**In minisymposium: Generalized Inverse and its Applications**

The reverse order law for the Moore-Penrose inverse has been first studied by Greville [1], who gave a necessary and sufficient condition for the reverse order law $$\label{e0} (AB)^{\dagger}=B^{\dagger} A^{\dagger}, $$ for matrices $A$ and $B$. Hartwig [2] considered triple reverse order law $(ABC)^{\dagger}=C^{\dagger} B^{\dagger} A^{\dagger}$. This was further extended to the reverse order law for $K$-inverses, where $K\subseteq\{1,2,3,4\}$ on the set of matrices, bounded linear operators or in $C^*$-algebras. Tian [3] considered the reverse order law for the Moore-Penrose inverse of product of $n$ matrices. By using rank of matrices, he derived necessary and sufficient conditions for $A_n^{\dagger}A_{n-1}^{\dagger}\cdots A_1^{\dagger}$ to be $\{1\}$-, $\{1,2\}$-, $\{1,3\}$-, $\{1,4\}$-, $\{1,2,3\}$-, $\{1,2,4\}$-inverse of $A_1A_2\cdots A_n$. Reverse order law for generalized inverses of product of $n$ matrices has also been studied by Wei [4].

We present new results related to the reverse order law for generalized inverses of multiple operator product. We derive necessary and sufficient conditions for the inclusions

- T.N.E. Greville,
*Note on the generalized inverse of a matrix product*, SIAM Rev. 8 (1966), pp. 518–521. - R. E. Hartwig,
*The reverse order law revisited*, Linear Algebra Appl. 76 (1986), 241–246. - Y. Tian,
*Reverse order laws for the generalized inverses of multiple matrix products*, Linear Algebra Appl. 211 (1994), pp. 85–100. - M. Wei,
*Reverse order laws for generalized inverses of multiple matrix products*, Linear Algebra Appl. 293 (1–3) (1999), pp. 273–288.