Variations on a theme of Beurling (Q554262)

By a Hilbert module \(H\) over \(\mathbb{C}[z]\), \(z = (z_1, \dots, z_m)\) with \(m \geq 1\), we mean a Hilbert space \(H\) equipped with a unital module action \(\mathbb{C}[z] \times H \to H\) such that the operator \(f \mapsto p.f\), \(f\in H\), is bounded for each \(p\in \mathbb{C}[z]\). The Beurling-Lax-Halmos Theorem states that, if \(\mathcal{S}\) is a non-zero submodule of \(H_2(\mathbb{D})\otimes \mathcal{E}\) for some Hilbert space \(\mathcal{E}\), then there exists a subspace \(\mathcal{E}_*\) of \(\mathcal{E}\) such that \(\mathcal{S}\) and \(H_2(\mathbb{D})\otimes \mathcal{E}_*\) are unitarily equivalent Hilbert modules. The author uses the language of Hilbert modules to interpret and investigate generalizations of the Beurling-Lax-Halmos Theorem on invariant subspaces of the unilateral shift in both the one and multivariate cases with an emphasis on the classical Hardy, Bergman and Drury-Arveson spaces.