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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 rangeadditivity, shorted operator and certain Hilbert space decomposition known ascompatibility. As an application, we extend to infinite dimensional Hilbertspace operators a formula by Fill and Fishkind related to the well-knownSherman-Morrison-Woodbury formula.

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