Extensions of $\mathcal{G}$-outer inverses

Abstract

Our first objective is to present equivalent conditions for the solvability of the system of matrix equations $ADA=A$, $DAB=B$ and $CAD=C$, where $D$ is unknown, $A,B,C$ are of appropriate dimensions, and to obtain its general solution in terms of appropriate inner inverses.
Our leading idea is to find characterizations and representations of a subclass of inner inverses that satisfy some properties of outer inverses.
A $\mathcal{G}$-$(B,C)$ inverse of $A$ is defined as a solution of this matrix system.
In this way, $\mathcal{G}$-$(B,C)$ inverses are defined and investigated as an extension of $\mathcal{G}$-outer inverses.
One-sided versions of $\mathcal{G}$-$(B,C)$ inverse are introduced as weaker kinds of $\mathcal{G}$-$(B,C)$ inverses and generalizations of one-sided versions of $\mathcal{G}$-outer inverse.
Applying the $\mathcal{G}$-$(B,C)$ inverse and its one-sided versions, we propose three new partial orders on the set of complex matrices.
These new partial orders extend the concepts of $\mathcal{G}$-outer $(T,S)$-partial order and one-sided $\mathcal{G}$-outer $(T,S)$-partial orders.