# Solutions of the system of operator equations $BXA=B=AXB$ via $*$-order

## TL;DR

This paper investigates conditions for solutions to a specific operator equation system involving bounded linear operators on a Hilbert space, introducing the concept of inverse along an operator and characterizing solutions via the *-order.

## Contribution

It establishes necessary and sufficient conditions for solutions, characterizes solutions using the *-order, and provides the general solution and characterizations of the *-order in this context.

## Key findings

- Solutions exist under specific conditions related to the *-order.
- An operator is a solution iff it satisfies a *-order inequality.
- The paper provides the general form of solutions.

## 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 $B \stackrel{*}{ \leq} AXA$, where the $*$-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.