Submodules of \(L^2(\mathbb R^2)\) (Q819502)

A closed subset \( {\mathcal M}\subset L^{2}({\mathbb R}^{2}) \) is called a submodule of \(L^{2}({\mathbb R}^{2})\) if \( e^{i\alpha x_{j}}{\mathcal M} \subset {\mathcal M} \) for any \(j=1, 2\), \(\alpha\geq0\). The author gives a new proof for the theorem (proved earlier in his paper [Can. Math. Bull. 47, No. 1, 100--107 (2004; Zbl 1062.47010)]) giving the structure of submodules of some special class. He also discusses some related questions.