Doubly commuting submodules of the Hardy module over polydiscs (Q2848715)

For \(n\in\mathbb{N}\) and \(E_*\) a Hilbert space, \(H^2_{E_*}(\mathbb{D}^n)\) is the set of all \(E_*\)-valued holomorphic functions \(f\) on the polydisk \(\mathbb{D}^n\), for which the norm NEWLINE\[NEWLINE \|f\|_2 = \sup_{0<r<1} \left( \int_{\mathbb{T}^n} \| f(rz) \|^2_{E_*} \,dz \right)^{\frac12}NEWLINE\]NEWLINE is finite. If \(E\) is another Hilbert space, then \(\mathcal L(E,E_*)\) denotes the set of all continuous linear maps from \(E\) to \(E_*\), and \(H^\infty_{E\to E_*}(\mathbb{D}^n)\) denotes the set of all \(\mathcal L(E,E_*)\)-valued holomorphic functions \(F\) for which the norm NEWLINE\[NEWLINE \| F \|_{\infty} = \sup_{z\in \mathbb{D}^n} \| f(z)\|_{\mathcal L(E,E_*)} NEWLINE\]NEWLINE is finite. An operator-valued function \(\Theta\in H^\infty_{E\to E_*}(\mathbb{D}^n)\) is inner if its pointwise boundary values are isometries almost everywhere on the torus \(\mathbb{T}^n\). A commuting family of bounded linear operators \(\{T_1,\dots,T_n\}\) on some Hilbert space \(\mathcal{H}\) is said to be doubly commuting if \(T_iT_j^* = T_j^*T_i\) whenever \(i\) and \(j\) are distinct indices. A closed subspace \(\mathcal S\) of \(H^2_{E_*}(\mathbb{D}^n)\) is said to be a doubly commuting submodule if it is invariant under each of the multiplication operators \(M_{z_j}\) corresponding to the coordinates, and the family consisting of the \(n\) restriction operators \(R_{z_j} = M_{z_j}|\mathcal{S}\) is doubly-commuting on \(\mathcal{S}\). The main result of the paper is as follows.NEWLINENEWLINE { Theorem.} Let \(\mathcal{S}\) be a closed nonzero subspace of \(H^2_{E_*}(\mathbb{D}^n)\). Then \(\mathcal{S}\) is a doubly commuting module if and only if there exists a Hilbert space \(E\subseteq E_*\) and an inner function \(\Theta\in H^\infty_{E\to E_*}(\mathbb{D}^n)\) such that NEWLINE\[NEWLINE \mathcal{S} = M_{\Theta}H^2_{E}(\mathbb{D}^n).NEWLINE\]NEWLINE This is a multidimensional extension of the Beurling-Lax-Halmos theorem in one dimension. The case \(n=2\) and \(E_*=\mathbb{C}\) was earlier proved by \textit{V. Mandrekar} [Proc. Am. Math. Soc. 103, No. 1, 145--148 (1988; Zbl 0658.47033)] by a different method. As an application, the authors give a partial extension of \textit{V.~Tolokonnikov}'s solution to the completion problem [Proc. Am. Math. Soc. 117, No. 4, 1023--1030 (1993; Zbl 0772.30044)] from one to several variables:NEWLINENEWLINE{Theorem.} Suppose that \(E\subset E_c\) are finite-dimensional Hilbert spaces. Let \(f\in H^\infty_{E\to E_c}(\mathbb{D}^n)\). Then the following are equivalent: (1) There exists \(g\in H^\infty_{E_c\to E}(\mathbb{D}^n)\) such that \(gf\equiv I\) in \(\mathbb{D}^n\) and the operators \(M_{z_1},\dots ,M_{z_n}\) doubly commute on \(\operatorname{ker}M_g\). (2) There exists \(F\in H^\infty_{E_c\to E_c}(\mathbb{D}^n)\) such that \(F|_E=f\), \(F|_{E_c\ominus E}\) is inner, and \(F^{-1}\in H^\infty_{E_c\to E_c}(\mathbb{D}^n)\).NEWLINENEWLINE The authors point out that this leaves open a few questions about the completion problem in polydisks.