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 $φ: \mathbb{D} \rightarrow \mathbb{D}$ be a holomorphic self map of $\mathbb{D}$. The composition operator $C_φ$ on $H^2$ is defined by $(C_φ f)(z)=f(φ(z)) \quad \quad (f \in H^2,\, z \in \mathbb{D}).$ 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 $\mathcal{S}(\mathbb{D}) = \{ψ\in H^\infty(\mathbb{D}): \|ψ\|_{\infty} := \sup_{z \in \mathbb{D}} |ψ(z)| \leq 1\}.$ 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 $φ$, holomorphic self maps of $\mathbb{D}$, and inner functions $θ\in H^\infty(\mathbb{D})$ such that the Beurling type invariant subspace $θH^2$ is an invariant subspace for $C_φ$. We prove the following result: $C_φ (θH^2) \subseteq θH^2$ if and only if $\frac{θ\circ φ}θ \in \mathcal{S}(\mathbb{D}).$ This classification also allows us to recover or improve some known results on Beurling type invariant subspaces of composition operators.

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