2020/04/01 by Snehasish Bose, Bose, Snehasish, P. Muthukumar +3 · 1 citation
Mathematics · #30D55 #46E15 #46E22 #47A15 #47B38 #Advanced Harmonic Analysis Research #Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Meromorphic and Entire Functions #Operator Algebras (math.OA) #Primary: 47B33 #Secondary: 30H10
paper · pdf · doi:10.48550/arxiv.2004.00264
openalex publication_date 2020/04/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let \mathbbD be the open unit disk in ℂ, let H2 denote the Hardy space on \mathbbD and let φ: \mathbbD → \mathbbD be a holomorphic self map of \mathbbD. The composition operator Cφ on H2 is defined by (Cφ f)(z)=f(φ(z)) (f ∈ H2, z ∈ \mathbbD). Denote by S(\mathbbD) the set of all functions that are holomorphic and bounded by one in modulus on \mathbbD, that is S(\mathbbD) = \ψ∈ H^∞(\mathbbD): ‖ψ‖∞ := sup_z ∈ \mathbbD |ψ(z)| ≤ 1\. The elements of S(\mathbbD) 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 \mathbbD, and inner functions θ∈ H^∞(\mathbbD) such that the Beurling type invariant subspace θH2 is an invariant subspace for Cφ. We prove the following result: Cφ (θH2) ⊆ θH2 if and only if \fracθ∘ φθ ∈ S(\mathbbD). This classification also allows us to recover or improve some known results on Beurling type invariant subspaces of composition operators.