On pseudoinverses of operator products

The analytical structure of the Moore-Penrose pseudoinverse of the product ab of any two operators over finite-dimensional unitary spaces is studied. The existence of the unique representation of the form (ab)⁺=b⁺(h+g)a⁺ is proved. Here h:= (a⁺abb⁺)⁺ is an (oblique) projector and g is an operator with a number of special properties. In particular, h+g is a projector, g is orthogonal to h in some metric, and g³=0. A necessary and sufficient condition for the case (ab)⁺=b⁺ha⁺ is established. This case contains the classical one (ab)⁺=b⁺a⁺ (the reverse-order law). For the latter a new necessary and sufficient condition is given.