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

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 $W^{1\mathrm{/}2}$ is in the p-Schatten class, for some $1 \leq p< \infty.$Given$A, B \in L($\mathcal{H}$)$withclosedrangeand$C \in L($\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 $\mathrm{\parallel }X{\mathrm{\parallel }}_{p,W}=\mathrm{\parallel }{W}^{1\mathrm{/}2}X{\mathrm{\parallel }}_p\mathrm{.}$ 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 $A^{\ast }W(AXB-C)=0.$ We also give sufficient conditions for the existence of the minimum of (1) and we characterize the operators where the minimum is attained.