Doubly commuting mixed invariant subspaces in the polydisc

TL;DR

This paper characterizes doubly commuting mixed invariant subspaces of the Hardy space over the polydisc, providing explicit forms, commutant descriptions, and concrete examples, advancing understanding of multivariable operator theory.

Contribution

It offers a complete characterization of doubly commuting mixed invariant subspaces in the polydisc Hardy space, including explicit structure and commutant representations.

Findings

Doubly commuting mixed invariant subspaces are characterized by tensor products involving inner functions and Jordan blocks.

Explicit formulas for the commutant of doubly commuting shift tuples are derived.

Concrete examples illustrating the structure of these subspaces are provided.

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 $1 \leq j \leq k$ 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.