Essentially orthogonal subspaces

Abstract

We study the set C consisting of pairs of orthogonal projectionsP,Q acting in a Hilbert space H such that PQ is a compact operator. Thesepairs have a rich geometric structure which we describe here. They are partitionedin three subclasses: C0 consists of pairs where P or Q have finite rank,C1 of pairs such that Q lies in the restricted Grassmannian (also called Sato-Grassmannian) of the polarization H = N(P)? R(P), and C?. We characterize the connected components of these classes: the components of C0 are parametrized by the rank, the components of C1 are parametrized by the Fredholm index of the pairs, and C? is connected. We show that these subsets are(non-complemented) differentiable submanifolds of B(H) x B(H).