Invariant subspace of composition operators on Hardy space

Abstract

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

23 pages