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 $R(B).$
Categories: math.FA