Weighted least square solutions of the equation AXB-C=0

TL;DR

This paper investigates weighted least squares solutions for operator equations in Hilbert spaces, establishing conditions for the existence of solutions and characterizing the minimizers in a weighted Schatten p-norm setting.

Contribution

It provides new conditions for the existence of solutions to weighted operator approximation problems and characterizes the operators where the minimum is attained.

Findings

Existence of solutions is linked to the normal equation $A^*W(AXB-C)=0$.

Sufficient conditions for the existence of minima are established.

Characterization of operators where the minimum is attained is provided.

Abstract

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 such that $W^{1/2}$ is in the p-Schatten class, for some $1 \leq p< \infty.$ Given $A, B \in L(\mathcal{H})$ with closed range and $C \in L(\mathcal{H}),$ we study the following weighted approximation problem: analize the existence of \begin{equation}\label{eqa1} \underset{X \in L(\mathcal{H})}{min}\Vert AXB-C \Vert_{p,W}, \ \ \ \ (1) \end{equation} where $\Vert X \Vert_{p,W}=\Vert W^{1/2}X \Vert_{p}.$ We also study the related operator approximation problem: analize the existence of \begin{equation} \label{eqa2} \underset{X \in L(\mathcal{H})}{min} (AXB-C)^{*}W(AXB-C), \ \ \ \ (2) \end{equation} where the order is the one induced in $L(\mathcal{H})$ by the cone of positive operators. In this paper we prove that the existence of the minimum of (2) 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 (1) and we characterize the operators where the minimum is attained.