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In the setting of arbitrary Hilbert spaces, we give a representation of M-P inverse of the sum of linear operators A + B under suitable conditions. Based on the full-rank decomposition of an operator, we prove that the extension of the Fill-Fishkind formula for A and B with closed ranges, remains valid, keeping the same conditions of Fill-Fishkind formula for two matrices, also we obtain an analogous formula under the Fill-Fishkind conditions, beyond we derive some representations of M-P inverse of a 2-by-2 block operator with disjoint ranges.

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