Wandering Subspaces of the Bergman Space and the Dirichlet Space Over ${{\mathbb{D}^{n}}}$

Abstract

Doubly commuting invariant subspaces of the Bergman space and the Dirichlet space over the unit polydisc ${\mathbb{D}^n}$ (with ${n \geq 2}$) are investigated. We show that for any non-empty subset ${\alpha=\{\alpha_1,\ldots,\alpha_k\}}$ of ${\{1,\ldots,n\}}$ and doubly commuting invariant subspace ${\mathcal{S}}$ of the Bergman space or the Dirichlet space over ${\mathbb{D}^n}$, restriction of the multiplication operator tuple on ${\mathcal{S}, M_{\alpha}|_\mathcal{S}:=(M_{z_{\alpha_1}}|_\mathcal{S},\ldots, M_{z_{\alpha_k}}|_\mathcal{S})}$, always possesses generating wandering subspace of the form

$$\bigcap_{i=1}^k(\mathcal{S}\ominus z_{\alpha_i}\mathcal{S})$$

.