Invariant subspaces of composition operators

The invariant subspace lattices of composition operators acting on H², the Hilbert-Hardy space over the unit disc, are characterized in select cases. The lattice of all spaces left invariant by both a composition operator and the unilateral shift M_z (the multiplication operator induced by the coordinate function), is shown to be nontrivial and is completely described in particular cases. Given an analytic selfmap ϕ of the unit disc, we prove that ϕ has an angular derivative at some point on the unit circle if and only if C_ϕ, the composition operator induced by ϕ, maps certain subspaces in the invariant subspace lattice of M_z into other such spaces. A similar characterization of the existence of angular derivatives of ϕ, this time in terms of A_ϕ, the Aleksandrov operator induced by ϕ, is obtained.