Korean Journal of Mathematics

The Kangwon-Kyungki Mathematical Society (강원경기수학회)
권호 표지
DOI QR코드

DOI QR Code

SOME TRACE INEQUALITIES FOR CONVEX FUNCTIONS OF SELFADJOINT OPERATORS IN HILBERT SPACES

  • Received : 2016.05.24
  • Accepted : 2016.06.18
  • Published : 2016.06.30
Download PDF
⟨ Previous Next ⟩

Abstract

Some new trace inequalities for convex functions of self-adjoint operators in Hilbert spaces are provided. The superadditivity and monotonicity of some associated functionals are investigated. Some trace inequalities for matrices are also derived. Examples for the operator power and logarithm are presented as well.

Keywords

References

  1. T. Ando, Matrix Young inequalities, Oper. Theory Adv. Appl. 75 (1995), 33-38.
  2. R. Bellman, Some inequalities for positive definite matrices, in: E.F. Beckenbach (Ed.), General Inequalities 2, Proceedings of the 2nd International Conference on General Inequalities, Birkhauser, Basel, 1980, pp. 89-90.
  3. E. V. Belmega, M. Jungers and S. Lasaulce, A generalization of a trace inequality for positive definite matrices, Aust. J. Math. Anal. Appl. 7 (2) (2010), Art. 26, 5 pp.
  4. E. A. Carlen, Trace inequalities and quantum entropy: an introductory course, Entropy and the quantum, 73-140, Contemp. Math., 529, Amer. Math. Soc., Providence, RI, 2010.
  5. D. Chang, A matrix trace inequality for products of Hermitian matrices, J. Math. Anal. Appl. 237 (1999), 721-725. https://doi.org/10.1006/jmaa.1999.6433
  6. L. Chen and C.Wong, Inequalities for singular values and traces, Linear Algebra Appl. 171 (1992), 109-120. https://doi.org/10.1016/0024-3795(92)90253-7
  7. I. D. Coop, On matrix trace inequalities and related topics for products of Hermitian matrix, J. Math. Anal. Appl. 188 (1994), 999-1001. https://doi.org/10.1006/jmaa.1994.1475
  8. S. S. Dragomir, A converse result for Jensen's discrete inequality via Gruss' inequality and applications in information theory, An. Univ. Oradea Fasc. Mat. 7 (1999/2000), 178-189.
  9. S. S. Dragomir, On a reverse of Jessen's inequality for isotonic linear functionals, J. Ineq. Pure & Appl. Math., 2 (3) (2001), Article 36.
  10. S. S. Dragomir, A Gruss type inequality for isotonic linear functionals and applications, Demonstratio Math. 36 (3) (2003), 551-562. Preprint RGMIA Res. Rep. Coll. 5(2002), Suplement, Art. 12. [ONLINE:http://rgmia.org/v5(E).php].
  11. S. S. Dragomir, Bounds for the normalized Jensen functional, Bull. Austral. Math. Soc. 74 (3) (2006), 471-476. https://doi.org/10.1017/S000497270004051X
  12. S. S. Dragomir, Bounds for the deviation of a function from the chord generated by its extremities, Bull. Aust. Math. Soc. 78 (2) (2008), 225-248. https://doi.org/10.1017/S0004972708000671
  13. S. S. Dragomir, Gruss' type inequalities for functions of self-adjoint operators in Hilbert spaces, Preprint, RGMIA Res. Rep. Coll., 11(e) (2008), Art. 11. [ONLINE: http://rgmia.org/v11(E).php].
  14. S. S. Dragomir, Some inequalities for convex functions of self-adjoint operators in Hilbert spaces, Filomat 23 (3) (2009), 81-92. Preprint RGMIA Res. Rep. Coll., 11(e) (2008), Art. 10. https://doi.org/10.2298/FIL0903081D
  15. S. S. Dragomir, Some Jensen's type inequalities for twice differentiable functions of self-adjoint operators in Hilbert spaces, Filomat 23 (3) (2009), 211-222. Preprint RGMIA Res. Rep. Coll., 11(e) (2008), Art. 13. https://doi.org/10.2298/FIL0903211D
  16. S. S. Dragomir, Some new Gruss' type inequalities for functions of self-adjoint operators in Hilbert spaces, Sarajevo J. Math. 6(18), (1) (2010), 89-107. Preprint RGMIA Res. Rep. Coll., 11(e) (2008), Art. 12. [ONLINE: http://rgmia.org/v11(E).php].
  17. S. S. Dragomir, New bounds for the Cebysev functional of two functions of self-adjoint operators in Hilbert spaces, Filomat 24 (2) (2010), 27-39. https://doi.org/10.2298/FIL1002027D
  18. S. S. Dragomir, Some Jensen's type inequalities for log-convex functions of self-adjoint operators in Hilbert spaces, Bull. Malays. Math. Sci. Soc. 34 (3) (2011), Preprint RGMIA Res. Rep. Coll., 13(2010), Sup. Art. 2.
  19. S. S. Dragomir, Some reverses of the Jensen inequality for functions of self-adjoint operators in Hilbert spaces, J. Ineq. & Appl., Vol. 2010, Article ID 496821. Preprint RGMIA Res. Rep. Coll., 11(e) (2008), Art. 15. [ONLINE: http://rgmia.org/v11(E).php].
  20. S. S. Dragomir, Some Slater's type inequalities for convex functions of self-adjoint operators in Hilbert spaces, Rev. Un. Mat. Argentina, 52 (1) (2011), 109-120. Preprint RGMIA Res. Rep. Coll., 11(e) (2008), Art. 7.
  21. S. S. Dragomir, Hermite-Hadamard's type inequalities for operator convex functions, Appl. Math. Comp. 218 (2011), 766-772. Preprint RGMIA Res. Rep. Coll., 13(2010), No. 1, Art. 7. https://doi.org/10.1016/j.amc.2011.01.056
  22. S. S. Dragomir, Hermite-Hadamard's type inequalities for convex functions of self-adjoint operators in Hilbert spaces, Preprint RGMIA Res. Rep. Coll., 13 (2) (2010), Art 1.
  23. S. S. Dragomir, New Jensen's type inequalities for differentiable log-convex functions of self-adjoint operators in Hilbert spaces, Sarajevo J. Math. 19 (1) (2011), 67-80. Preprint RGMIA Res. Rep. Coll., 13(2010), Sup. Art. 2.
  24. S. S. Dragomir, Operator Inequalities of the Jensen, Cebysev and Gruss Type. Springer Briefs in Mathematics. Springer, New York, 2012. xii+121 pp. ISBN: 978-1-4614-1520-6.
  25. S. S. Dragomir, Operator Inequalities of Ostrowski and Trapezoidal Type. Springer Briefs in Mathematics. Springer, New York, 2012. x+112 pp. ISBN: 978-1-4614-1778-1
  26. S. S. Dragomir and N. M. Ionescu, Some converse of Jensen's inequality and applications, Rev. Anal. Numer. Theor. Approx. 23 (1) (1994), 71-78. MR:1325895 (96c:26012).
  27. S. Furuichi and M. Lin, Refinements of the trace inequality of Belmega, Lasaulce and Debbah, Aust. J. Math. Anal. Appl. 7 (2) (2010), Art. 23, 4 pp.
  28. T. Furuta, J. Micic Hot, J. Pecaric and Y. Seo, Mond-Pecaric Method in Operator Inequalities. Inequalities for Bounded self-adjoint Operators on a Hilbert Space, Element, Zagreb, 2005.
  29. G. Helmberg,Introduction to Spectral Theory in Hilbert Space, John Wiley, New York, 1969.
  30. H. D. Lee, On some matrix inequalities, Korean J. Math. 16 (2) (2008), 565-571.
  31. L. Liu, A trace class operator inequality, J. Math. Anal. Appl. 328 (2007), 1484-1486. https://doi.org/10.1016/j.jmaa.2006.04.092
  32. S. Manjegani, Holder and Young inequalities for the trace of operators, Positivity 11 (2007), 239-250. https://doi.org/10.1007/s11117-006-2054-6
  33. H. Neudecker, A matrix trace inequality, J. Math. Anal. Appl. 166 (1992), 302-303. https://doi.org/10.1016/0022-247X(92)90344-D
  34. K. Shebrawi and H. Albadawi, Operator norm inequalities of Minkowski type, J. Inequal. Pure Appl. Math. 9 (1) (2008), 1-10, article 26.
  35. K. Shebrawi and H. Albadawi, Trace inequalities for matrices, Bull. Aust. Math. Soc. 87 (2013), 139-148. https://doi.org/10.1017/S0004972712000627
  36. B. Simon, Trace Ideals and Their Applications, Cambridge University Press, Cambridge, 1979.
  37. A. Matkovic, J. Pecaric and I. Peric, A variant of Jensen's inequality of Mercer's type for operators with applications., Linear Algebra Appl. 418 (2006), no. 2-3, 551-564. https://doi.org/10.1016/j.laa.2006.02.030
  38. C. A. McCarthy, $c_p$, Israel J. Math., 5 (1967), 249-271. https://doi.org/10.1007/BF02771613
  39. J. Micic, Y. Seo, S.-E. Takahasi and M. Tominaga, Inequalities of Furuta and Mond-Pecaric, Math. Ineq. Appl., 2(1999), 83-111.
  40. B. Mond and J. Pecaric, Convex inequalities in Hilbert space, Houston J. Math., 19 (1993), 405-420.
  41. B. Mond and J. Pecaric, On some operator inequalities, Indian J. Math. 35 (1993), 221-232.
  42. B. Mond and J. Pecaric, Classical inequalities for matrix functions, Utilitas Math., 46 (1994), 155-166.
  43. S. Simic, On a global upper bound for Jensen's inequality, J. Math. Anal. Appl. 343 (2008), 414-419. https://doi.org/10.1016/j.jmaa.2008.01.060
  44. Z. Ulukok and R. Turkmen, On some matrix trace inequalities. J. Inequal. Appl. 2010, Art. ID 201486, 8 pp.
  45. X. Yang, A matrix trace inequality, J. Math. Anal. Appl. 250 (2000), 372-374. https://doi.org/10.1006/jmaa.2000.7068
  46. X. M. Yang, X. Q. Yang and K. L. Teo, A matrix trace inequality, J. Math. Anal. Appl. 263 (2001), 327-331. https://doi.org/10.1006/jmaa.2001.7613
  47. Y. Yang, A matrix trace inequality, J. Math. Anal. Appl. 133 (1988), 573-574. https://doi.org/10.1016/0022-247X(88)90423-4

Cited by

  1. The right Riemann-Liouville fractional Hermite-Hadamard type inequalities derived from Green’s function vol.10, pp.4, 2016, https://doi.org/10.1063/1.5143908
  2. Random matrix theory of the isospectral twirling vol.10, pp.3, 2016, https://doi.org/10.21468/scipostphys.10.3.076