Shorted operators and minus order

Let $\mathcal{H}$ be a Hilbert space, $L(\mathcal{H})$ the algebra of bounded linear operators on $\mathcal{H}$ and $W\in L(\mathcal{H})$ a positive operator. Given a closed subspace $\mathcal{S}$ of $\mathcal{H}$, we characterize the shorted operator $W_{\mathrm{/}\mathcal{S}}$ of $W$ to $\mathcal{S}$ as the maximum and as the infimum of certain sets, for the minus order $\mathrm{.}$ Also, given $A\in L(\mathcal{H})$ with closed range, we study the following operator approximation problem considering the minus order: $$mi{n}_{}\{(AX-I{)}^{\ast }W(AX-I):X\in L(\mathcal{H}),subjecttoN(A^{\ast }W)\subseteq N(X)\}\mathrm{.}$$ We show that, under certain conditions, the shorted operator $W_{\mathrm{/}R(A)}$ (of $W$ to the range of $A$) is the minimum of this problem and we characterize the set of solutions.