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 W_{ψ, φ} defined by W_{ψ, φ} f = ψ f ○ 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 W_{ψ, φ} to be normal (assuming φ fixes a point in U). In general, we show that if W_{ψ, 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 W_{ψ, φ} : H² (U) → H² (U) when it is unitary or when it is normal and φ fixes a point in U.