  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.
