  We say that two operators A, B have the range additivity property if 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 by Fill and Fishkind related to the well-known
Sherman-Morrison-Woodbury formula.
