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 $\varphi$ 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 $\parallel\varphi\parallel_\infty<1$.