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

DOI: 10.14232/actasm-015-773-y

AMS Subject Classification (1991): 47A13, 47A15, 46E20, 46M05

Keyword(s): Hardy space, inner sequence, operator-valued inner function, invariant subspace, unitary equivalence

Received March 17, 2015. (Registered under 23/2015.)