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Operator inequalities of Hölder type for unitarily invariant norms

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The aim of this work is to present some Hölder-type inequalities for sums and products of operators related to unitarily invariant norms. These results generalize some known Hölder inequalities for operators.

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References

  1. Albadawi, H.: Hölder-type inequalities invlving unitarily invariant norms. Positivity 16, 255–270 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  2. Albadawi, H.: Matrix inequalities related to Hölder inequality. Banach J. Math. Anal. 7, 162–171 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  3. Ando, T., Hiai, F.: Hölder-type inequalities for matrices. Math. Inequal. Appl. 1, 1–30 (1998)

    MathSciNet  MATH  Google Scholar 

  4. Arambašić, L., Bakić, D., Moslehian, M.: A treatment of the Cauchy–Schwarz inequality in \(C^{*}\)-modules. J. Math. Anal. Appl. 381, 546–556 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bhatia, R.: Positive Definite Matrices. Princeton Press, Princeton (2007)

    MATH  Google Scholar 

  6. Bhatia, R.: Matrix Analysis. Springer, Berlin (1997)

    Book  MATH  Google Scholar 

  7. Bhatia, R., Davis, C.: A Cauchy–Schwarz inequality for operators with applications. Linear Algebra Appl. 223(224), 119–129 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bourin, J., Lee, E.: Concave functions of positive operators, sums and congruences. J. Oper. Theory 63, 151–157 (2010)

    MathSciNet  MATH  Google Scholar 

  9. Fujii, J.: Operator-valued inner product and operator inequalities. Banach J. Math. Anal. 2, 59–67 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  10. Fujii, J., Fujii, M., Moslehian, M., Seo, Y.: Cauchy–Schwarz inequality in semi-inner product \(C^{*}\)-modules. J. Math. Anal. Appl. 394, 835–840 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  11. Hiai, F.: Log-majorizations and norm inequalities for exponential operators. Linear Oper Banach Cent. Publ. 38, 119–181 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  12. Hiai, F., Zhan, X.: Inequalities involving unitarily invariant norms and operator monotone functions. Linear Algebra Appl. 341, 151–169 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  13. Hirzallah, O., Kittaneh, F.: Inequalities for sums and direct sums of Hilbert space operators. Linear Algebra Appl. 424, 71–82 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Horn, R., Mathias, R.: Cauchy–Schwarz inequalities associated with positive semidefinite matrices. Linear Algebra Appl. 142, 63–82 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  15. Horn, R., Zhan, X.: Inequalities for C-S semi norms and Lieb functions. Linear Algebra Appl. 291, 103–113 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  16. Ilisević, D., Varośanec, S.: On the Cauchy–Schwarz inequality and its reverse in semi inner product \(C^{*}\)-modules. Banach J. Math. Anal. 1, 78–84 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  17. Kittaneh, F.: Notes on some inequalities for Hilbert space operators. Publ. Res. Inst. Math. Sci. 24, 283–293 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  18. Shebrawi, K., Albadawi, H.: Norms inequalities for the absolute value of Hilbert space operators. Linear Multilinear Algebra 58, 453–463 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  19. Zhan, X.: Matrix Theory. Higher Education Press, Beijing (2008). In Chinese

    MATH  Google Scholar 

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Acknowledgements

The authors would like to express their thanks to editor and reviewer(s) for their careful reading and valuable comments and suggestions, which lead a great improvement of this paper. This work is partially supported by the National Natural Science Foundation of China (NSFC) Grant 11161040.

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Authors and Affiliations

  1. College of Science, Shihezi University, Shihezi, 832003, Xinjiang, People’s Republic of China

    Jianguo Zhao

  2. College of Mathematics and Statistics, Chongqing University, Chongqing, 401331, People’s Republic of China

    Junliang Wu

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  1. Jianguo Zhao
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  2. Junliang Wu
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Correspondence to Jianguo Zhao.

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Zhao, J., Wu, J. Operator inequalities of Hölder type for unitarily invariant norms. Positivity 21, 1495–1506 (2017). https://doi.org/10.1007/s11117-017-0481-1

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  • DOI: https://doi.org/10.1007/s11117-017-0481-1

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