SHIFT-INVARIANT SUBSPACES INVARIANT FOR COMPOSITION OPERATORS ON THE HARDY-HILBERT SPACE

If phi is an analytic map of the unit disk D into itself, the composition operator C-phi on a Hardy space H-2 is defined by C-phi(f) = f omicron phi. The unilateral shift on H-2 is the operator of multiplication by z. Beurling (1949) characterized the invariant subspaces for the shift. In this paper, we consider the shift-invariant subspaces that are invariant for composition operators. More specifically, necessary and sufficient conditions are provided for an atomic inner function with a single atom to be invariant for a composition operator, and the Blaschke product invariant subspaces for a composition operator are described. We show that if phi has Denjoy-Wolff point a on the unit circle, the atomic inner function subspaces with a single atom at a are invariant subspaces for the composition operator C-phi.