K^p_\Theta \to H^p(\mathbb{D}^n)$, $1<p<\infty$, where $H^p(\mathbb{D}^n)$ denotes the Hardy space.