Weighted least squares solutions of the equation $AXB-C=0$

CONTINO MAXIMILIANO; GIRIBET J. I.; A. MAESTRIPIERI

LINEAR ALGEBRA AND ITS APPLICATIONS

ELSEVIER SCIENCE INC

Lugar: Amsterdam; Año: 2017

0024-3795

Let $\HH$ be a Hilbert space, $L(\HH)$ the algebra of bounded linear operators on $\HH$ and $W \in L(\HH)$ a positive operator such that $W^{1/2}$ is in the p-Schatten class, for some $1 \leq p< \infty.$ Given $A, B \in L(\HH)$ with closed range and $C \in L(\HH),$ we study the following weighted approximation problem: analize the existence of \begin{equation}\underset{X \in L(\HH)}{min}\Nphissw{AXB-C}, \label{eqa1}\end{equation}where $\Nphissw{X}=\Nphiss{W^{1/2}X}.$ We also study the related operator approximation problem: analize the existence of \begin{equation}\underset{X \in L(\HH)}{min} (AXB-C)^{*}W(AXB-C), \label{eqa2}\end{equation} where the order is the one induced in $L(\HH)$ by the cone of positive operators.In this paper we prove that the existence of the minimum of \eqref{eqa2} is equivalent to the existence of a solution of the normal equation $A^*W(AXB-C)=0.$ We also give sufficient conditions for the existence of the minimum of \eqref{eqa1} and we characterize the operators where the minimum is attained.