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Vector Inequalities for a Projection in Hilbert Spaces and Applications

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Progress in Approximation Theory and Applicable Complex Analysis

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 117))

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Abstract

In this paper we establish some vector inequalities related to Schwarz and Buzano results. Applications for norm and numerical radius inequalities of two bounded operators are given as well.

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References

  1. Aldaz, J.M.: Strengthened Cauchy-Schwarz and Hölder inequalities. J. Inequal. Pure Appl. Math. 10 (4), Article 116, 6pp. (2009)

    Google Scholar 

  2. Buzano, M.L.: Generalizzazione della diseguaglianza di Cauchy-Schwarz (Italian). Rend. Sem. Mat. Univ. e Politech. Torino 31, 405–409 (1974) (1971/73)

    Google Scholar 

  3. Davidson, K.R., Holbrook, J.A.R.: Numerical radii of zero-one matrices. Mich. Math. J. 35, 261–267 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  4. De Rossi, A.: A strengthened Cauchy-Schwarz inequality for biorthogonal wavelets. Math. Inequal. Appl. 2 (2), 263–282 (1999)

    MathSciNet  MATH  Google Scholar 

  5. Dragomir, S.S.: Some refinements of Schwartz inequality, Simpozionul de Matematici şi Aplicaţii, Timişoara, Romania, 1–2 Noiembrie 1985, pp. 13–16

    Google Scholar 

  6. Dragomir, S.S.: A generalization of Grüss’ inequality in inner product spaces and applications. J. Math. Anal. Appl. 237, 74–82 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  7. Dragomir, S.S.: Some Grüss type inequalities in inner product spaces. J. Inequal. Pure Appl. Math. 4 (2), Art. 42 (2003)

    Google Scholar 

  8. Dragomir, S.S.: New reverses of Schwarz, triangle and Bessel inequalities in inner product spaces. Aust. J. Math. Anal. Appl. 1 (1), Art. 1, 18pp. (2004)

    Google Scholar 

  9. Dragomir, S.S.: Reverses of Schwarz, triangle and Bessel inequalities in inner product spaces. J. Inequal. Pure Appl. Math. 5 (3), Article 76, 18pp. (2004)

    Google Scholar 

  10. Dragomir, S.S.: A potpourri of Schwarz related inequalities in inner product spaces. I. J. Inequal. Pure Appl. Math. 6 (3), Article 59, 15pp. (2005)

    Google Scholar 

  11. Dragomir, S.S.: Advances in Inequalities of the Schwarz, Grüss and Bessel Type in Inner Product Spaces. Nova Science Publishers, Hauppauge, NY (2005). viii+249 pp. ISBN: 1-59454-202-3

    Google Scholar 

  12. Dragomir, S.S.: A potpourri of Schwarz related inequalities in inner product spaces. II. J. Inequal. Pure Appl. Math. 7 (1), Article 14, 11pp. (2006)

    Google Scholar 

  13. Dragomir, S.S.: Inequalities for the norm and the numerical radius of linear operators in Hilbert spaces. Demonstratio Math. 40 (2), 411–417 (2007)

    MathSciNet  MATH  Google Scholar 

  14. Dragomir, S.S.: New inequalities of the Kantorovich type for bounded linear operators in Hilbert spaces. Linear Algebra Appl. 428, 2750–2760 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. Dragomir, S.S.: Some inequalities for the norm and the numerical radius of linear operators in Hilbert spaces. Tamkang J. Math. 39 (1), 1–7 (2008)

    MathSciNet  MATH  Google Scholar 

  16. Dragomir, S.S.: Advances in Inequalities of the Schwarz, Triangle and Heisenberg Type in Inner Product Spaces. Nova Science Publishers, Hauppauge, NY (2007)

    MATH  Google Scholar 

  17. Dragomir, S.S.: Some inequalities for power series of selfadjoint operators in Hilbert spaces via reverses of the Schwarz inequality. Integral Transforms Spec. Funct. 20 (9–10), 757–767 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  18. Dragomir, S.S.: Inequalities for the Numerical Radius of Linear Operators in Hilbert Spaces. Springer Briefs in Mathematics. Springer, Berlin (2013). x+120 pp. ISBN: 978-3-319-01447-0; 978-3-319-01448-7

    Google Scholar 

  19. Dragomir, S.S.: Some inequalities in inner product spaces related to Buzano’s and Grüss’ results. Preprint. RGMIA Res. Rep. Coll. 18, Art. 16 (2015). [Online http://rgmia.org/papers/v18/v18a16.pdf]

  20. Dragomir, S.S.: Some inequalities in inner product spaces. Preprint. RGMIA Res. Rep. Coll. 18, Art. 19 (2015). [Online http://rgmia.org/papers/v18/v18a19.pdf]

  21. Dragomir, S.S., Goşa, A.C.: Quasilinearity of some composite functionals associated to Schwarz’s inequality for inner products. Period. Math. Hung. 64 (1), 11–24 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  22. Dragomir, S.S., Mond, B.: Some mappings associated with Cauchy-Buniakowski-Schwarz’s inequality in inner product spaces. Soochow J. Math. 21 (4), 413–426 (1995)

    MathSciNet  MATH  Google Scholar 

  23. Dragomir, S.S., Sándor, I.: Some inequalities in pre-Hilbertian spaces. Studia Univ. Babeş-Bolyai Math. 32 (1), 71–78 (1987)

    MathSciNet  MATH  Google Scholar 

  24. Fong, C.K., Holbrook, J.A.R.: Unitarily invariant operators norms. Can. J. Math. 35, 274–299 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  25. Gunawan, H.: On n-inner products, n-norms, and the Cauchy-Schwarz inequality. Sci. Math. Jpn. 55 (1), 53–60 (2002)

    MathSciNet  MATH  Google Scholar 

  26. Gustafson, K.E., Rao, D.K.M.: Numerical Range. Springer, New York (1997)

    Book  MATH  Google Scholar 

  27. Lorch, E.R.: The Cauchy-Schwarz inequality and self-adjoint spaces. Ann. Math. (2) 46, 468–473 (1945)

    Google Scholar 

  28. Lupu, C., Schwarz, D.: Another look at some new Cauchy-Schwarz type inner product inequalities. Appl. Math. Comput. 231, 463–477 (2014)

    MathSciNet  Google Scholar 

  29. Marcus, M.: The Cauchy-Schwarz inequality in the exterior algebra. Q. J. Math. Oxf. Ser. (2) 17, 61–63 (1966)

    Google Scholar 

  30. Mercer, P.R.: A refined Cauchy-Schwarz inequality. Int. J. Math. Educ. Sci. Tech. 38 (6), 839–842 (2007)

    Article  MathSciNet  Google Scholar 

  31. Metcalf, F.T.: A Bessel-Schwarz inequality for Gramians and related bounds for determinants. Ann. Mat. Pura Appl. (4) 68, 201–232 (1965)

    Google Scholar 

  32. Müller, V.: The numerical radius of a commuting product. Mich. Math. J. 39, 255–260 (1988)

    MathSciNet  MATH  Google Scholar 

  33. Okubo, K., Ando, T.: Operator radii of commuting products. Proc. Am. Math. Soc. 56, 203–210 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  34. Precupanu T.: On a generalization of Cauchy-Buniakowski-Schwarz inequality. An. Şti. Univ. “Al. I. Cuza” Iaşi Secţ. I a Mat. (N.S.) 22 (2), 173–175 (1976)

    Google Scholar 

  35. Trenčevski, K., Malčeski, R.: On a generalized n-inner product and the corresponding Cauchy-Schwarz inequality. J. Inequal. Pure Appl. Math. 7 (2), Article 53, 10pp. (2006)

    Google Scholar 

  36. Wang, G.-B., Ma, J.-P.: Some results on reverses of Cauchy-Schwarz inequality in inner product spaces. Northeast. Math. J. 21 (2), 207–211 (2005)

    MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Mathematics, College of Engineering & Science, Victoria University, 14428, Melbourne city, MC 8001, VIC, Australia

    Silvestru Sever Dragomir

  2. School of Computer Science & Applied Mathematics, University of the Witwatersrand, Private Bag 3, Johannesburg, 2050, South Africa

    Silvestru Sever Dragomir

Authors
  1. Silvestru Sever Dragomir
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Correspondence to Silvestru Sever Dragomir .

Editor information

Editors and Affiliations

  1. Department of Mathematics and Statistics, Auburn University, Auburn, Alabama, USA

    Narendra Kumar Govil

  2. Department of Mathematics, University of Central Florida, Orlando, Florida, USA

    Ram Mohapatra

  3. Department of Mathematics, Tuskegee University, Tuskegee, Alabama, USA

    Mohammed A. Qazi

  4. Department of Mathematics, University of Erlangen-Nuremberg, Erlangen, Germany

    Gerhard Schmeisser

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Dragomir, S.S. (2017). Vector Inequalities for a Projection in Hilbert Spaces and Applications. In: Govil, N., Mohapatra, R., Qazi, M., Schmeisser, G. (eds) Progress in Approximation Theory and Applicable Complex Analysis. Springer Optimization and Its Applications, vol 117. Springer, Cham. https://doi.org/10.1007/978-3-319-49242-1_9

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