Shorted operators and minus order

CONTINO MAXIMILIANO; GIRIBET JUAN; ALEJANDRA MAESTRIPIERI

LINEAR AND MULTILINEAR ALGEBRA

TAYLOR & FRANCIS LTD

Lugar: Londres; Año: 2018 vol. 67 p. 2173 - 2190

0308-1087

Let $HH$ be a Hilbert space, $L(HH)$ the algebra of bounded linear operators on $HH$ and $W in L(HH)$ a positive operator. Given a closed subspace $St$ of $HH$, we characterize the shorted operator $W_{/ St}$ of $W$ to $St$ as the maximum and as the infimum of certain sets, for the {it minus order} $minus$. Also, given $A in L(HH)$ with closed range, we study the following operator approximation problem considering the minus order: $$min_{minus} {(AX-I)^*W(AX-I) : X in L(HH) ,label{eqa2} mbox{ subject to } N(A^*W)subseteq N(X) }.$$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.