Operators which are the difference of two projections

We study the set D of differences D = { 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 D c ⊂ D F , where D c are the compact elements in D , and D F are the differences A = P − Q such that the pair ( P , Q ) is a Fredholm pair ( ( P , Q ) is a Fredholm pair if Q P | R ( P ) : R ( P ) → R ( Q ) is a Fredholm operator).