Understand
We refine Epstein's method to prove joint concavity/convexity of matrix trace functions of the extended Lieb type $Tr{\Phi(A^p)^{1/2}\Psi(B^q)\Phi(A^p)^{1/2}}^s$, where $\Phi$ and $\Psi$ are positive linear maps.
- By the same method combined with majorization technique, similar properties are proved for symmetric (anti-) norm functions of the form $||{\Phi(A^p)\sigma\Psi(B^q)}^s||$ involving an operator mean $\sigma$.
- Carlen and Lieb's variational method is also used to improve the convexity property of norm functions $||\Phi(A^p)^s||$.
Built on
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E. A. Carlen and E. H. Lieb, A Minkowski type trace inequality and strong subadditivity of quantum entropy, Advances in the Mathematical Sciences, Amer. Math. Soc. Transl. Ser. 2
1999
Cited alongside, same era.
F. Hiai, Concavity of certain matrix trace functions, Taiwanese J. Math
2001
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T. N. Bekjan, On joint convexity of trace functions, Linear Algebra Appl
2004
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T. Furuta, J. Mićić Hot, J. Pečarić and Y. Seo, Mond-Pečarić Method in Operator Inequalities
2005
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E. A. Carlen and E. H. Lieb, A Minkowski type trace inequality and strong subadditivity of quantum entropy II: convexity and concavity, Lett. Math. Phys
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F. Hiai, Matrix Analysis: Matrix Monotone Functions, Matrix Means, and Majorization (GSIS selected lectures), Interdisciplinary Information Sciences
2010
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A. Jenčová and M. B. Ruskai, A unified treatment of convexity of relative entropy and related trace functions, with conditions for equality, Rev. Math. Phys
2010
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J.-C. Bourin and F. Hiai, Norm and anti-norm inequalities for positive semi-definite matrices, Internat. J. Math
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J. Mićić, Z. Pavić and J. Pečarić, Jensen’s inequality for operators without operator convexity, Linear Algebra Appl
2011
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2013
Closest in time.
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