The range and pseudo-inverse of a product (Q1079162)

A basic problem in the theory of pseudo-inverse is to determine when the range of a product is closed and the pseudo-inverse of a product is the product of the pseudo-inverses. The result we obtained is the following. Let H,K,L be Hilbert spaces over complex field. Assume that \(A\in B(K,H)\) and \(B\in B(L,K)\) have closed range. Then AB has closed range iff the angle between Ker A and B((Ker AB)\({}^{\perp})\) is positive; \((AB)^+=B^+A^+\) iff B((Ker AB)\({}^{\perp})\subset (Ker A)^{\perp}\), \(A^*((Ker B^*A^*)^{\perp}\subset (Ker B^*)^{\perp}\). Clearly the conditions we present have a unified symmetric form and apparent geometric sense. Moreover, the proofs are much simpler and clearer.