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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16303742&_mathId=si1.gif&_user=111111111&_pii=S0022247X16303742&_rdoc=1&_issn=0022247X&md5=2642dd6c0a1142a947d44e9d01c2d3b9" title="Click to view the MathML source">Hclass="mathContainer hidden">class="mathCode">$\mathcal{H}$ be a Hilbert space, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16303742&_mathId=si139.gif&_user=111111111&_pii=S0022247X16303742&_rdoc=1&_issn=0022247X&md5=9eef978474e7c1ad8117c2518b8311b9" title="Click to view the MathML source">L(H)class="mathContainer hidden">class="mathCode">$L(\mathcal{H})$ the algebra of bounded linear operators on class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16303742&_mathId=si1.gif&_user=111111111&_pii=S0022247X16303742&_rdoc=1&_issn=0022247X&md5=2642dd6c0a1142a947d44e9d01c2d3b9" title="Click to view the MathML source">Hclass="mathContainer hidden">class="mathCode">$\mathcal{H}$ and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16303742&_mathId=si3.gif&_user=111111111&_pii=S0022247X16303742&_rdoc=1&_issn=0022247X&md5=d17fda4ed8ac8094608d1f2b46e0327d" title="Click to view the MathML source">W∈L(H)class="mathContainer hidden">class="mathCode">$W\in L(\mathcal{H})$ a positive operator such that class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16303742&_mathId=si4.gif&_user=111111111&_pii=S0022247X16303742&_rdoc=1&_issn=0022247X&md5=7b44df5c427e5b10d04111fcb66e2b8a" title="Click to view the MathML source">W^1/2class="mathContainer hidden">class="mathCode">$W^{1/2}$ is in the p-Schatten class, for some class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16303742&_mathId=si158.gif&_user=111111111&_pii=S0022247X16303742&_rdoc=1&_issn=0022247X&md5=43e7ecdb29927a185e2121440427cfab" title="Click to view the MathML source">1≤p<∞class="mathContainer hidden">class="mathCode">$1\le p<\infty$. Given class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16303742&_mathId=si6.gif&_user=111111111&_pii=S0022247X16303742&_rdoc=1&_issn=0022247X&md5=2fa9d05490322ea9bce5c62697165f30" title="Click to view the MathML source">A∈L(H)class="mathContainer hidden">class="mathCode">$A\in L(\mathcal{H})$ with closed range and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16303742&_mathId=si100.gif&_user=111111111&_pii=S0022247X16303742&_rdoc=1&_issn=0022247X&md5=039deedf4f9e53dd24a664e9fd65a97a" title="Click to view the MathML source">B∈L(H)class="mathContainer hidden">class="mathCode">$B\in L(\mathcal{H})$, we study the following weighted approximation problem: analyze the existence of where class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16303742&_mathId=si9.gif&_user=111111111&_pii=S0022247X16303742&_rdoc=1&_issn=0022247X&md5=7c243f37d1a7a4b5f0fc75b0ecb2a078" title="Click to view the MathML source">‖X‖_p,W=‖W^1/2X‖_pclass="mathContainer hidden">class="mathCode">${\Vert X\Vert }_{p,W}={\Vert W^{1/2}X\Vert }_p$. In this paper we prove that the existence of this minimum is equivalent to a compatibility condition between class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16303742&_mathId=si10.gif&_user=111111111&_pii=S0022247X16303742&_rdoc=1&_issn=0022247X&md5=ffaf32fc4ac5e32349d77858cf2d538b" title="Click to view the MathML source">R(B)class="mathContainer hidden">class="mathCode">$R(B)$ and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16303742&_mathId=si11.gif&_user=111111111&_pii=S0022247X16303742&_rdoc=1&_issn=0022247X&md5=81c22a282eaff9513080ec093b91c49a" title="Click to view the MathML source">R(A)class="mathContainer hidden">class="mathCode">$R(A)$ involving the weight W, and we characterize the operators which minimize this problem as W-inverses of A   in class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16303742&_mathId=si10.gif&_user=111111111&_pii=S0022247X16303742&_rdoc=1&_issn=0022247X&md5=ffaf32fc4ac5e32349d77858cf2d538b" title="Click to view the MathML source">R(B)class="mathContainer hidden">class="mathCode">$R(B)$.