Normal weighted composition operators on weighted Dirichlet spaces

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