Concavity of certain matrix trace and norm functions. II

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• 作者：Fumio Hiai ^{hiai.fumio@gmail.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 15A60 ; 47A30 ; 47A60
• 刊名：Linear Algebra and its Applications
• 出版年：2016
• 出版时间：1 May 2016
• 年：2016
• 卷：496
• 期：Complete
• 页码：193-220
• 全文大小：504 K

文摘

We refine Epstein's method to prove joint concavity/convexity of matrix trace functions of Lieb type a409a2cb63b30686555e376153">$\mathrm{Tr}f(\mathrm{\Phi }{(A^p)}^{1/2}\mathrm{\Psi }(B^q)\mathrm{\Phi }{(A^p)}^{1/2})$ and symmetric (anti-) norm functions of the form $\Vert f(\mathrm{\Phi }(A^p)\sigma \mathrm{\Psi }(B^q))\Vert$, where Φ and Ψ are positive linear maps, σ   is an operator mean, and f(x^γ)$f(x^{\gamma })$ with a certain power γ   is an operator monotone function on (0,∞)$(0,\infty )$. Moreover, the variational method of Carlen, Frank and Lieb is extended to general non-decreasing convex/concave functions on (0,∞)$(0,\infty )$ so that we prove joint concavity/convexity of more trace functions of Lieb type.