Weighted least squares solutions of the equation AXB − C = 0

Let H be a Hilbert space, L(H) the algebra of bounded linear operators on H and W∈L(H) a positive operator such that W1/2 is in the p-Schatten class, for some 1≤p<∞. Given A,B∈L(H) with closed range and C∈L(H), we study the following weighted approximation problem: analyze the existence of(0.1)minX∈L(H)‖AXB−C‖p,W, where ‖X‖p,W=‖W1/2X‖p. We also study the related operator approximation problem: analyze the existence of(0.2)minX∈L(H)(AXB−C)⁎W(AXB−C), where the order is the one induced in L(H) by the cone of positive operators. In this paper we prove that the existence of the minimum of (0.2) 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 (0.1) and we characterize the operators where the minimum is attained.