Space-preserving composition operators (Q752229)

Let H be the family of holomorphic functions on the unit disk \(\delta\) and let V be a subspace of H. What are the holomorphic functions \(\phi\) mapping \(\delta\) into \(\delta\) such that whenever \(f\in H\) and \((f\circ \phi)\in V\) it follows that \(f\in V?\) A function satisfying this condition is said to possess property (*) relative to the subspace V. In this paper we construct a family of functions \(\phi\) for which (*) holds for the subspaces of \(H^ p\) (the Hardy spaces), BMOA, and the disk algebra. The functions \(\phi\) can be factored as a finite Blaschke product times an outer function, and hence have modulus less than one on arcs of the unit circle.