A Note on the Reverse Order Law for g-Inverse of Operator Product

The generalized inverse has many important applications in aspects of the theoretical research of matrices and statistics. One of the core problems of the generalized inverse is finding the necessary and sufficient conditions of the reverse order laws for the generalized inverse of the operator product. In this paper, we study the reverse order law for the g-inverse of an operator product T1T2T3 using the technique of matrix form of bounded linear operators. In particular, some necessary and sufficient conditions for the inclusion T-3{1}T-2{1}T-1{1} subset of (T1T2T3){1} is presented. Moreover, some finite dimensional results are extended to infinite dimensional settings.