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1994 | 30 | 1 | 287-298
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A survey of certain trace inequalities

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Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Słowa kluczowe
Rocznik
Tom
30
Numer
1
Strony
287-298
Opis fizyczny
Daty
wydano
1994
Twórcy
autor
  • Department of Mathematics, Faculty of Chemical Engineering, Technical University Budapest, Sztoczek u. 2, H ép II. 25, H-1521 Budapest XI, Hungary
Bibliografia
  • [1] P. M. Alberti and A. Uhlmann, Stochasticity and Partial Order. Doubly Stochastic Maps and Unitary Mixing, Deutscher Verlag Wiss., Berlin, 1981.
  • [2] T. Ando, Majorization, doubly stochastic matrices and comparison of eigenvalues, Linear Algebra Appl. 118 (1989), 163-248.
  • [3] T. Ando, private communication, 1992.
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  • [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.
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  • [8] M. Breitenbecker and H. R. Grümm, Note on trace inequalities, Comm. Math. Phys. 26 (1972), 276-279.
  • [9] J. E. Cohen, Inequalities for matrix exponentials, Linear Algebra Appl. 111 (1988), 25-28.
  • [10] J. E. Cohen, S. Friedland, T. Kato and F. P. Kelly, Eigenvalue inequalities for products of matrix exponentials, ibid. 45 (1982), 55-95.
  • [11] S. Friedland and W. So, On the product of matrix exponentials, ibid. 196 (1994), 193-205.
  • [12] J. I. Fujii and E. Kamei, Relative operator entropy in noncommutative information theory, Math. Japon. 34 (1989), 341-348.
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  • [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] F. Hiai and D. Petz, The proper formula for relative entropy and its asymptotics in quantum probability, Comm. Math. Phys. 143 (1991), 99-114.
  • [16] F. Hiai and D. Petz, The Golden-Thompson trace inequality is complemented, Linear Algebra Appl. 181 (1993), 153-185.
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  • [19] E. H. Lieb, Convex trace functions and the Wigner-Yanase-Dyson conjecture, Adv. in Math. 11 (1973), 267-288.
  • [20] A. W. Marshall and I. Olkin, Inequalities: Theory of Majorization and Its Applications, Academic Press, New York, 1979.
  • [21] A. W. Marshall and I. Olkin, Inequalities for the trace function, Aequationes Math. 29 (1985), 36-39.
  • [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.
  • [23] D. Petz, Sufficient subalgebras and the relative entropy of states on a von Neumann algebra, Comm. Math. Phys. 105 (1986), 123-131.
  • [24] D. Petz, A variational expression for the relative entropy, ibid. 114 (1988), 345-348.
  • [25] D. Petz and J. Zemánek, Characterizations of the trace, Linear Algebra Appl. 111 (1988), 43-52.
  • [26] W. Pusz and S. L. Woronowicz, Functional calculus for sesquilinear forms and the purification map, Rep. Math. Phys. 8 (1975), 159-170.
  • [27] D. Ruelle, Statistical Mechanics. Rigorous Results, Benjamin, New York, 1969.
  • [28] M. B. Ruskai, Inequalities for traces on von Neumann algebras, Comm. Math. Phys. 26 (1972), 280-289.
  • [29] M. B. Ruskai and F. K. Stillinger, Convexity inequalities for estimating free energy and relative entropy, J. Phys. A 23 (1990), 2421-2437.
  • [30] W. So, Equality cases in matrix exponential inequalities, SIAM J. Matrix Anal. Appl. 13 (1992), 1154-1158.
  • [31] R. F. Streater, Convergence of the quantum Boltzmann map, Comm. Math. Phys. 98 (1985), 177-185.
  • [32] C. J. Thompson, Inequality with applications in statistical mechanics, J. Math. Phys. 6 (1965), 1812-1813.
  • [33] R. C. Thompson, High, low and qualitative roads in linear algebra, Linear Algebra Appl. 162-164 (1992), 23-64.
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Bibliografia
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