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, 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 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.
- Composition operator
- Hardy space
- Normal operator
- Weighted composition operator