Maps preserving the Douglas solution of operator equations

We consider bijective maps $ϕ$ 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 $Y=ϕ(X)$ is the Douglas solution of the equation $ϕ(A)=ϕ(B)Y$. We prove that those maps are implemented by a unitary or anti-unitary map $U$, i.e., $ϕ(A)=UAU^*$.

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