Inner multipliers and Rudin type invariant subspaces

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 $H^2(\mathbb{D}^n)$.Comment: 8 page

On certain Toeplitz operators and associated completely positive maps

We study Toeplitz operators with respect to a commuting $n$-tuple of bounded operators which satisfies some additional conditions coming from complex geometry. Then we consider a particular such tuple on a function space. The algebra of Toeplitz operators with respect to that particular tuple becomes naturally homeomorphic to $L^\infty$ of a certain compact subset of $\mathbb C^n$. Dual Toeplitz operators are characterized. En route, we prove an extension type theorem which is not only important for studying Toeplitz operators, but also has an independent interest because dilation theorems do not hold in general for $n>2$.Comment: 25 pages. arXiv admin note: text overlap with arXiv:1706.0346