On the product of projectors and generalized inverses

We consider generalized inverses of linear operators on arbitrary vector spaces and study the question when their product in reverse order is again a generalized inverse. It turns out that this problem is equivalent to the question when the product of projectors is again a projector, and we discuss necessary and sufficient conditions in terms of the defining spaces. We present a new representation of the product of generalized inverses that does not require explicit knowledge of the factors. Our approach is based on implicit representations of subspaces via their orthogonals in the dual space, and we formulate a duality principle for statements about generalized inverses. For Fredholm operators, the corresponding computations reduce to finite-dimensional problems. We also illustrate our results with examples for matrices.