Range additivity, shorted operator and the Sherman–Morrison–Woodbury formula

We say that two operators A, B   have the range additivity property if R(A+B)=R(A)+R(B)R(A+B)=R(A)+R(B). In this article we study the relationship between range additivity, shorted operator and certain Hilbert space decomposition known as compatibility. As an application, we extend to infinite dimensional Hilbert space operators a formula of Fill and Fishkind related to the well-known Sherman–Morrison–Woodbury formula.