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 C_ on the Hardy space H² and nonzero singular measures ₁, ₂ on the unit circle such that C_ (S䃑 H²) S䃒 H², where S㶁 denotes the singular inner function corresponding to the measure ᵢ, i=1, 2. In this article, we consider this problem in a more general setting. We characterize holomorphic self maps of the unit disk D and inner functions ₁, ₂ such that C_ (₁ Hᵖ) ₂ 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 is an elliptic automorphism. For a given Blaschke product, we discuss about finding all self maps such that Hᵖ is invariant under C_.