Invariant subspace of composition operators on Hardy space

TL;DR

This paper investigates the structure of invariant subspaces of composition operators on Hardy spaces, providing characterizations for Beurling type subspaces and conditions under which certain polynomial subspaces are invariant.

Contribution

It offers new criteria for invariant subspaces of composition operators on Hardy spaces, including Beurling type and polynomial subspaces, under various conditions.

Findings

Invariant subspaces characterized for specific composition operators

Beurling type invariant subspaces identified via inner functions

Polynomial subspaces invariant under certain compact composition operators

Abstract

We consider the invariant subspace of composition operators on Hardy space $H^p$ where the composition operators corresponding to a function $\varphi$ that is a holomorphic self-map of $\mathbb D$. Firstly, we discuss composition operators $C_\varphi$ on subspace $H_{\alpha,\beta}^p$ of Hardy space $H^p$. We will explore the invariant subspaces for $C_\varphi$ in various special cases. Secondly, we consider Beurling type invariant subspace for $C_\varphi$. When $\theta$ is a inner function, we prove that $\theta H^p$ is invariant for $C_\varphi$ if and only if $\displaystyle{\frac{\theta\circ\varphi}{\theta}}$ belongs to $\mathcal S(\mathbb D)$. Thirdly, we obtain that $z^nH^p$ is nontrivial invariant subspace for Deddends algebras $\mathcal D_{C_\varphi}$ when $C_\varphi$ is a compact composition operator and $\varphi$ satisfies that $\varphi(0)=0$ and $\parallel\varphi\parallel_\infty<1$.