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Invariant subspace of composition operators on Hardy space

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