Pairs of orthogonal projections with a fixed difference

Let A   be a self-adjoint contraction on a Hilbert space HH. In 2014, E. Andruchow proved that if A   is a difference of two orthogonal projections, with non-trivial generic part, then there exist infinitely many pairs (P,Q)(P,Q) of orthogonal projections such that A=P−QA=P−Q. In the present paper, we give a sufficient and necessary condition for A   to be a difference of a pair of orthogonal projections. Then we give a characterization of all pairs (P,Q)(P,Q) of orthogonal projections such that A=P−QA=P−Q. Moreover, we characterize the von Neumann algebra generated by such pairs (P,Q)(P,Q), and consider the connected components of the set {(P,Q):A=P−Q}{(P,Q):A=P−Q}.