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.
Categories: math.FA