Application of the Pick Function in the Lieb Concavity Theorem for Deformed Exponentials
The Lieb concavity theorem, successfully solved in the Wigner–Yanase–Dyson conjecture, is an important application of matrix concave functions. Recently, the Thompson–Golden theorem, a corollary of the Lieb concavity theorem, was extended to deformed exponentials. Hence, it is worthwhile to study th...
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| Published in | Fractal and fractional Vol. 6; no. 1; p. 20 |
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| Abstract | The Lieb concavity theorem, successfully solved in the Wigner–Yanase–Dyson conjecture, is an important application of matrix concave functions. Recently, the Thompson–Golden theorem, a corollary of the Lieb concavity theorem, was extended to deformed exponentials. Hence, it is worthwhile to study the Lieb concavity theorem for deformed exponentials. In this paper, the Pick function is used to obtain a generalization of the Lieb concavity theorem for deformed exponentials, and some corollaries associated with exterior algebra are obtained. |
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| AbstractList | The Lieb concavity theorem, successfully solved in the Wigner–Yanase–Dyson conjecture, is an important application of matrix concave functions. Recently, the Thompson–Golden theorem, a corollary of the Lieb concavity theorem, was extended to deformed exponentials. Hence, it is worthwhile to study the Lieb concavity theorem for deformed exponentials. In this paper, the Pick function is used to obtain a generalization of the Lieb concavity theorem for deformed exponentials, and some corollaries associated with exterior algebra are obtained. |
| Author | Sun, Huafei Yang, Guozeng Wang, Jing Li, Yonggang |
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| Cites_doi | 10.1561/9781601988393 10.1090/pspum/007/0155837 10.1016/0024-3795(79)90179-4 10.1016/j.aim.2013.07.019 10.1090/conm/529/10428 10.1063/1.3573594 10.1007/978-0-387-68276-1 10.1007/s11005-020-01289-7 10.4153/CJM-1964-041-x 10.1016/j.laa.2019.06.013 10.1073/pnas.0807965106 10.1007/s10955-013-0890-x 10.1090/S0002-9947-1955-0082655-4 10.1007/978-3-642-65755-9 10.1016/0001-8708(73)90011-X 10.1215/ijm/1256051007 10.1007/978-1-4612-0653-8 10.1007/BF01646492 |
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| References_xml | – ident: ref_1 doi: 10.1561/9781601988393 – ident: ref_15 doi: 10.1090/pspum/007/0155837 – volume: 26 start-page: 203 year: 1979 ident: ref_10 article-title: Concavity of certain maps on positive definite matrices and applications to Hadamard products publication-title: Linear Algebra Appl. doi: 10.1016/0024-3795(79)90179-4 – volume: 248 start-page: 531 year: 2013 ident: ref_23 article-title: The simplest proof of Lieb concavity theorem publication-title: Adv. Math. doi: 10.1016/j.aim.2013.07.019 – volume: 529 start-page: 73 year: 2010 ident: ref_4 article-title: Trace inequalities and quantum entropy: An introductory course publication-title: Contemp. Math. doi: 10.1090/conm/529/10428 – ident: ref_6 – ident: ref_9 – ident: ref_5 – ident: ref_3 – ident: ref_2 – volume: 52 start-page: 043505 year: 2011 ident: ref_22 article-title: A simple proof of Lieb concavity theorem publication-title: J. Math. Phys. doi: 10.1063/1.3573594 – ident: ref_20 doi: 10.1007/978-0-387-68276-1 – volume: 110 start-page: 2203 year: 2020 ident: ref_13 article-title: Variational representations related to Tsallis relative entropy publication-title: Lett. Math. Phys. doi: 10.1007/s11005-020-01289-7 – volume: 16 start-page: 397 year: 1964 ident: ref_7 article-title: On the positive definite nature of certain matrix expressions publication-title: Cunud. J. Math. doi: 10.4153/CJM-1964-041-x – volume: 579 start-page: 419 year: 2019 ident: ref_19 article-title: A generalized Lieb’s theorem and its applications to spectrum estimates for a sum of random matrices publication-title: Linear Algebra Appl. doi: 10.1016/j.laa.2019.06.013 – volume: 106 start-page: 1006 year: 2009 ident: ref_11 article-title: A matrix convexity approach to some celebrated quantum inequalities publication-title: Proc. Natl. Acad. Sci. USA doi: 10.1073/pnas.0807965106 – volume: 154 start-page: 807 year: 2014 ident: ref_18 article-title: Trace Functions with Applications in Quantum Physics publication-title: J. Stat. Phys. doi: 10.1007/s10955-013-0890-x – volume: 79 start-page: 58 year: 1955 ident: ref_17 article-title: Monotone and convex operator functions publication-title: Trans. Am. Math. Soc. doi: 10.1090/S0002-9947-1955-0082655-4 – ident: ref_14 doi: 10.1007/978-3-642-65755-9 – volume: 11 start-page: 267 year: 1973 ident: ref_8 article-title: Convex trace functions and the Wigner–Yanase–Dyson conjecture publication-title: Adv. Math. doi: 10.1016/0001-8708(73)90011-X – volume: 18 start-page: 565 year: 1974 ident: ref_16 article-title: A Schwarz inequality for positive linear maps on C★-algebras publication-title: Illinois J. Math. doi: 10.1215/ijm/1256051007 – ident: ref_21 doi: 10.1007/978-1-4612-0653-8 – volume: 31 start-page: 317 year: 1973 ident: ref_12 article-title: Remarks on two theorems of E. Lieb publication-title: Comm. Math. Phys. doi: 10.1007/BF01646492 |
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| Title | Application of the Pick Function in the Lieb Concavity Theorem for Deformed Exponentials |
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