Beurling type invariant subspaces of composition operators

Let $\mathbb{D}$ be the open unit disk in $\mathbb{C}$, let $H^2$ denote the Hardy space on $\mathbb{D}$ and let $\varphi : \mathbb{D} \rightarrow \mathbb{D}$ be a holomorphic self map of $\mathbb{D}$. The composition operator $C_{\varphi}$ on $H^2$ is defined by $$(C_{\varphi} f)(z)=f(\varphi(z)) \quad \quad (f \in H^2,\, z \in \mathbb{D}).$$ Denote by $\mathcal{S}(\mathbb{D})$ the set of all functions that are holomorphic and bounded by one in modulus on $\mathbb{D}$, that is $$\mathcal{S}(\mathbb{D}) = \{\psi \in H^\infty(\mathbb{D}): \|\psi\|_{\infty} := \sup_{z \in \mathbb{D}} |\psi(z)| \leq 1\}.$$ The elements of $\mathcal{S}(\mathbb{D})$ are called Schur functions. The aim of this paper is to answer the following question concerning invariant subspaces of composition operators: Characterize $\varphi$, holomorphic self maps of $\mathbb{D}$, and inner functions $\theta \in H^\infty(\mathbb{D})$ such that the Beurling type invariant subspace $\theta H^2$ is an invariant subspace for $C_{\varphi}$. We prove the following result: $C_{\varphi} (\theta H^2) \subseteq \theta H^2$ if and only if $$\frac{\theta \circ \varphi}{\theta} \in \mathcal{S}(\mathbb{D}).$$ This classification also allows us to recover or improve some known results on Beurling type invariant subspaces of composition operators.