Weighted Procrustes problems

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 $1\le p<\infty .$ 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 $R(B).$

Cited by in corpus (2)

• Weighted least square solutions of the equation AXB-C=0
• Weighted operator least squares problems and the J-trace in Krein spaces