Singular inner eigenfunctions of composition operators

Abstract

This paper characterizes all the singular inner eigenfunctions of the composition operators $C_ϕ$ that arise from discrete measures, when $ϕ$ is an automorphism of unit disk. By establishing a connection between Beurling and model invariant subspaces, we classify all the inner functions so that the corresponding Beurling subspace is invariant under the composition operators induced by non-elliptic automorphisms. This classification involves solving the eigenfunction equation for the composition operator. Further, we present some applications of the above-mentioned connection.