Generalization of the Lieb–Thirring–Araki Inequality and Its Applications
The matrix eigenvalue is very important in matrix analysis, and it has been applied to matrix trace inequalities, such as the Lieb–Thirring–Araki theorem and Thompson–Golden theorem. In this manuscript, we obtain a matrix eigenvalue inequality by using the Stein–Hirschman operator interpolation ineq...
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| Published in | Mathematics (Basel) Vol. 9; no. 7; p. 723 |
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| Abstract | The matrix eigenvalue is very important in matrix analysis, and it has been applied to matrix trace inequalities, such as the Lieb–Thirring–Araki theorem and Thompson–Golden theorem. In this manuscript, we obtain a matrix eigenvalue inequality by using the Stein–Hirschman operator interpolation inequality; then, according to the properties of exterior algebra and the Schur-convex function, we provide a new proof for the generalization of the Lieb–Thirring–Araki theorem and Furuta theorem. |
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| AbstractList | The matrix eigenvalue is very important in matrix analysis, and it has been applied to matrix trace inequalities, such as the Lieb–Thirring–Araki theorem and Thompson–Golden theorem. In this manuscript, we obtain a matrix eigenvalue inequality by using the Stein–Hirschman operator interpolation inequality; then, according to the properties of exterior algebra and the Schur-convex function, we provide a new proof for the generalization of the Lieb–Thirring–Araki theorem and Furuta theorem. |
| Author | Wang, Jing Sun, Huafei Li, Yonggang |
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| Cites_doi | 10.1016/0024-3795(94)90484-7 10.1090/conm/529/10428 10.1090/pspum/007/0155837 10.1063/1.1704727 10.1016/j.aim.2020.107208 10.1561/2200000048 10.1137/0120053 10.1007/BF01045887 10.1007/s10955-015-1237-6 10.1007/978-0-387-68276-1 10.1016/j.na.2019.111640 10.1512/iumj.1972.21.21036 10.1007/978-0-387-21554-9 10.1007/978-1-4612-0653-8 10.1103/PhysRev.137.B1127 10.1007/978-3-642-65755-9 |
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| Copyright | 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License. |
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| References_xml | – volume: 197–198 start-page: 113 year: 1994 ident: ref_8 article-title: Log majorization and complementary Golden-Thompson type inequalities publication-title: Linear Algebra Appl. doi: 10.1016/0024-3795(94)90484-7 – 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 – volume: Volume 7 start-page: 187 year: 1963 ident: ref_20 article-title: Notions generalizing convexity for functions defined on spaces of matrices publication-title: Proceedings of the Symposia in Pure Mathematics doi: 10.1090/pspum/007/0155837 – volume: 6 start-page: 1812 year: 1965 ident: ref_6 article-title: Inequality with applications in statistical mechanics publication-title: J. Math. Phys. doi: 10.1063/1.1704727 – volume: 369 start-page: 107208 year: 2020 ident: ref_12 article-title: Generalizing Lieb’s concavity theorem via operator interpolation publication-title: Adv. Math. doi: 10.1016/j.aim.2020.107208 – ident: ref_2 – ident: ref_10 – volume: 8 start-page: 1 year: 2015 ident: ref_19 article-title: An introduction to matrix concentration inequalities publication-title: Found. Trends Mach. Learn. doi: 10.1561/2200000048 – volume: 20 start-page: 520 year: 2006 ident: ref_3 article-title: Shorted operators publication-title: SIAM J. Appl. Math. doi: 10.1137/0120053 – ident: ref_15 – volume: 19 start-page: 167 year: 1990 ident: ref_9 article-title: On an inequality of Lieb and Thirring publication-title: Lett. Math. Phys. doi: 10.1007/BF01045887 – volume: 159 start-page: 1300 year: 2015 ident: ref_23 article-title: Golden-Thompson’s inequality for deformed exponentials publication-title: J. Stat. Phys. doi: 10.1007/s10955-015-1237-6 – ident: ref_11 doi: 10.1007/978-0-387-68276-1 – volume: 191 start-page: 111640 year: 2020 ident: ref_1 article-title: A necessary and sufficient condition on the equivalence between local and global optimal solutions in variational control problems publication-title: Nonlinear Anal. doi: 10.1016/j.na.2019.111640 – volume: 21 start-page: 457 year: 1971 ident: ref_7 article-title: Generalization of the Golden-Thompson inequality Tr(eAeB) ≥ TreA+B publication-title: Zndiana Univ. Math. doi: 10.1512/iumj.1972.21.21036 – ident: ref_16 doi: 10.1007/978-0-387-21554-9 – ident: ref_17 – volume: 101 start-page: 85 year: 1987 ident: ref_13 article-title: A ≥ B ≥ 0 assures (BrApBr)1q ≥ Bp+2rq for r ≥ 0, p > 0, q ≥ 1 with 1+2rq ≥ p + 2r publication-title: Proc. Am. Math. Sot. – ident: ref_14 doi: 10.1007/978-1-4612-0653-8 – ident: ref_18 – ident: ref_22 – volume: 137 start-page: 1127 year: 1965 ident: ref_5 article-title: Lower bounds for Helmholtz function publication-title: Phys. Rev. doi: 10.1103/PhysRev.137.B1127 – ident: ref_21 doi: 10.1007/978-3-642-65755-9 |
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| SubjectTerms | Algebra Eigenvalues Inequality Interpolation Lieb–Thirring–Araki theorem Mathematical functions matrix eigenvalue Matrix methods Operators (mathematics) Theorems Thompson–Golden theorem |
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