Pairs of Projections: Geodesics, Fredholm and Compact Pairs

Abstract

A pair (P,Q) of orthogonal projections in a Hilbert space H is called a Fredholm pair if QP:R(P)→R(Q)is a Fredholm operator. Let F be the set of all Fredholm pairs. A pair is called compact if P−Q is compact. Let C be the set of all compact pairs. Clearly C⊂F properly. In this paper it is shown that both sets are differentiable manifolds, whose connected components are parametrized by the Fredholm index. In the process, pairs P,Q that can be joined by a geodesic (or equivalently, a minimal geodesic) of the Grassmannian of H are characterized: this happens if and only if dim(R(P)∩N(Q))=dim(R(Q)∩N(P)).