Weighted Procrustes problems

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• 作者：Maximiliano Contino^a ; ^{mcontino@fi.uba.ar" class="auth_mail" title="E-mail the corresponding author} ; Juan Ignacio Giribet^a ; ^b ; ^{jgiribet@fi.uba.ar" class="auth_mail" title="E-mail the corresponding author} ; Alejandra Maestripieri^a ; ^b ; ^{amaestri@fi.uba.ar" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Operator approximation ; Schatten p classes ; Oblique projections
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：1 January 2017
• 年：2017
• 卷：445
• 期：1
• 页码：443-458
• 全文大小：391 K

文摘

Let H$\mathcal{H}$ be a Hilbert space, L(H)$L(\mathcal{H})$ the algebra of bounded linear operators on H$\mathcal{H}$ and W∈L(H)$W\in L(\mathcal{H})$ a positive operator such that W^1/2$W^{1/2}$ is in the p-Schatten class, for some 1≤p<∞$1\le p<\infty$. Given A∈L(H)$A\in L(\mathcal{H})$ with closed range and B∈L(H)$B\in L(\mathcal{H})$, we study the following weighted approximation problem: analyze the existence of where ‖X‖_p,W=‖W^1/2X‖_p${\Vert X\Vert }_{p,W}={\Vert W^{1/2}X\Vert }_p$. In this paper we prove that the existence of this minimum is equivalent to a compatibility condition between R(B)$R(B)$ 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)$R(B)$.