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 $C_{\phi }$ on the Hardy space $H^2$ and nonzero singular measures ${\mu }_1$, ${\mu }_2$ on the unit circle such that $C_{\phi }(S_{{\mu }_1}{H}^2)\subseteq S_{{\mu }_2}{H}^2,$ where $S_{{\mu }_i}$ denotes the singular inner function corresponding to the measure ${\mu }_i,i=1,2$. In this article, we consider this problem in a more general setting. We characterize holomorphic self maps $\phi$ of the unit disk $\mathbb{D}$ and inner functions ${\theta }_1,{\theta }_2$ such that $C_{\phi }({\theta }_1{H}^p)\subseteq {\theta }_2{H}^p,$ 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 $\phi$ is an elliptic automorphism. For a given Blaschke product $\theta$, we discuss about finding all self maps $\phi$ such that $\theta H^p$ is invariant under $C_{\phi }$.