A generalization of Araki's log-majorization

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• 作者：Fumio Hiai ^{hiai.fumio@gmail.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：15A42 ; 15A60 ; 47A30
• 刊名：Linear Algebra and its Applications
• 出版年：2016
• 出版时间：15 July 2016
• 年：2016
• 卷：501
• 期：Complete
• 页码：1-16
• 全文大小：366 K

文摘

We generalize Araki's log-majorization to the log-convexity theorem for the eigenvalues of Φ(A^p)^1/2Ψ(B^p)Φ(A^p)^1/2$\mathrm{\Phi }{(A^p)}^{1/2}\mathrm{\Psi }(B^p)\mathrm{\Phi }{(A^p)}^{1/2}$ as a function of p≥0$p\ge 0$, where A, B   are positive semidefinite matrices and Φ, Ψ are positive linear maps between matrix algebras. Similar generalizations of the log-majorization of Ando–Hiai type for the weighted geometric mean $A{\mathrm{\#}}_{\alpha }B$ are also given, including a lemma on general operator means $A\sigma E$ of A and a projection E.