Shorted operators and minus order

MAESTRIPIERI, A.; GIRIBET, J. I.; CONTINO, M.

LINEAR AND MULTILINEAR ALGEBRA

TAYLOR & FRANCIS LTD

Año: 2018 p. 1 - 18

0308-1087

Let H be a Hilbert space, L(H) the algebra of bounded linear opera- tors 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.