Lieb concavity theorem, successfully solved the Wigner-Yanase-Dyson conjecture, which is a very important theorem, and there are many proofs of it. Generalization of the Lieb concavity theorem has been obtained by Huang, which implies that it is jointly concave for any nonnegative matrix monotone function $ f(x) $ over $ \left(\operatorname{Tr}\left[\wedge^{k}(A^{\frac{qs}{2}}K^{\ast}B^{sp}KA^{\frac{sq}{2}})^{\frac{1}{s}}\right]\right)^{\frac{1}{k}} $. In this manuscript, we obtained $ \left(\operatorname{Tr}\left[\wedge^{k}(f(A^{\frac{qs}{2}})K^{\ast}f(B^{sp})Kf(A^{\frac{sq}{2}}))^{\frac{1}{s}}\right]\right)^{\frac{1}{k}} $ was jointly concave for any nonnegative matrix monotone function $ f(x) $ by using Epstein's theorem, and some more general results were obtained.
Citation: Qiujin He, Chunxia Bu, Rongling Yang. A Generalization of Lieb concavity theorem[J]. AIMS Mathematics, 2024, 9(5): 12305-12314. doi: 10.3934/math.2024601
Lieb concavity theorem, successfully solved the Wigner-Yanase-Dyson conjecture, which is a very important theorem, and there are many proofs of it. Generalization of the Lieb concavity theorem has been obtained by Huang, which implies that it is jointly concave for any nonnegative matrix monotone function $ f(x) $ over $ \left(\operatorname{Tr}\left[\wedge^{k}(A^{\frac{qs}{2}}K^{\ast}B^{sp}KA^{\frac{sq}{2}})^{\frac{1}{s}}\right]\right)^{\frac{1}{k}} $. In this manuscript, we obtained $ \left(\operatorname{Tr}\left[\wedge^{k}(f(A^{\frac{qs}{2}})K^{\ast}f(B^{sp})Kf(A^{\frac{sq}{2}}))^{\frac{1}{s}}\right]\right)^{\frac{1}{k}} $ was jointly concave for any nonnegative matrix monotone function $ f(x) $ by using Epstein's theorem, and some more general results were obtained.
| [1] |
E. P. Wigner, M. M. Yanase, On the positive definite nature of certain matrix expressions, Cunud. J. Math., 16 (1964), 397–406. https://doi.org/10.4153/CJM-1964-041-x doi: 10.4153/CJM-1964-041-x
|
| [2] |
T. Ando, Concavity of certain maps on positive definite matrices and applications to Hadamard products, Linear Alge. Appl., 26 (1979), 203–241. https://doi.org/10.1016/0024-3795(79)90179-4 doi: 10.1016/0024-3795(79)90179-4
|
| [3] |
J. S. Aujla, A simple proof of Lieb concavity theorem, J. Math. Phys., 52 (2011), 043505. https://doi.org/10.1063/1.3573594 doi: 10.1063/1.3573594
|
| [4] | B. Bhatia, Positive definite matrices, Princeton University Press, 2007. https://doi.org/10.1515/9781400827787 |
| [5] |
E. Lieb, Convex trace functions and the Wigner-Yanase-Dyson conjecture, Adv. Math., 11 (1973), 267–288. https://doi.org/10.1016/0001-8708(73)90011-X doi: 10.1016/0001-8708(73)90011-X
|
| [6] | R. Bhatia, Matrix Analysis, Springer, 1997. https://doi.org/10.1007/978-1-4612-0653-8 |
| [7] |
E. G. Effros, A matrix convexity approach to some celebrated quantum inequalities, Proc. Natl. Acad. Sci., 106 (2009), 1006–1008. https://doi.org/10.1073/pnas.0807965106 doi: 10.1073/pnas.0807965106
|
| [8] |
I. Nikoufar, A. Ebadian, M. E. Gordji, The simplest proof of Lieb concavity theorem, Adv. Math., 248 (2013), 531–533. https://doi.org/10.1016/j.aim.2013.07.019 doi: 10.1016/j.aim.2013.07.019
|
| [9] | D. Huang, Generalizing Lieb's concavity theorem via operator interpolation, arXiv, 03304. https://doi.org/10.1016/j.aim.2020.107208 |
| [10] | F. Zhang, Matrix theory: Basic results and techniques, Springer, 1999. https://doi.org/10.1007/978-1-4614-1099-7 |
| [11] | B. Simon, Trace ideals and their applications, 2 Eds., American Mathematical Soc., 2005. https://doi.org/10.1090/surv/120 |
| [12] | E. F. Beckenbach, R. Bellman, Inequalities, Springer Science, 1961. https://doi.org/10.1007/978-3-642-64971-4 |
| [13] | R. Bellman, Introduction to matrix analysis, Classics in Applied Mathematics, 1960. https://doi.org/10.1137/1.9781611971170 |
| [14] | A. W. Marshall, I. Olkin, B. C. Arnold, Inequalities: Theory of majorization and its applications, New York: Springer, 2011. https://doi.org/10.1007/978-0-387-68276-1 |
| [15] | E. Carlen, Trace inequalities and quantum entropy: An introductory course, Hill Center, 2010. https://doi.org/10.1090/conm/529/10428 |
| [16] | C. Davis, Notions generalizing convexity for functions defined on spaces of matrices, Amer. Math. Sot., 1963,187–201. |
| [17] | F. Hansen, Correction to: Trace functions with applications in quantum physics, J. Stat. Phys., 188 (2022). https://doi.org/10.1007/s10955-022-02929-z |
| [18] |
H. Epstein, Remarks on two theorems of E. Lieb, Comm. Math. Phys., 31 (1973), 317–325. https://doi.org/10.1007/BF01646492 doi: 10.1007/BF01646492
|
| [19] | W. Donogue, Monotone matrix functions and analytic continuution, New York: Springer, 1974. https://doi.org/10.1007/978-3-642-65755-9 |
| [20] |
D. Huang, A generalized Lieb's theorem and its applications to spectrum estimates for a sum of random matrices, Linear Algebra Appl., 579 (2019), 419–448. https://doi.org/10.1016/j.laa.2019.06.013 doi: 10.1016/j.laa.2019.06.013
|
Qiujin He, Chunxia Bu, Rongling Yang. A Generalization of Lieb concavity theorem[J]. AIMS Mathematics, 2024, 9(5): 12305-12314. doi: 10.3934/math.2024601
DownLoad: