Normal weighted composition operators on the Hardy space

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 H 2 ( U ) . We characterize those weighted composition operators on H 2 ( 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 H 2 ( 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 H 2 ( 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 : H 2 ( U ) → H 2 ( U ) when it is unitary or when it is normal and φ fixes a point in U .