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

Abstract

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 $B\stackrel{\ast }{\le }AXA$, where the $\ast$-order $C\stackrel{\ast }{\le }D$ means $C{C}^{\ast }=D{C}^{\ast },C^{\ast }C=C^{\ast }D$. Moreover we present the general solution of the equation above. Finally, we present some characterizations of $C\stackrel{\ast }{\le }D$ via other operator equations.

13 pages, to appear in Electron. J. Linear Algebra (ELA)