HILBERT COMPLEXES

A Hilbert complex is just a complex {A figure is presented} where the Dj are closed operators between Hilbert spaces with domain Dj and Dj + 1 {ring operator} Dj = 0. Although this is a fairly simple object, it reflects surprisingly much of the structure known from elliptic complexes on noncompact manifolds, the main application we have in mind. In this paper we undertake a systematic study of Hilbert complexes and their relationship with elliptic complexes. It turns out that this perspective gives a common structure to various known theorems along with generalizations and extensions. We apply the abstract machinery to the de Rham complex in several singular situations. © 1992.