Weighted Procrustes problems

Let HH be a Hilbert space, L(H)L(H) the algebra of bounded linear operators on HH and W∈L(H)W∈L(H) a positive operator such that W1/2W1/2 is in the p-Schatten class, for some 1≤p<∞1≤p<∞. Given A∈L(H)A∈L(H) with closed range and B∈L(H)B∈L(H), we study the following weighted approximation problem: analyze the existence ofminX∈L(H)‖AX−B‖p,W, where ‖X‖p,W=‖W1/2X‖p‖X‖p,W=‖W1/2X‖p. In this paper we prove that the existence of this minimum is equivalent to a compatibility condition between R(B)R(B) and R(A)R(A) involving the weight W, and we characterize the operators which minimize this problem as W-inverses of A   in R(B)R(B).