Invariant subspaces of parabolic self-maps in the Hardy space

Abstract

It is shown that the lattice of invariant subspaces of the operator of multiplication by a cyclic element of a Banach algebra consists of the closed ideals of this algebra. As an application, with the help of some elements of the Gelfand Theory of Banach algebras, the lattice of invariant subspaces of composition operators acting on the Hardy space, whose inducing symbol is a parabolic non-automorphism, is found. In particular, each invariant subspace always consists of the closed span of a set of eigenfunctions. As a consequence, such composition operators have no non-trivial reducing subspaces.