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 $W_{/ \mathcal{S}}$ of $W$ to $\mathcal{S}$ as the maximum and as the infimum of certain sets, for the minus order $\stackrel{-}{\leq}.$ Also, given $A \in L(\mathcal{H})$ with closed range, we study the following operator approximation problem considering the minus order: $$min_{\stackrel{-}{\leq}} \ \{(AX-I)^*W(AX-I) : X \in L(\mathcal{H}), \mbox{ subject to } N(A^*W)\subseteq N(X) \}.$$ We show that, under certain conditions, the shorted operator $W_{/R(A)}$ (of $W$ to the range of $A$) is the minimum of this problem and we characterize the set of solutions.