, Volume 63, Issue 2, pp 377–389 | Cite as

Some singular value and unitarily invariant norm inequalities for Hilbert space operators

  • A. Taghavi
  • V. Darvish
  • H. M. Nazari
  • S. S. Dragomir


In this paper, we prove some singular value inequalities for sum and product of operators. Also, we obtain several generalizations of recent inequalities. Moreover, as applications we establish some unitarily invariant norm and trace inequalities for operators which provide refinements of previous results.


Singular value inequality Unitarlily invariant norm Hermite-Hadamard inequality 

Mathematics Subject Classification

47A63 15A60 47B05 47B10 



This work was written whilst the second author was visiting Victoria University. He is grateful for the support and hospitality.


  1. 1.
    Bhatia, R.: Matrix Analysis, GTM 169. Springer, New York (1997)CrossRefGoogle Scholar
  2. 2.
    Zhan, X.: Matrix Inequalities. Springer, Berlin (2002)CrossRefMATHGoogle Scholar
  3. 3.
    Dragomir, S.S.: Hermite–Hadamard’s type inequalities for operator convex functions. Appl. Math. Comput. 218, 766–772 (2011)MathSciNetMATHGoogle Scholar
  4. 4.
    Ghazanfari, A.G.: The Hermite–Hadamard type inequalities for operator s-convex functions. J. Adv. Res. Pure Math. 6(3), 52–61 (2014)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Uchiyama, M.: Commutativity of self-adjoint operators. Pac. J. Math. 161, 385–392 (1993)CrossRefMATHGoogle Scholar
  6. 6.
    Nagisa, M., Ueda, M., Wada, S.: Commutativity of operators. Nihonkai Math. J. 17, 1–8 (2006)MathSciNetMATHGoogle Scholar
  7. 7.
    Bhatia, R., Kittaneh, F.: Notes on matrix arithmetic–geometric mean inequalities. Linear Algebra Appl. 308, 77–84 (2000)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Drury, S.W.: On a question of Bhatia and Kittaneh. Linear Algebra Appl. 437, 1955–1960 (2012)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Bhatia, R., Kittaneh, F.: On singular values of a product of operators. SIAM J. Matrix Anal. Appl. 11, 271–277 (1990)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Hirzallah, O., Kittaneh, F.: Inequalities for sums and direct sums of Hilbert space operators. Linear Algebra Appl. 424, 71–82 (2007)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Bhatia, R., Kittaneh, F.: Norm inequalities for positive operators. Lett. Math. Phys. 43, 225–231 (1998)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Ando, T., Zhan, X.: Norm inequalities related to operator monotone functions. Math. Ann. 315, 771–780 (1999)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Aujla, J.S., Silva, F.C.: Weak majorization inequalities and convex functions. Linear Algebra Appl. 369, 217–233 (2003)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Hiai, F., Zhan, X.: Inequalities involving unitarily invariant norms and operator monotone functions. Linear Algebra Appl. 341, 151–169 (2002)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Bhatia, R., Kittaneh, F.: The matrix arithmetic-geometric mean inequality revisited. Linear Algebra Appl. 428, 2177–2191 (2008)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Simon, B.: Trace Ideals and Their Applications. Cambridge University Press, Cambridge (1979)MATHGoogle Scholar
  17. 17.
    Shebrawi, Kh, Albadawi, H.: Trace inequalities for matrices. Bull. Aust. Math. Soc. 87, 139–148 (2013)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (2012)CrossRefGoogle Scholar

Copyright information

© Università degli Studi di Ferrara 2017

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Mathematical SciencesUniversity of MazandaranBabolsarIran
  2. 2.Mathematics, School of Engineering and ScienceVictoria UniversityMelbourne City MCAustralia

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