Shorted operators and minus order
Maximiliano Contino, Juan I. Giribet, Alejandra Maestripieri · Linear and Multilinear Algebra · 2018
Let H be a Hilbert space, L(H) the algebra of bounded linear operators on H and W∈L(H) a positive operator. Given a closed subspace S of H, we characterize the shorted operator W/S of W to S as the maximum and as the infimum of certain sets, for the minus order ≤−. Also, given A∈L(H) with closed range, we study the following operator approximation problem considering the minus order: min≤− {(AX−I)∗W(AX−I):X∈L(H),subject to N(A∗W)⊆N(X)}. We show that, under certain conditions, the shorted operator of W/R(A) is the minimum of this problem and we characterize the set of solutions.