Some representations of moore-penrose inverse for the sum of two operators and the extension of the fill-fishkind formula

In the setting of arbitrary Hilbert spaces, we give a representation of M-P inverse of the sum of linear operators \begin{document}$ A+B $\end{document} under suitable conditions. Based on the full-rank decomposition of an operator, we prove that the extension of the Fill-Fishkind formula for \begin{document}$ A $\end{document} and \begin{document}$ B $\end{document} 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.