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 $\clq$ of $H^2(\mathbb{ D}^n)$ ($n \geq 2$) is doubly commuting if and only if \[\clq = \clq_{ \Theta_1} \otimes \cdots \otimes \clq_{ \Theta_n},\]where each $\clq_{ \Theta_i}$ is either a one variable Jordan block $H^2(\mathbb{ D})/\Theta_i H^2(\mathbb{ D})$ for some inner function $\Theta_i$ or the Hardy module $H^2(\mathbb{ D})$ on the unit disk for all $i = 1, \ldots, n$. We say that a submodule $\cls$ of $H^2(\mathbb{ D}^n)$ is a co-doubly commuting if the quotient module $H^2(\mathbb{ D}^n)/\cls$ 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 $\cls$ of $H^2(\mathbb{ D}^n)$ is co-doubly commuting if and only if \[\cls = \mathop{ \sum}_{ i=1}^m \Theta_i H^2(\mathbb{ D}^n),\]for some integer $m \leq n$ and one variable inner functions $\{ \Theta_i\}_{ i=1}^m$.
Categories: math.FA math.OA