Shift invariant subspaces of composition operators on H^p

Abstract.

Let C_φ denote the composition operator defined on the standard Hardy spaces H^p as $f \mapsto f \circ \varphi ,$ where φ is an analytic self-map of the unit disk in the complex plane. In this paper we discuss those invariant subspaces of C_φ in H^p which are invariant under the shift operator, $\mathfrak{G}f(z) = zf(z).$ We restrict our attention to the case where φ is an inner function. Our main result characterises these invariant subspaces. We also consider C_φ when restricted to such an invariant subspace and we describe the structure of the operator and find a formula for the essential spectral radius.