Beurling type invariant subspaces of composition operators

Abstract

Let $\mathbb{D}$ be the open unit disk in $\mathbb{C}$, let $H^2$ denote the Hardy space on $\mathbb{D}$ and let $\ddot{I}†:\mathbb{D}\to \mathbb{D}$ be a holomorphic self map of $\mathbb{D}$. The composition operator $C_{\ddot{I}}†$ 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 $\ddot{I}†$, holomorphic self maps of $\mathbb{D}$, and inner functions $\hat{I}¸\in H^{\infty }(\mathbb{D})$ such that the Beurling type invariant subspace $\hat{I}¸H^2$ is an invariant subspace for $C_{\ddot{I}}†$. We prove the following result: $C_{\ddot{I}}†(\hat{I}¸H^2)\subseteq \hat{I}¸H^2$ 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.

13 pages, revised. To appear in Journal of Operator Theory