Invertible composition operators on Hp

The weakly closed algebras generated by certain sets of composition operators are shown to be reflexive. A structure theorem for invertible composition operators on H² is obtained and used to show that such operators are reflexive. The structure theorem shows that invertible hyperbolic composition operators are similar to cosubnormal operators built up from bilateral weighted shifts. Another consequence of the structure theorem is that the composition operators induced by hyperbolic disc automorphisms are universal. Thus the general invariant subspace problem for Hilbert space operators is contained in the problem of determining the invariant subspace lattices of these operators.