PL EN


Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
1997 | 38 | 1 | 119-181
Tytuł artykułu

Log-majorizations and norm inequalities for exponential operators

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
Concise but self-contained reviews are given on theories of majorization and symmetrically normed ideals, including the proofs of the Lidskii-Wielandt and the Gelfand-Naimark theorems. Based on these reviews, we discuss logarithmic majorizations and norm inequalities of Golden-Thompson type and its complementary type for exponential operators on a Hilbert space. Furthermore, we obtain norm convergences for the exponential product formula as well as for that involving operator means.
Słowa kluczowe
Rocznik
Tom
38
Numer
1
Strony
119-181
Opis fizyczny
Daty
wydano
1997
Twórcy
autor
  • Department of Mathematics, Ibaraki University, Mito, Ibaraki 310, Japan
Bibliografia
  • [1] T. Ando, Topics on Operator Inequalities, Lecture notes (mimeographed), Hokkaido Univ., Sapporo, 1978.
  • [2] T. Ando, On some operator inequalities, Math. Ann. 279 (1987), 157-159.
  • [3] T. Ando, Comparison of norms |||f(A)-f(B)||| and |||f(|A-B|)|||, Math. Z. 197 (1988), 403-409.
  • [4] T. Ando, Majorization, doubly stochastic matrices and comparison of eigenvalues, Linear Algebra Appl. 118 (1989), 163-248.
  • [5] T. Ando, Majorizations and inequalities in matrix theory, ibid. 199 (1994), 17-67.
  • [6] T. Ando and F. Hiai, Log majorization and complementary Golden-Thompson type inequalities, ibid. 197/198 (1994), 113-131.
  • [7] H. Araki, Relative Hamiltonian for faithful normal states of a von Neumann algebra, Publ. Res. Inst. Math. Sci. 9 (1973), 165-209.
  • [8] H. Araki, Golden-Thompson and Peierls-Bogolubov inequalities for a general von Neumann algebras, Comm. Math. Phys. 34 (1973), 167-178.
  • [9] H. Araki, Some properties of modular conjugation operator of von Neumann algebras and a non-commutative Radon-Nikodym theorem with a chain rule, Pacific J. Math. 50 (1974), 309-354.
  • [10] H. Araki, Relative entropy of states of von Neumann algebras, Publ. Res. Inst. Math. Sci. 11 (1976), 809-833.
  • [11] H. Araki, Relative entropy for states of von Neumann algebras II, ibid. 13 (1977), 173-192.
  • [12] H. Araki, On an inequality of Lieb and Thirring, Lett. Math. Phys. 19 (1990), 167-170.
  • [13] B. Beauzamy, Introduction to Banach Spaces and Their Geometry, North-Holland, Amsterdam, 1982.
  • [14] D. S. Bernstein, Inequalities for matrix exponentials, SIAM J. Matrix Anal. Appl. 9 (1988), 156-158.
  • [15] R. Bhatia, Perturbation Bounds for Matrix Eigenvalues, Pitman Res. Notes in Math. Ser. 162, Longman, Harlow, 1987.
  • [16] R. Bhatia and C. Davis, A Cauchy-Schwarz inequality for operators with applications, Linear Algebra Appl. 223/224 (1995), 119-129.
  • [17] G. Birkhoff, Tres observaciones sobre el algebra lineal, Univ. Nac. Tucuman Rev. Ser. A 5 (1946), 147-151.
  • [18] M. Sh. Birman, L. S. Koplienko, and M. Z. Solomyak, Estimates for the spectrum of the difference between fractional powers of two self-adjoint operators, Soviet Math. (Iz. VUZ) 19, no. 3 (1975), 1-6.
  • [19] P. R. Chernoff, Note on product formulas for operator semigroups, J. Funct. Anal. 2 (1968), 238-242.
  • [20] J. Dixmier, Formes linéaires sur un anneau d'opérateurs, Bull. Soc. Math. France 81 (1953), 9-39.
  • [21] P. G. Dodds, T. K. Dodds, and B. de Pagter, Non-commutative Banach function spaces, Math. Z. 201 (1989), 583-597.
  • [22] P. G. Dodds, T. K. Dodds, and B. de Pagter, Non-commutative Köthe duality, Trans. Amer. Math. Soc. 339 (1993), 717-750.
  • [23] M. J. Donald, Relative hamiltonians which are not bounded from above, J. Funct. Anal. 91 (1990), 143-173.
  • [24] W. Donoghue, Monotone Matrix Functions and Analytic Continuation, Springer, New York, 1974.
  • [25] T. Fack, Sur la notion de valeur caractéristique, J. Operator Theory 7 (1982), 307-333.
  • [26] T. Fack and H. Kosaki, Generalized s-numbers of τ-measurable operators, Pacific J. Math. 123 (1986), 269-300.
  • [27] M. Fujii, T. Furuta, and E. Kamei, Furuta's inequality and its application to Ando's theorem, Linear Algebra Appl. 179 (1993), 161-169.
  • [28] T. Furuta, A ≥ B ≥ 0 assures $(B^{r} A^{p} B^{r})^{1/q} ≥ B^{(p+2r)/q}$ for r ≥ 0, p ≥ 0, q ≥ 1 with (1+2r)q ≥ p+2r, Proc. Amer. Math. Soc. 101 (1987), 85-88.
  • [29] T. Furuta, Applications of order preserving operator inequalities, in: Operator Theory: Adv. Appl. 59, Birkhäuser, Basel, 1992, 180-190.
  • [30] T. Furuta, Extension of the Furuta inequality and Ando-Hiai log-majorization, Linear Algebra Appl. 219 (1995), 139-155.
  • [31] I. Gohberg, S. Goldberg, and M. A. Kaashoek, Classes of Linear Operators I, Birkhäuser, Basel, 1990.
  • [32] I. C. Gohberg and M. G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators, Trnansl. Math. Monographs 18, Amer. Math. Soc., 1969.
  • [33] S. Golden, Lower bounds for Helmholtz function, Phys. Rev. 137 (1965), B1127-B1128.
  • [34] U. Haagerup, The standard form of von Neumann algebras, Math. Scand. 37 (1975), 271-283.
  • [35] U. Haagerup, $L^p$-spaces associated with an arbitrary von Neumann algebra, in: Colloq. Internat. CNRS 274, 1979, 175-184.
  • [36] F. Hansen and G. K. Pedersen, Jensen's inequality for operators and Löwner's theorem, Math. Ann. 258 (1982), 229-241.
  • [37] B. Helffer, Around the transfer operator and the Trotter-Kato formula, in: Operator Theory: Adv. Appl. 78, Birkhäuser, Basel, 1995, 161-174.
  • [38] F. Hiai, Majorization and stochastic maps in von Neumann algebras, J. Math. Anal. Appl. 127 (1987), 18-48.
  • [39] F. Hiai, Spectral majorization between normal operators in von Neumann algebras, in: Operator Algebras and Operator Theory, W. B. Arveson et al. (eds.), Pitman Res. Notes in Math. Ser. 271, Longman, Harlow, 1992, 78-115.
  • [40] F. Hiai, Trace norm convergence of exponential product formula, Lett. Math. Phys. 33 (1995), 147-158.
  • [41] F. Hiai and Y. Nakamura, Majorization for generalized s-numbers in semifinite von Neumann algebras, Math. Z. 195 (1987), 17-27.
  • [42] F. Hiai and Y. Nakamura, Distance between unitary orbits in von Neumann algebras, Pacific J. Math. 138 (1989), 259-294.
  • [43] F. Hiai and Y. Nakamura, Closed convex hulls of unitary orbits in von Neumann algebras, Trans. Amer. Math. Soc. 323 (1991), 1-38.
  • [44] F. Hiai and D. Petz, The Golden-Thompson trace inequality is complemented, Linear Algebra Appl. 181 (1993), 153-185.
  • [45] M. Hilsum, Les espaces $L^p$ d'une algèbre de von Neumann définies par la derivée spatiale, J. Funct. Anal. 40 (1981), 151-169.
  • [46] T. Ichinose and H. Tamura, Error estimate in operator norm for Trotter-Kato product formula, Integral Equations Operator Theory 27 (1997), 195-207.
  • [47] T. Ichinose and H. Tamura, Error bound in trace norm for Trotter-Kato product formula of Gibbs semigroups, preprint, 1996.
  • [48] E. Kamei, Double stochasticity in finite factors, Math. Japon. 29 (1984), 903-907.
  • [49] E. Kamei, An order on statistical operators implicitly introduced by von Neumann, Math. Japon. 30 (1985), 891-895.
  • [50] T. Kato, A generalization of the Heinz inequality, Proc. Japan Acad. 37 (1961), 305-308.
  • [51] T. Kato, Trotter's product formula for an arbitrary pair of self-adjoint contraction semigroups, in: Topics in Functional Analysis, I. Gohberg and M. Kac (eds.), Adv. Math. Suppl. Stud. 3, Academic Press, New York, 1978, 185-195.
  • [52] F. Kittaneh, Norm inequalities for fractional powers of positive operators, Lett. Math. Phys. 27 (1993), 279-285.
  • [53] H. Kosaki, Non-commutative Lorentz spaces associated with a semi-finite von Neumann algebra and applications, Proc. Japan Acad. Ser. A Math. Sci. 57 (1981), 303-306.
  • [54] H. Kosaki, Relative entropy of states: a variational expression, J. Operator Theory 16 (1986), 335-348.
  • [55] H. Kosaki, An inequality of Araki-Lieb-Thirring (von Neumann algebra case), Proc. Amer. Math. Soc. 114 (1992), 477-481.
  • [56] H. Kosaki, private communication, 1995.
  • [57] F. Kubo and T. Ando, Means of positive linear operators, Math. Ann. 246 (1980), 205-224.
  • [58] A. Lenard, Generalization of the Golden-Thompson inequality $Tr(e^{A}e^{B}) ≥ Tre^{A+B}$, Indiana Univ. Math. J. 21 (1971), 457-467.
  • [59] E. H. Lieb, Convex trace functions and the Wigner-Yanase-Dyson conjecture, Adv. in Math. 11 (1973), 267-288.
  • [60] E. H. Lieb and W. Thirring, in: Studies in Mathematical Physics, E. H. Lieb, B. Simon, and A. S. Wightman (eds.), Princeton Univ. Press, Princeton, 1976, 301-302.
  • [61] A. S. Markus, The eigen- and singular values of the sum and product of linear operators, Russian Math. Surveys 19 (4) (1964), 91-120.
  • [62] A. W. Marshall and I. Olkin, Inequalities: Theory of Majorization and Its Applications, Academic Press, New York, 1979.
  • [63] L. Mirsky, Symmetric gauge functions and unitarily invariant norms, Quart. J. Math. Oxford Ser. (2) 11 (1960), 50-59.
  • [64] Y. Nakamura, An inequality for generalized s-numbers, Integral Equations Operator Theory 10 (1987), 140-145.
  • [65] H. Neidhardt and V. A. Zagrebnov, The Trotter-Kato product formula for Gibbs semigroups, Comm. Math. Phys. 131 (1990), 333-346.
  • [66] E. Nelson, Notes on non-commutative integration, J. Funct. Anal. 15 (1974), 103-116.
  • [67] J. von Neumann, Some matrix-inequalities and metrization of matrix-space, Tomsk Univ. Rev. 1 (1937), 286-299 (Collected Works, Vol. IV, 205-218).
  • [68] M. Ohya and D. Petz, Quantum Entropy and Its Use, Springer, Berlin, 1993.
  • [69] D. Petz, Spectral scale of self-adjoint operators and trace inequalities, J. Math. Anal. Appl. 109 (1985), 74-82.
  • [70] D. Petz, A variational expression for the relative entropy, Comm. Math. Phys. 114 (1988), 345-349.
  • [71] D. Petz, A survey of certain trace inequalities, in: Functional Analysis and Operator Theory, Banach Center Publ. 30, Inst. Math., Polish Acad. Sci., Warszawa, 1994, 287-298.
  • [72] R. T. Powers and E. Størmer, Free states of the canonical anticommutation relations, Comm. Math. Phys. 16 (1970), 1-33.
  • [73] M. Reed and B. Simon, Methods of Modern Mathematical Physics II, Academic Press, New York, 1975.
  • [74] M. Reed and B. Simon, Methods of Modern Mathematical Physics IV, Academic Press, New York, 1978.
  • [75] D. L. Rogava, Error bounds for Trotter-type formulas for self-adjoint operators, Functional Anal. Appl. 27 (1993), 217-219.
  • [76] M. B. Ruskai, Inequalities for traces on von Neumann algebras, Comm. Math. Phys. 26 (1972), 280-289.
  • [77] R. Schatten, Norm Ideals of Completely Continuous Operators, Springer, Berlin, 1970.
  • [78] I. Segal, A non-commutative extension of abstract integration, Ann. of Math. (2) 57 (1953), 401-457.
  • [79] B. Simon, Trace Ideals and Their Applications, London Math. Soc. Lecture Note Ser. 35, Cambridge Univ. Press, Cambridge, 1979.
  • [80] S. Strătilă, Modular Theory in Operator Algebras, Editura Academiei and Abacus Press, Tunbridge Wells, 1981.
  • [81] M. Suzuki, Quantum statistical Monte Carlo methods and applications to spin systems, J. Statist. Phys. 43 (1986), 883-909.
  • [82] M. Suzuki, Convergence of general decompositions of exponential operators, Comm. Math. Phys. 163 (1994), 491-508.
  • [83] K. Symanzik, Proof and refinements of an inequality of Feynman, J. Math. Phys. 6 (1965), 1155-1156.
  • [84] C. J. Thompson, Inequality with applications in statistical mechanics, ibid. 6 (1965), 1812-1813.
  • [85] C. J. Thompson, Inequalities and partial orders on matrix spaces, Indiana Univ. Math. J. 21 (1971), 469-480.
  • [86] A. Uhlmann, Relative entropy and the Wigner-Yanase-Dyson-Lieb concavity in an interpolation theory, Comm. Math. Phys. 54 (1977), 21-32.
  • [87] H. Umegaki, Conditional expectation in an operator algebra, IV (entropy and information), Kōdai Math. Sem. Rep. 14 (1962), 59-85.
  • [88] B.-Y. Wang and M.-P. Gong, Some eigenvalue inequalities for positive semidefinite matrix power products, Linear Algebra Appl. 184 (1993), 249-260.
  • [89] K. Watanabe, Some results on non-commutative Banach function spaces, Math. Z. 210 (1992), 555-572.
  • [90] K. Watanabe, Some results on non-commutative Banach function spaces II (infinite cases), Hokkaido Math. J. 22 (1993), 349-364.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-bcpv38i1p119bwm
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.