  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 $A \in L(\mathcal{H})$ with closed range and $B \in
L(\mathcal{H}),$ we study the following weighted approximation problem: analize
the existence of $$\underset{X \in L(\mathcal{H})}{min}\Vert AX-B
\Vert_{p,W},$$ where $\Vert X \Vert_{p,W}=\Vert W^{1/2}X \Vert_{p}.$
  In this paper we prove that the existence of this minimum is equivalent to a
compatibility condition between $R(B)$ and $R(A)$ involving the weight $W,$ and
we characterize the operators which minimize this problem as $W$-inverses of
$A$ in $R(B).$

Functional Analysis