Operators which are the difference of two projections

We study the set D of differencesD={A=P−Q:P,Q∈P}, where P denotes the set of orthogonal projections in H. We describe models and factorizations for elements in D, which are related to the geometry of P. The study of D throws new light on the geodesic structure of P (we show that two projections in generic position are joined by a unique minimal geodesic). The topology of D is examined, particularly its connected components are studied. Also we study the subsets Dc⊂DF, where Dc are the compact elements in D, and DF are the differences A=P−Q such that the pair (P,Q) is a Fredholm pair ((P,Q) is a Fredholm pair if QP|R(P):R(P)→R(Q) is a Fredholm operator).