Shift invariant subspaces of composition operators.

Let C denote the composition operator defined on the standard Hardy spaces Hp as where is an analytic self-map of the unit disk in the complex plane. In this paper we discuss those invariant subspaces of C in Hp which are invariant under the shift operator, We restrict our attention to the case where is an inner function. Our main result characterises these invariant subspaces. We also consider C when restricted to such an invariant subspace and we describe the structure of the operator and find a formula for the essential spectral radius.