On the wandering property in Dirichlet spaces

Abstract

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 $|\alpha |\le \frac{\log (2)}{\log (k+1)}$, the chosen norm is the usual one in $D_{\alpha }$.