A survey of certain trace inequalities

Dénes Petz

Banach Center Publications (1994)

  • Volume: 30, Issue: 1, page 287-298
  • ISSN: 0137-6934

Abstract

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This paper concerns inequalities like TrA ≤ TrB, where A and B are certain Hermitian complex matrices and Tr stands for the trace. In most cases A and B will be exponential or logarithmic expressions of some other matrices. Due to the interest of the author in quantum statistical mechanics, the possible applications of the trace inequalities will be commented from time to time. Several inequalities treated below have been established in the context of Hilbert space operators or operator algebras. Notwithstanding these extensions our discussion will be limited to matrices.

How to cite

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Petz, Dénes. "A survey of certain trace inequalities." Banach Center Publications 30.1 (1994): 287-298. <http://eudml.org/doc/262566>.

@article{Petz1994,
abstract = {This paper concerns inequalities like TrA ≤ TrB, where A and B are certain Hermitian complex matrices and Tr stands for the trace. In most cases A and B will be exponential or logarithmic expressions of some other matrices. Due to the interest of the author in quantum statistical mechanics, the possible applications of the trace inequalities will be commented from time to time. Several inequalities treated below have been established in the context of Hilbert space operators or operator algebras. Notwithstanding these extensions our discussion will be limited to matrices.},
author = {Petz, Dénes},
journal = {Banach Center Publications},
keywords = {trace inequalities; Hermitian complex matrices; Hilbert space operators; operator algebras},
language = {eng},
number = {1},
pages = {287-298},
title = {A survey of certain trace inequalities},
url = {http://eudml.org/doc/262566},
volume = {30},
year = {1994},
}

TY - JOUR
AU - Petz, Dénes
TI - A survey of certain trace inequalities
JO - Banach Center Publications
PY - 1994
VL - 30
IS - 1
SP - 287
EP - 298
AB - This paper concerns inequalities like TrA ≤ TrB, where A and B are certain Hermitian complex matrices and Tr stands for the trace. In most cases A and B will be exponential or logarithmic expressions of some other matrices. Due to the interest of the author in quantum statistical mechanics, the possible applications of the trace inequalities will be commented from time to time. Several inequalities treated below have been established in the context of Hilbert space operators or operator algebras. Notwithstanding these extensions our discussion will be limited to matrices.
LA - eng
KW - trace inequalities; Hermitian complex matrices; Hilbert space operators; operator algebras
UR - http://eudml.org/doc/262566
ER -

References

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  1. [1] P. M. Alberti and A. Uhlmann, Stochasticity and Partial Order. Doubly Stochastic Maps and Unitary Mixing, Deutscher Verlag Wiss., Berlin, 1981. Zbl0474.46047
  2. [2] T. Ando, Majorization, doubly stochastic matrices and comparison of eigenvalues, Linear Algebra Appl. 118 (1989), 163-248. Zbl0673.15011
  3. [3] T. Ando, private communication, 1992. 
  4. [4] H. Araki, Golden-Thompson and Peierls-Bogoliubov inequalities for a general von Neumann algebra, Comm. Math. Phys. 34 (1973), 167-178. 
  5. [5] H. Araki, On an inequality of Lieb and Thirring, Lett. Math. Phys. 19 (1990), 167-170. Zbl0705.47020
  6. [6] V. P. Belavkin and P. Staszewski, C*-algebraic generalization of relative entropy and entropy, Ann. Inst. Henri Poincaré Sect. A 37 (1982), 51-58. Zbl0526.46060
  7. [7] D. S. Bernstein, Inequalities for trace of matrix exponentials, SIAM J. Matrix Anal. Appl. 9 (1988), 156-158. Zbl0658.15018
  8. [8] M. Breitenbecker and H. R. Grümm, Note on trace inequalities, Comm. Math. Phys. 26 (1972), 276-279. Zbl0254.47036
  9. [9] J. E. Cohen, Inequalities for matrix exponentials, Linear Algebra Appl. 111 (1988), 25-28. Zbl0662.15012
  10. [10] J. E. Cohen, S. Friedland, T. Kato and F. P. Kelly, Eigenvalue inequalities for products of matrix exponentials, ibid. 45 (1982), 55-95. Zbl0489.15007
  11. [11] S. Friedland and W. So, On the product of matrix exponentials, ibid. 196 (1994), 193-205. Zbl0792.15012
  12. [12] J. I. Fujii and E. Kamei, Relative operator entropy in noncommutative information theory, Math. Japon. 34 (1989), 341-348. Zbl0699.46048
  13. [13] S. Golden, Lower bounds for the Helmholtz function, Phys. Rev. 137 (1965), B1127-B1128. 
  14. [14] F. Hiai, Some remarks on the trace operator logarithm and relative entropy, in: Quantum Probability and Related Topics VIII, World Sci., Singapore, to appear. 
  15. [15] F. Hiai and D. Petz, The proper formula for relative entropy and its asymptotics in quantum probability, Comm. Math. Phys. 143 (1991), 99-114. Zbl0756.46043
  16. [16] F. Hiai and D. Petz, The Golden-Thompson trace inequality is complemented, Linear Algebra Appl. 181 (1993), 153-185. Zbl0784.15011
  17. [17] S. Klimek and A. Lesniewski, A Golden-Thompson inequality in supersymmetric quantum mechanics, Lett. Math. Phys. 21 (1991), 237-244. Zbl0723.58054
  18. [18] H. Kosaki, An inequality of Araki-Lieb-Thirring (von Neumann algebra case), Proc. Amer. Math. Soc. 114 (1992), 477-481. Zbl0762.46060
  19. [19] E. H. Lieb, Convex trace functions and the Wigner-Yanase-Dyson conjecture, Adv. in Math. 11 (1973), 267-288. Zbl0267.46055
  20. [20] A. W. Marshall and I. Olkin, Inequalities: Theory of Majorization and Its Applications, Academic Press, New York, 1979. Zbl0437.26007
  21. [21] A. W. Marshall and I. Olkin, Inequalities for the trace function, Aequationes Math. 29 (1985), 36-39. 
  22. [22] D. Petz, Properties of quantum entropy, in: Quantum Probability and Applications II, L. Accardi and W. von Waldenfels (eds.), Lecture Notes in Math. 1136, Springer, 1985, 428-441. Zbl0583.46053
  23. [23] D. Petz, Sufficient subalgebras and the relative entropy of states on a von Neumann algebra, Comm. Math. Phys. 105 (1986), 123-131. Zbl0597.46067
  24. [24] D. Petz, A variational expression for the relative entropy, ibid. 114 (1988), 345-348. 
  25. [25] D. Petz and J. Zemánek, Characterizations of the trace, Linear Algebra Appl. 111 (1988), 43-52. Zbl0661.15017
  26. [26] W. Pusz and S. L. Woronowicz, Functional calculus for sesquilinear forms and the purification map, Rep. Math. Phys. 8 (1975), 159-170. Zbl0327.46032
  27. [27] D. Ruelle, Statistical Mechanics. Rigorous Results, Benjamin, New York, 1969. 
  28. [28] M. B. Ruskai, Inequalities for traces on von Neumann algebras, Comm. Math. Phys. 26 (1972), 280-289. Zbl0257.46101
  29. [29] M. B. Ruskai and F. K. Stillinger, Convexity inequalities for estimating free energy and relative entropy, J. Phys. A 23 (1990), 2421-2437. Zbl0709.60549
  30. [30] W. So, Equality cases in matrix exponential inequalities, SIAM J. Matrix Anal. Appl. 13 (1992), 1154-1158. Zbl0759.15017
  31. [31] R. F. Streater, Convergence of the quantum Boltzmann map, Comm. Math. Phys. 98 (1985), 177-185. Zbl0573.60094
  32. [32] C. J. Thompson, Inequality with applications in statistical mechanics, J. Math. Phys. 6 (1965), 1812-1813. 
  33. [33] R. C. Thompson, High, low and qualitative roads in linear algebra, Linear Algebra Appl. 162-164 (1992), 23-64. 

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