Normal weighted composition operators on the Hardy space H2(U)

Let φ be an analytic function on the open unit disc U such that φ(U)⊆U, and let ψ be an analytic function on U such that the weighted composition operator Wψ,φ defined by Wψ,φf=ψf○φ is bounded on the Hardy space H2(U). We characterize those weighted composition operators on H2(U) that are unitary, showing that in contrast to the unweighted case (ψ≡1), every automorphism of U induces a unitary weighted composition operator. A conjugation argument, using these unitary operators, allows us to describe all normal weighted composition operators on H2(U) for which the inducing map φ fixes a point in U. This description shows both ψ and φ must be linear fractional in order for Wψ,φ to be normal (assuming φ fixes a point in U). In general, we show that if Wψ,φ is normal on H2(U) and ψ≢0, then φ must be either univalent on U or constant. Descriptions of spectra are provided for the operator Wψ,φ:H2(U)→H2(U) when it is unitary or when it is normal and φ fixes a point in U.