Weighted Procrustes problems

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$A \in L($\mathcal{H}$)$withclosedrangeand$B \in L($\mathcal{H}$),$ we study the following weighted approximation problem: analize the existence of $$\mathrm{\parallel }AX-B{\mathrm{\parallel }}_{p,W},$$ where $\mathrm{\parallel }X{\mathrm{\parallel }}_{p,W}=\mathrm{\parallel }{W}^{1\mathrm{/}2}X{\mathrm{\parallel }}_p\mathrm{.}$ 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 $R(B)\mathrm{.}$