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

Abstract: 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). ]