ILAS2016 — 11–15 July 2016 — KU Leuven, Belgium

## 20th ILAS Conference

In minisymposium: Generalized Inverse and its Applications

Various results concerning the reverse order law for generalized inverses of operators
Jovana Nikolov Radenković (Faculty of Science and Mathematics, University of Nis)

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

\begin{eqnarray*} A_n\{1\}\cdot A_{n-1}\{1\}\cdots A_1\{1\}&\subseteq &(A_1A_2\cdots A_n)\{1\},\\ A_n\{1,2\}\cdot A_{n-1}\{1,2\}\cdots A_1\{1,2\}&\subseteq &(A_1A_2\cdots A_n)\{1,2\},\\ A_n\{1,2\}\cdot A_{n-1}\{1,2\}\cdots A_1\{1,2\}&=&(A_1A_2\cdots A_n)\{1,2\},\\ A_n\{1,3\}\cdot A_{n-1}\{1,3\}\cdots A_1\{1,3\}&\subseteq &(A_1A_2\cdots A_n)\{1,3\},\\ A_n\{1,4\}\cdot A_{n-1}\{1,4\}\cdots A_1\{1,4\}&\subseteq &(A_1A_2\cdots A_n)\{1,4\}. \end{eqnarray*}

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