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Inequalities for Two Operators

  • Silvestru Sever Dragomir
Chapter
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Part of the SpringerBriefs in Mathematics book series (BRIEFSMATH)

Abstract

In this chapter we present recent results obtained by the author concerning the norms and the numerical radii of two bounded linear operators. The proofs of the results are elementary. Some vector inequalities in inner product spaces as well as inequalities for means of nonnegative real numbers are also employed.

Keywords

Numerical Radius Vector Inequalities Nonnegative Real Numbers Kittaneh Cartesian Decomposition 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Dragomir, S.S.: A generalisation of Grüss’ inequality in inner product spaces and applications. J. Math. Anal. Applic. 237, 74–82 (1999)CrossRefzbMATHGoogle Scholar
  2. 2.
    Dragomir, S.S.: Some Grüss type inequalities in inner product spaces. J. Inequal. Pure & Appl. Math. 4(2), Article 42 (2003)Google Scholar
  3. 3.
    Dragomir, S.S.: New reverses of Schwarz, triangle and Bessel inequalities in inner product spaces. Australian J. Math. Anal. & Appl. 1(1), Article 1, 1–18 (2004)Google Scholar
  4. 4.
    Dragomir, S.S.: Reverses of Schwarz, triangle and Bessel inequalities in inner product spaces. J. Inequal. Pure & Appl. Math., 5(3), Article 76 (2004)Google Scholar
  5. 5.
    Dragomir, S.S.: A counterpart of Schwarz’s inequality in inner product spaces. East Asian Math. J. 20(1), 1–10 (2004)zbMATHGoogle Scholar
  6. 6.
    Dragomir, S.S.: Advances in Inequalities of the Schwarz, Gruss and Bessel Type in Inner Product Spaces, Nova Science Publishers, Inc., New York (2005)Google Scholar
  7. 7.
    Dragomir, S.S.: Inequalities for the norm and the numerical radius of composite operators in Hilbert spaces. Inequalities and applications. Internat. Ser. Numer. Math. 157, 135–146 Birkhäuser, Basel, 2009. Preprint available in RGMIA Res. Rep. Coll. 8 (2005), Supplement, Article 11Google Scholar
  8. 8.
    Dragomir, S.S.: Some inequalities for the Euclidean operator radius of two operators in Hilbert spaces. Linear Algebra Appl. 419(1), 256–264 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Dragomir, S.S.: Reverse inequalities for the numerical radius of linear operators in Hilbert spaces. Bull. Austral. Math. Soc. 73(2), 255–262 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Dragomir, S.S.: Reverses of the Schwarz inequality generalising a Klamkin-McLenaghan result. Bull. Austral. Math. Soc.73(1), 69–78 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Dragomir, S.S.: Inequalities for the norm and the numerical radius of linear operators in Hilbert spaces. Demonstratio Math. 40(2), 411–417 (2007)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Dragomir, S.S.: Norm and numerical radius inequalities for a product of two linear operators in Hilbert spaces. J. Math. Inequal. 2(4), 499–510 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Dragomir, S.S.: New inequalities of the Kantorovich type for bounded linear operators in Hilbert spaces. Linear Algebra Appl. 428(11–12), 2750–2760 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Dragomir, S.S.: Some inequalities of the Grüss type for the numerical radius of bounded linear operators in Hilbert spaces. J. Inequal. Appl. 2008, Art. ID 763102, 9 pp. Preprint, RGMIA Res. Rep. Coll. 11(1) (2008)Google Scholar
  15. 15.
    Dragomir, S.S.: Some inequalities for commutators of bounded linear operators in Hilbert spaces, Preprint. RGMIA Res. Rep. Coll.11(1), Article 7 (2008)Google Scholar
  16. 16.
    Dragomir, S.S.: Power inequalities for the numerical radius of a product of two operators in Hilbert spaces.Sarajevo J. Math. 5(18)(2), 269–278 (2009)Google Scholar
  17. 17.
    Dragomir, S.S.: A functional associated with two bounded linear operators in Hilbert spaces and related inequalities. Ital. J. Pure Appl. Math. No. 27, 225–240 (2010)zbMATHGoogle Scholar
  18. 18.
    Dragomir, S.S., SÁNDOR, J.: Some inequalities in prehilbertian spaces. Studia Univ. “Babeş-Bolyai” - Mathematica 32(1), 71–78 (1987)Google Scholar
  19. 19.
    Goldstein, A., Ryff, J.V., Clarke, L.E.: Problem 5473. Amer. Math. Monthly 75(3), 309 (1968)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Gustafson, K.E., Rao, D.K.M. Numerical Range Springer, New York, Inc. (1997)CrossRefGoogle Scholar
  21. 21.
    Halmos, P.R.: A Hilbert Space Problem Book, 2nd edn. Springer-Verlag, New York, Heidelberg, Berlin (1982)CrossRefzbMATHGoogle Scholar
  22. 22.
    Kittaneh, F.: Notes on some inequalities for Hilbert space operators. Publ. Res. Inst. Math. Sci. 24, 283–293 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Kittaneh, F.: A numerical radius inequality and an estimate for the numerical radius of the Frobenius companion matrix. Studia Math. 158(1), 11–17 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Kittaneh, F.: Numerical radius inequalities for Hilbert space operators. Studia Math. 168(1), 73–80 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Pearcy, C.: An elementary proof of the power inequality for the numerical radius. Michigan Math. J. 13, 289–291 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Popescu, G.: Unitary invariants in multivariable operator theory. Mem. Amer. Math. Soc. 200(941), vi+91 (2009) ISBN: 978-0-8218-4396-3. Preprint, Arχiv.math.OA/0410492Google Scholar

Copyright information

© Silvestru Sever Dragomir 2013

Authors and Affiliations

  • Silvestru Sever Dragomir
    • 1
    • 2
  1. 1.College of Engineering and ScienceVictoria UniversityMelbourneAustralia
  2. 2.School of Computational and Applied MathematicsUniversity of the Withwatersrand BraamfonteinJohannesburgSouth Africa

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