Projected Composition Operators On the Hardy Space

Let P be the projection from L2, the Lebesgue space of the circle, Γ, to H2, the Hardy space. Given a complex valued function φ on Γ, define the operator Kφ : H2 → H2 by Kφ(f) = P(f ο φ). If φ is the boundary value function of a holomorphic function, then the projection is unnecessary and we have a classical composition operator. In that case the conditions under which Kφ is bounded are known. In this paper we give some results on the boundedness of Kφ for general φ.