Inner multipliers and Rudin type invariant subspaces

Abstract

Let $\mathcal{E}$ be a Hilbert space and $H_{\mathcal{E}}^2(\mathbb{D})$ be the $\mathcal{E}$-valued Hardy space over the unit disc $\mathbb{D}$ in $\mathbb{C}$. The well known Beurling-Lax-Halmos theorem states that every shift invariant subspace of $H_{\mathcal{E}}^2(\mathbb{D})$ other than $\{0\}$ has the form $\Theta H_{{\mathcal{E}}_{\ast }}^2(\mathbb{D})$, where $\Theta$ is an operator-valued inner multiplier in $H_{B({\mathcal{E}}_{\ast },\mathcal{E})}^{\infty }(\mathbb{D})$ for some Hilbert space ${\mathcal{E}}_{\ast }$. In this paper we identify $H^2({\mathbb{D}}^n)$ with $H^2({\mathbb{D}}^{n-1})$-valued Hardy space $H_{H^2({\mathbb{D}}^{n-1})}^2(\mathbb{D})$ and classify all such inner multiplier $\Theta \in H_{\mathcal{B}(H^2({\mathbb{D}}^{n-1}))}^{\infty }(\mathbb{D})$ for which $\Theta H_{H^2({\mathbb{D}}^{n-1})}^2(\mathbb{D})$ is a Rudin type invariant subspace of $H^2({\mathbb{D}}^n)$.

8 pages