Normal weighted composition operators on weighted Dirichlet spaces

Abstract

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 W ψ , φ defined by W ψ , φ f = ψ ⋅ ( f ∘ φ ) . On the Hardy space and Bergman space, it is known that W ψ , φ is bounded and normal precisely when ψ = c K p / ( K p ∘ φ ) and φ = α p ∘ ( δ α p ) , where K p is the reproducing kernel for the space, α p ( 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 W ψ , 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.