Weighted Procrustes problems

Abstract

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∈L(H) with closed range and 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. 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).