A characterization of submodules via the Beurling-Lax-Halmos theorem

Abstract:

Shift invariant subspaces in the vector-valued Hardy space $H^2(E)$ play important roles in Nagy-Foias operator model theory. A theorem by Beurling, Lax and Halmos characterizes such invariant subspaces by operator-valued inner functions $\Theta (z)$. When $E=H^2(\mathbb {D})$, $H^2(E)$ is the Hardy space over the bidisk $H^2(\mathbb {D}^2)$. This paper shows that for some well-known examples of invariant subspaces in $H^{2}({\mathbb {D}}^2)$, the function $\Theta (z)$ turns out to be strikingly simple.