Wandering subspaces of the Bergman space and the Dirichlet space over polydisc

Doubly commutativity of invariant subspaces of the Bergman space and the Dirichlet space over the unit polydisc $\mathbb{D}^n$ (with $n \geq 2$) is investigated. We show that for any non-empty subset $\alpha=\{\alpha_1,\dots,\alpha_k\}$ of $\{1,\dots,n\}$ and doubly commuting invariant subspace $\s$ of the Bergman space or the Dirichlet space over $\D^n$, the tuple consists of restrictions of co-ordinate multiplication operators $M_{\alpha}|_\s:=(M_{z_{\alpha_1}}|_\s,\dots, M_{z_{\alpha_k}}|_\s)$ always possesses wandering subspace of the form $$\bigcap_{i=1}^k(\s\ominus z_{\alpha_i}\s).$$