Inner multipliers and Rudin type invariant subspaces

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 $\mathrm{\Theta }{H}_{{\mathcal{E}}_{\ast }}^2(\mathbb{D})$, where $\mathrm{\Theta }$ is an operator-valued inner multiplier in $H^\infty_{B($\mathcal{E}$*, $\mathcal{E}$)}($\mathbb{D}$)$forsomeHilbertspace$$\mathcal{E}$*$. 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 $\Theta \in H^\infty_{$\mathcal{B}$(H^2($\mathbb{D}$^{n-1}))}($\mathbb{D}$)$forwhich$\Theta H^2_{H^2($\mathbb{D}$^{n-1})}($\mathbb{D}$)$isaRudintypeinvariantsubspaceof$H^2($\mathbb{D}$^n)$.