Maps preserving the Douglas solution of operator equations

We consider bijective maps $\varphi$ on the full operator algebra $\mathcal{B}(\mathcal{H})$ of an infinite dimensional Hilbert space with the property that, for every $A,B,X\in \mathcal{B}(\mathcal{H})$, $X$ is the Douglas solution of the equation $A=BX$ if and only if $Y=\varphi (X)$ is the Douglas solution of the equation $\varphi (A)=\varphi (B)Y$. We prove that those maps are implemented by a unitary or anti-unitary map $U$, i.e., $\varphi (A)=UA{U}^{\ast }$.