Weighted Procrustes problems

TL;DR

This paper investigates a weighted operator approximation problem in Hilbert spaces, establishing conditions for the existence of solutions and characterizing minimizers as W-inverses, extending classical least squares concepts.

Contribution

It provides necessary and sufficient conditions for the existence of solutions to weighted Procrustes problems and characterizes minimizers as W-inverses, generalizing previous unweighted results.

Findings

Existence of minimum linked to a compatibility condition involving the weight W.

Characterization of minimizers as W-inverses of A in R(B).

Extension of classical least squares to weighted operator settings.

Abstract

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 $R(B).$