Doubly commuting mixed invariant subspaces in the polydisc

Abstract

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 $1\le j\le k$ and $M_{z_j}^{\ast }(\mathcal{Q})\subseteq \mathcal{Q}$, $k+1\le j\le n$ for some integer $k\in \{1,2,\dots ,n-1\}$. We prove that a mixed invariant subspace $\mathcal{Q}$ of $H^2({\mathbb{D}}^n)$ is doubly commuting if and only if \[ \mathcal{Q} = ΘH^2(\mathbb{D}^k) \otimes \mathcal{Q}_{θ_1} \otimes \cdots \otimes \mathcal{Q}_{θ_{n-k}}, \] 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.

19 pages, revised version