We show that in a scale of weighted Dirichlet spaces $D_{ \alpha}$, including the Bergman space, given any finite Blaschke product $B$ there exists an equivalent norm in $D_{ \alpha}$ such that $B$ satisfies the wandering subspace property with respect to such norm. This extends, in some sense, previous results by Carswell, Duren and Stessin. As a particular instance, when $B(z)=z^k$ and $|\alpha| \leq \frac{ \log (2)}{ \log(k+1)}$, the chosen norm is the usual one in $D_\alpha$.
Categories: math.CV