Normal weighted composition operators on weighted Dirichlet spaces

Let φ be an analytic self-map of D with φ(p)=p for some p∈D, let ψ be bounded and analytic on D, and consider the weighted composition operator Wψ,φ defined by Wψ,φf=ψ⋅(f∘φ). On the Hardy space and Bergman space, it is known that Wψ,φ is bounded and normal precisely when ψ=cKp/(Kp∘φ) and φ=αp∘(δαp), where Kp is the reproducing kernel for the space, αp(z)=(p−z)/(1−p¯z), and δ and c are constants with |δ|≤1. In particular, in this setting, φ is necessarily linear-fractional. Motivated by this result, we characterize the bounded, normal weighted composition operators Wψ,φ on the Dirichlet space D in the case when φ is linear-fractional with fixed point p∈D, showing that no nontrivial normal weighted composition operators of this form exist on D. Our methods also allow us to extend this result to certain weighted Dirichlet spaces in the case when φ is not an automorphism.