Applications of completions of operator matrices to reverse order law for {1}-inverses of operators on Hilbert spaces

Although the reverse order law for {1}-generalized inverses of matrices has been completely resolved, in this paper we do it for the first time for operators acting on separable Hilbert spaces. The approach taken here involves some of the previous research on completions of operator matrices to left and right invertibility. We thus demonstrate usability of some of the results of that type by showing how they can be applied to one of the topics in generalized inverses of operators that has seen a great interest over the years.