Some operator inequalities
Hu, Yao-Zhong
Séminaire de probabilités de Strasbourg, Tome 28 (1994), p. 316-333 / Harvested from Numdam
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Publié le : 1994-01-01
@article{SPS_1994__28__316_0,
     author = {Hu, Yao-Zhong},
     title = {Some operator inequalities},
     journal = {S\'eminaire de probabilit\'es de Strasbourg},
     volume = {28},
     year = {1994},
     pages = {316-333},
     mrnumber = {1329122},
     zbl = {0820.47020},
     language = {en},
     url = {http://dml.mathdoc.fr/item/SPS_1994__28__316_0}
}

Hu, Yao-Zhong. Some operator inequalities. Séminaire de probabilités de Strasbourg, Tome 28 (1994) pp. 316-333. http://gdmltest.u-ga.fr/item/SPS_1994__28__316_0/

[1] T. Ando. Topics on Operator Inequalities, mimeographed lecture notes, Hokkaido Univ., Sapporo, 1978. | Zbl 0388.47024

[2] T. Ando. Comparison of the norms ∥f(A)-f(B)∥ and ∥f(|A-B|)∥, Math. Z., 197, 1988, 403-409. | MR 926848 | Zbl 0618.47007

[3] T. Ando. On some operator inequalities, Math. Ann., 279, 1987, 157-159. | MR 912844 | Zbl 0613.47020

[4] T. Ando. Concavity of certain maps on positive definite matrices and applications to Hadamard products, Linear Alg. Appl., 26, 1979, 203-241. | MR 535686 | Zbl 0495.15018

[5] H. Araki. On quasi-free states of CAR and Bogoliubov automorphisms, Publ. RIMS Kyoto Univ., 6, 1971, 385-442. | MR 295702 | Zbl 0227.46061

[6] H. Araki and S. Yamagami. An inequality for Hilbert-Schmidt norm, Comm. Math. Phys., 81, 1981, 89-96. | MR 630332 | Zbl 0468.47013

[7] J. Bendat and S. Sherman. Monotone and convex operator functions, Trans. Amer. Math. Soc., 79, 1955, 58-71. | MR 82655 | Zbl 0064.36901

[8] F.A. Berezin. Convex operator functions, Math. Sbornik, 17, 1972, 269-277. | MR 300121 | Zbl 0279.47006

[9] R.C. Blei. An elementary proof of the Grothendieck inequality, Proc. Amer. Math. Soc., 100, 1987, 58-60. | MR 883401 | Zbl 0637.46017

[10] P.J. Bushell and G.B. Trustrum. Trace inequalities for positive definitive power products, Linear Alg. Appl., 132, 1990, 173-178. | MR 1058100 | Zbl 0701.15015

[11] M. Breitenecker and H.R. Grümm. Note on trace inequalities, Comm. Math. Phys., 26, 1972, 276-279. | MR 312309 | Zbl 0254.47036

[12] K.J. Couteur. Representation of the function Tr(exp(A - λB)) as a Laplace transformation with positive weight and some matrix inequalities, J. Phys. A., 13, 1980, 3147-3149. | MR 587741 | Zbl 0446.44002

[13] G. Davis. Notions generalizing convexity for functions defined on space of matrices, Proc. Symposia in Pure Math. : "Convexity", 1963, 159-170. | MR 155837 | Zbl 0196.30303

[14] P. Deift. Applications of a commutation formula, Duke Math. J., 45, 1978, 267-310. | MR 495676 | Zbl 0392.47013

[15] W. Donoghue. Monotone Matrix Functions and Analytic Continuation, Springer, 1974. | MR 486556 | Zbl 0278.30004

[16] H. Epstein. Remarks on two theorems of E. Lieb, Comm. Math. Phys., 31, 1973, 317-325. | MR 343073 | Zbl 0257.46089

[17] S. Friedland and M. Katz. On a matrix inequality, Linear Alg. Appl., 85, 1987, 185-190. | MR 869352 | Zbl 0612.15010

[18] S. Golden. Lower bounds for the Herglotz functions, Phys. Rev., 137 B, 1965, 1127-1128. | MR 189691 | Zbl 0125.23204

[19] A. Grothendieck. Résumé de la théorie métrique des produits tensoriels topologiques, Bol. Mat. Sao Paulo, 8, 1956, 1-79. | MR 94682 | Zbl 0074.32303

[20] U. Haagerup. The Grothendieck inequality for the bilinear forms on C* - algebras, Adv. in Math., 56, 1985, 93-116. | MR 788936 | Zbl 0593.46052

[21] U. Haagerup. A new upper bound for the complex Grothendieck constant, Israel J. Math., 60, 1987, 199-224. | MR 931877 | Zbl 0646.46019

[22] F. Hansen and G.K. Pedersen. Jensen's inequality for operators and Löwner's theorem. Math. Ann., 258, 1982, 229-241. | MR 649196 | Zbl 0473.47011

[23] R.A. Horn. The Hadamard product, Proc. Symp. in Applied Math., AMS, 40, 1990, 87-169. | MR 1059485 | Zbl 0712.15011

[24] Y.Z. Hu. Calculs formels sur les E.D.S. de Stratonovitch, Sém. Prob. 'XXIV,s LNM 1426, Springer, 1990, 453-460. | Numdam | MR 1071561 | Zbl 0702.60055

[25] J.L. Krivine. Constante de Grothendieck et fonctions de type positif sur les sphères, Adv. Math., 31, 1979, 16-30. | MR 521464 | Zbl 0413.46054

[26] A. Korányi. On a theorem of Löwner and its connection with resolvents of selfadjoint transformations, Acta Sci. Math. (Szeged), 17, 1956, 63-70. | MR 82656 | Zbl 0074.10202

[27] A. Lenard. Generalization of the Golden-Thompson inequality Tr(eAeB) ≥ Tr(eA+B), Indiana Univ. Math. J., 21, 1971, 457-467. | MR 289536 | Zbl 0215.08606

[28] E.H. Lieb. Convex trace functions and the Wigner-Yanase-Dyson conjecture, Adv. in Math., 11, 1973, 267-288. | MR 332080 | Zbl 0267.46055

[29] E.H. Lieb and W.E. Thirring. Inequalities for the moments of the eigenvalues of the Schrödinger Hamiltonian and their relation to Sobolev inequalities, Essays in honour of V. Bargman, Princeton, N.J., 1976. | Zbl 0342.35044

[30] K. Löwner. Über monotone Matrixfunctionen, Math. Z., 38, 1934, 117-216. | JFM 60.0055.01 | Zbl 0008.11301

[31] G. Pisier. Grothendieck's theorem for noncommutative C*-algebras, with an appendix on Grothendieck's constants, J. Funct. Anal., 29, 1978, 397-415. | MR 512252 | Zbl 0388.46043

[32] G. Pisier. Factorization of Linear Operators and Geometry of Banach Spaces, CBMS Regional Conf. Series in Math., 60, AMS, 1986. | MR 829919 | Zbl 0588.46010

[33] R. Reitz. A proof of the Grothendieck inequality, Israel J. Math., 19, 1974, 271-276. | MR 367628 | Zbl 0321.46018

[34] M. Rosenblum and J. Rovnyak. Hardy Classes and Operator Theory, Oxford Univ. Press, 1985. | MR 822228 | Zbl 0586.47020

[35] B. Simon. Trace Ideals and Their Applications, London Math. Soc. Lect. Notes Series 35, Cambridge Univ. Press, 1979. | MR 541149 | Zbl 0423.47001

[36] G. Sparr. A new proof of Löwner's theorem on monotone matrix functions, Math. Scand., 47, 1980, 266-274. | MR 612700 | Zbl 0433.15012

[37] B.Sz. Nagy. Remarks on the preceding paper of A. Korányi, Acta Sci. Math. (Szeged) 17, 1956, 71-75. | MR 82657 | Zbl 0074.10203

[38] C. Thompson. Inequalities with applications in statistical mechanics, J. Math. Phys., 6, 1965, 1812-1813. | MR 189688

[39] C. Thompson. Inequalities and partial orders on matrix spaces, Indiana Univ. Math. J., 21, 1971, 469-480. | MR 294372 | Zbl 0227.15005

The following papers have been known to us after the paper was written. J. Lindenstrauss and A. Pelczynski. Absolutely summing operators in Lp spaces and their applications, Studia Math., 19, 1968, 275-326. | Zbl 0183.40501

K. Ball, E.A. Carlen and E.H. Lieb. Sharp uniform convexity and smoothness inequalities for trace norms, preprint, 1993. | MR 1262940

T. Furuta. A proof by operator means of an order preserving inequality. Linear Alg. Appl., 113 :129-130 (1989). | MR 978587 | Zbl 0770.47003

M. Fujii, T. Furuta, K. Kamei. Operator functions associated with Furuta's inequality, Linear Alg. Appl., 149 :91-96 (1991). | MR 1092871 | Zbl 0725.47023

U. Haagerup. Solution of the similarity próblem for cyclic representations of C* - algebras, Ann. of M., 118, 1983, 215-240. | MR 717823 | Zbl 0543.46033

U. Haagerup and G. Pisier. Bounded linear operators between C*-algebras, preprint (1993). | MR 1240608