Composition operators between Beurling subspaces of Hardy space

TL;DR

This paper characterizes when composition operators map certain Beurling subspaces of Hardy spaces into each other, extending previous work to more general inner functions and including special cases like Blaschke products.

Contribution

It provides a comprehensive characterization of composition operators acting on Beurling subspaces associated with inner functions, generalizing prior results and including measure-theoretic conditions for elliptic automorphisms.

Findings

Characterization of holomorphic self maps for subspace inclusion

Special focus on Blaschke products and singular inner functions

Measure-theoretic description for elliptic automorphisms

Abstract

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 $C_\phi$.