On the Wandering Property in Dirichlet spaces

We show that in a scale of weighted Dirichlet spaces Dα, including the Bergman space, given any finite Blaschke product B there exists an equivalent norm in Dα such that B satisfies the wandering subspace property with respect to such norm. This extends, in some sense, previous results by Carswell et al. (Indiana Univ Math J 51(4):931–961, 2002). As a particular instance, when B(z)=zk and |α|≤log(2)log(k+1), the chosen norm is the usual one in Dα.