Normal weighted composition operators on weighted Dirichlet spaces

Let phi be an analytic self-map of D with phi(p) = p for some p E D, let be bounded and analytic on D, and consider the weighted composition operator W-psi,W-phi defined by f = psi.(f o phi). On the Hardy space and Bergman space, it is known that W-psi,W-phi is bounded and normal precisely when phi= cK(p) (K(p)o phi) and phi = alpha(p),o(delta alpha(p)), where Kp is the reproducing kernel for the space, alpha(p)(z) = (p - z)1(1 - pz), and 8 and c are constants with l l < 1. In particular, in this setting, cp is necessarily linear-fractional. Motivated by this result, we characterize the bounded, normal weighted composition operators on the Dirichlet space D in the case when co is linear-fractional with fixed point p E 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 cp is not an automorphism. (C) 2014 Elsevier Inc. All rights reserved.