Solutions of the system of operator equations $BXA=B=AXB$ via $\ast$-order

In this paper, we establish some necessary and sufficient conditions for the existence of solutions to the system of operator equations $BXA=B=AXB$ in the setting of bounded linear operators on a Hilbert space, where the unknown operator $X$ is called the inverse of $A$ along $B$. After that, under some mild conditions we prove that an operator $X$ is a solution of $BXA=B=AXB$ if and only if $BAXA$, where the $\ast$-order $C\stackrel{}{ \leq} D$means$CC^=DC^*, C^*C=C^D$. Moreover we present the general solution of the equation above. Finally, we present some characterizations of $C \stackrel{}{ \leq} D$ via other operator equations.