Composition operators between Beurling subspaces of Hardy space

Abstract

V. Matache (J. Operator Theory 73(1):243--264, 2015) raised an open problem about characterizing composition operators $C_{\ddot{I}}•$ on the Hardy space $H^2$ and nonzero singular measures $\hat{I}{¼}_1$, $\hat{I}{¼}_2$ on the unit circle such that $C_{\ddot{I}}•(S_{\hat{I}{¼}_1}{H}^2)\subseteq S_{\hat{I}{¼}_2}{H}^2,$ where $S_{\hat{I}{¼}_i}$ denotes the singular inner function corresponding to the measure $\hat{I}{¼}_i,i=1,2$. In this article, we consider this problem in a more general setting. We characterize holomorphic self maps $\ddot{I}•$ of the unit disk $\mathbb{D}$ and inner functions $\hat{I}{¸}_1,\hat{I}{¸}_2$ such that $C_{\ddot{I}}•(\hat{I}{¸}_1{H}^p)\subseteq \hat{I}{¸}_2{H}^p,$ 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 $\ddot{I}•$ is an elliptic automorphism. For a given Blaschke product $\hat{I}¸$, we discuss about finding all self maps $\ddot{I}•$ such that $\hat{I}¸H^p$ is invariant under $C_{\ddot{I}}•$.

15 pages