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

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• 作者：Maximiliano Contino^b ; ^{mcontino@fi.uba.ar} ; Juan Giribet^a ; ^b ; ^{jgiribet@fi.uba.ar} ; Alejandra Maestripieri^a ; ^b ; ^{amaestri@fi.uba.ar}
• 关键词：47A58 ; 47B10 ; 41A65
• 刊名：Linear Algebra and its Applications
• 出版年：2017
• 出版时间：1 April 2017
• 年：2017
• 卷：518
• 期：Complete
• 页码：177-197
• 全文大小：399 K
• 卷排序：518

文摘

Let HH be a Hilbert space, L(H)L(H) the algebra of bounded linear operators on HH and W∈L(H)W∈L(H) a positive operator such that W1/2W1/2 is in the p-Schatten class, for some 1≤p<∞1≤p<∞. Given A,B∈L(H)A,B∈L(H) with closed range and C∈L(H)C∈L(H), we study the following weighted approximation problem: analyze the existence ofequation(0.1)minX∈L(H)‖AXB−C‖p,W, where ‖X‖_{p,W=‖W1/2X‖p‖X‖p,W=‖W1/2X‖p. We also study the related operator approximation problem: analyze the existence ofequation(0.2)minX∈L(H)(AXB−C)⁎W(AXB−C), where the order is the one induced in L(H)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)=0A⁎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.}