  Let $\mathcal{E}$ be a Hilbert space and $H^2_{\mathcal{E}}(\mathbb{D})$ 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 $H^2_{\mathcal{E}}(\mathbb{D})$ other than $\{0\}$
has the form $\Theta H^2_{\mathcal{E}_*}(\mathbb{D})$, where $\Theta$ is an
operator-valued inner multiplier in $H^\infty_{B(\mathcal{E}_*,
\mathcal{E})}(\mathbb{D})$ for some Hilbert space $\mathcal{E}_*$. In this
paper we identify $H^2(\mathbb{D}^n)$ with $H^2(\mathbb{D}^{n-1})$-valued Hardy
space $H^2_{H^2(\mathbb{D}^{n-1})}(\mathbb{D})$ and classify all such inner
multiplier $\Theta \in
H^\infty_{\mathcal{B}(H^2(\mathbb{D}^{n-1}))}(\mathbb{D})$ for which $\Theta
H^2_{H^2(\mathbb{D}^{n-1})}(\mathbb{D})$ is a Rudin type invariant subspace of
$H^2(\mathbb{D}^n)$.
