Jordan Blocks of H^2(D^n)

We develop a several variables analog of the Jordan blocks of the Hardy space $H^2(\mathbb{D})$. In this consideration, we obtain a complete characterization of the doubly commuting quotient modules of the Hardy module $H^2({\mathbb{D}}^n)$. We prove that a quotient module $\text{clq}$ of $H^2({\mathbb{D}}^n)$ ($n\ge 2$) is doubly commuting if and only if $$\text{clq}={\text{clq}}_{{\mathrm{\Theta }}_1}\otimes \cdots \otimes {\text{clq}}_{{\mathrm{\Theta }}_n},$$where each ${\text{clq}}_{{\mathrm{\Theta }}_i}$ is either a one variable Jordan block $H^2(\mathbb{D})\mathrm{/}{\mathrm{\Theta }}_i{H}^2(\mathbb{D})$ for some inner function ${\mathrm{\Theta }}_i$ or the Hardy module $H^2(\mathbb{D})$ on the unit disk for all $i = 1, \ldots, n$\mathrm{.}Wesaythatasubmodule$\cls$of$H^2($\mathbb{D}$^n)$ is a co-doubly commuting if the quotient module $H^2({\mathbb{D}}^n)\mathrm{/}\text{cls}$ is doubly commuting. We obtain a Beurling like theorem for the class of co-doubly commuting submodules of $H^2({\mathbb{D}}^n)$. We prove that a submodule $\text{cls}$ of $H^2({\mathbb{D}}^n)$ is co-doubly commuting if and only if $$\text{cls}={}_{i=1}^m{\mathrm{\Theta }}_i{H}^2({\mathbb{D}}^n),$$for some integer $m\le n$ and one variable inner functions $\{{\mathrm{\Theta }}_i{\}}_{i=1}^m$.