Doubly commuting mixed invariant subspaces in the polydisc

We obtain a complete characterization for doubly commuting mixed invariant subspaces of the Hardy space over the unit polydisc. We say a closed subspace $\mathcal{Q}$ of $H^2(\mathbb{D}^n)$ is mixed invariant if $M_{z_{j}}(\mathcal{Q}) \subseteq \mathcal{Q}$ for $1 \leq j \leq k$ and $M_{z_{j}}^*(\mathcal{Q}) \subseteq \mathcal{Q}$, $k+1 \leq j \leq n$ for some integer $k \in \{1, 2, \ldots, n-1 \}$. We prove that a mixed invariant subspace $\mathcal{Q}$ of $H^2(\mathbb{D}^n)$ is doubly commuting if and only if $$\mathcal{Q} = \Theta H^2(\mathbb{D}^k) \otimes \mathcal{Q}_{\theta_1} \otimes \cdots \otimes \mathcal{Q}_{\theta_{n-k}},$$ where $\Theta \in H^{\infty}(\mathbb{D}^k)$ is some inner function and $\mathcal{Q}_{\theta_j}$ is either a Jordan block $H^2(\mathbb{D})\ominus \theta_j H^2(\mathbb{D})$ for some inner function $\theta_j$ or the Hardy space $H^2(\mathbb{D})$. Furthermore, an explicit representation for the commutant of an $n$-tuple of doubly commuting shifts as well as a representation for the commutant of a doubly commuting tuple of shifts and co-shifts are obtained. Finally, we discuss some concrete examples of mixed invariant subspaces.