Composition operators between Beurling subspaces of Hardy space

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 D 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 θ Hᵖ is invariant under C_φ.