Maps preserving the Douglas solution of operator equations

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