Hilbert complexes
A Hilbert complex is just a complex \[ 0\to {\mathcal D}_0@> D_0>> {\mathcal D_1}@> D_1>>\cdots @> D_{N- 1}>> {\mathcal D}_N\to 0, \] where the \(D_j\) are closed operators between Hilbert spaces with domain \({\mathcal D}_j\) and \(D_{j+ 1}\circ D_j= 0\). Although this is a fairly simple object, it reflects surprisingly much of the structure known from elliptic complexes on noncompact manifolds, the main application the authors have in mind. In this paper they 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. The authors apply the abstract machinery to the de Rham complex in several singular situations.
elliptic complexes on noncompact manifolds
scientific article; zbMATH DE number 89204
scientific article