2024/08/19 by V. A. Anjali, Anjali, V. A., P. Muthukumar +3
Mathematics · #30H10 #46E15 #46E22 #47B38 #Advanced Harmonic Analysis Research #Complex Variables (math.CV) #Differential Equations and Boundary Problems #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Primary 47B33 #Secondary 47A15
paper · pdf · doi:10.48550/arxiv.2408.09759
openalex publication_date 2024/08/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
V. Matache (J. Operator Theory 73(1):243--264, 2015) raised an open problem about characterizing composition operators Cϕ on the Hardy space H2 and nonzero singular measures μ1, μ2 on the unit circle such that Cϕ(Sμ1 H2)⊆ Sμ2 H2, where Sμi denotes the singular inner function corresponding to the measure μi,i=1,2. In this article, we consider this problem in a more general setting. We characterize holomorphic self maps ϕ of the unit disk \mathbbD and inner functions θ1, θ2 such that Cϕ(θ1 Hp)⊆ θ2 Hp, for p>0. Emphasis is given to Blaschke products and singular inner functions as a special case. We also give an another measure-theoretic characterization to above question when ϕ is an elliptic automorphism. For a given Blaschke product θ, we discuss about finding all self maps ϕ such that θHp is invariant under Cϕ.