Some vector inequalities for two operators in Hilbert spaces with applications

Sever S. Dragomir

doi:10.1515/ausm-2016-0005

.blurhash-client-img { display: none !important; }

Some vector inequalities for two operators in Hilbert spaces with applications

Acta Universitatis Sapientiae, Mathematica

Volume 8 (2016): Issue 1 (August 2016)

By: Sever S. Dragomir

Open Access

|Aug 2016

Abstract

In this paper we establish some vector inequalities for two operators related to Schwarz and Buzano results. We show amongst others that in a Hilbert space H we have the inequality 12[〈|A|2+|B|22x,x〉1/2〈|A|2+|B|22y,y〉1/2+|〈|A|2+|B|22x,y〉|]≥|〈Re (B*A) x,y〉|$${1 \over 2}\left[ {\left\langle {{{\left| {\rm{A}} \right|^2 + \left| {\rm{B}} \right|^2 } \over 2}{\rm{x}},{\rm{x}}} \right\rangle ^{1/2} \left\langle {{{\left| {\rm{A}} \right|^2 + \left| {\rm{B}} \right|^2 } \over 2}{\rm{y}},{\rm{y}}} \right\rangle ^{1/2} + \left| {\left\langle {{{\left| {\rm{A}} \right|^2 + \left| {\rm{B}} \right|^2 } \over {\rm{2}}}} {\rm{x}},{\rm{y}}\right\rangle } \right|} \right] \ge \left| {\left\langle {{\mathop{\rm Re}\nolimits} ({\rm{B}}*{\rm{A}})\,{\rm{x}},{\rm{y}}} \right\rangle } \right|$$ for A, B two bounded linear operators on H such that Re (B*A) is a nonnegative operator and any vectors x, y ∈ H.

Applications for norm and numerical radius inequalities are given as well.

References

[1] J. M. Aldaz, Strengthened Cauchy-Schwarz and Hölder inequalities. J. Inequal. Pure Appl. Math.10 (4) (2009), Article 116, 6 pp.
Search in Google Scholar
[2] M. L. Buzano, Generalizzazione della diseguaglianza di Cauchy-Schwarz. (Italian), Rend. Sem. Mat. Univ. e Politech. Torino, 31 (1971/73) (1974), 405–409.
Search in Google Scholar
[3] K. R. Davidson, J. A. R. Holbrook, Numerical radii of zero-one matricies, Michigan Math. J.35 (1988), 261–267.10.1307/mmj/1029003753
Open DOI Search in Google Scholar
[4] A. De Rossi, A strengthened Cauchy-Schwarz inequality for biorthogonal wavelets. Math. Inequal. Appl., 2 (2) (1999), 263–282.10.7153/mia-02-24
Open DOI Search in Google Scholar
[5] S. S. Dragomir, Some refinements of Schwartz inequality, Simpozionul de Matematici şi Aplicaţii, Timişoara, Romania, 1-2 Noiembrie 1985, 13–16.
Search in Google Scholar
[6] S. S. Dragomir, A generalization of Grüss’ inequality in inner product spaces and applications, J. Math. Anal. Appl., 237 (1999), 74–82.10.1006/jmaa.1999.6452
Open DOI Search in Google Scholar
[7] S. S. Dragomir, Some Grüss type inequalities in inner product spaces, J. Inequal. Pure and Appl. Math., 4 (2) Art. 42 (2003).
Search in Google Scholar
[8] S. S. Dragomir, Some inequalities for power series of selfadjoint operators in Hilbert spaces via reverses of the Schwarz inequality. Integral Transforms Spec. Funct.20 (9-10) (2009), 757–767.10.1080/10652460902910054
Open DOI Search in Google Scholar
[9] S. S. Dragomir, A potpourri of Schwarz related inequalities in inner product spaces. I. J. Inequal. Pure Appl. Math.6 (3) (2005), Article 59, 15 pp.
Search in Google Scholar
[10] S. S. Dragomir, A potpourri of Schwarz related inequalities in inner product spaces. II. J. Inequal. Pure Appl. Math.7 (1) (2006), Article 14, 11 pp.
Search in Google Scholar
[11] S. S. Dragomir, Reverses of Schwarz, triangle and Bessel inequalities in inner product spaces. J. Inequal. Pure Appl. Math.5 (3) (2004), Article 76, 18 pp.
Search in Google Scholar
[12] S. S. Dragomir, New reverses of Schwarz, triangle and Bessel inequalities in inner product spaces. Aust. J. Math. Anal. Appl.1 (1) (2004), Art. 1, 18 pp.
Search in Google Scholar
[13] S. S. Dragomir, Inequalities for the norm and the numerical radius of linear operators in Hilbert spaces. Demonstratio Math.40 (2) (2007), 411–417.10.1515/dema-2007-0213
Open DOI Search in Google Scholar
[14] S. S. Dragomir, Some inequalities for the norm and the numerical radius of linear operators in Hilbert spaces. Tamkang J. Math.39 (1) (2008), 1–7.10.5556/j.tkjm.39.2008.40
Open DOI Search in Google Scholar
[15] S. S. Dragomir, New inequalities of the Kantorovich type for bounded linear operators in Hilbert spaces, Linear Algebra and its Applications428 (2008), 2750–2760.10.1016/j.laa.2007.12.025
Open DOI Search in Google Scholar
[16] S. S. Dragomir, Advances in Inequalities of the Schwarz, Grüss and Bessel Type in Inner Product Spaces. Nova Science Publishers, Inc., Hauppauge, NY, 2005. viii+249 pp. ISBN: 1-59454-202-3.
Search in Google Scholar
[17] S. S. Dragomir, Advances in Inequalities of the Schwarz, Triangle and Heisenberg Type in Inner Product Spaces. Nova Science Publishers, Inc.,
Search in Google Scholar
[18] S. S. Dragomir, Inequalities for the Numerical Radius of Linear Operators in Hilbert Spaces. Springer Briefs in Mathematics. Springer, 2013. x+120 pp. ISBN: 978-3-319-01447-0; 978-3-319-01448-7.
Search in Google Scholar
[19] S. S. Dragomir, Some inequalities in inner product spaces related to Buzano’s and Grüss’ results, Preprint, RGMIA Res. Rep. Coll., 18 (2015), Art. 16. [Online http://rgmia.org/papers/v18/v18a16.pdf].
Search in Google Scholar
[20] S. S. Dragomir, Some inequalities in inner product spaces, Preprint, RGMIA Res. Rep. Coll., 18 (2015), Art. 19. [Online http://rgmia.org/papers/v18/v18a19.pdf]
Search in Google Scholar
[21] S. S. Dragomir, Vector inequalities for a projection in Hilbert spaces and applications, Preprint, RGMIA Res. Rep. Coll., 18 (2015), Art.
Search in Google Scholar
[22] S. S. Dragomir, Anca C. Goşa, Quasilinearity of some composite functionals associated to Schwarz’s inequality for inner products. Period. Math. Hungar.64 (1) (2012), 11–24.10.1007/s10998-012-9011-x
Open DOI Search in Google Scholar
[23] S. S. Dragomir, B. Mond, Some mappings associated with Cauchy-Buniakowski-Schwarz’s inequality in inner product spaces. Soochow J. Math.21 (4) (1995), 413–426.
Search in Google Scholar
[24] S. S. Dragomir, I. Sándor, Some inequalities in pre-Hilbertian spaces. Studia Univ. Babeş-Bolyai Math.32 (1) (1987), 71–78.
Search in Google Scholar
[25] C. K. Fong, J. A. R. Holbrook, Unitarily invariant operators norms, Canad. J. Math.35 (1983), 274–299.10.4153/CJM-1983-015-3
Open DOI Search in Google Scholar
[26] H. Gunawan, On n-inner products, n-norms, and the Cauchy-Schwarz inequality. Sci. Math. Jpn.55 (1) (2002), 53–60
Search in Google Scholar
[27] K. E. Gustafson, D. K. M. Rao, Numerical Range, Springer-Verlag, New York, Inc., 1997.10.1007/978-1-4613-8498-4
Open DOI Search in Google Scholar
[28] P. R. Halmos, A Hilbert Space Problem Book, Springer-Verlag, New York, Heidelberg, Berlin, Second edition, 1982.10.1007/978-1-4684-9330-6
Open DOI Search in Google Scholar
[29] J. A. R. Holbrook, Multiplicative properties of the numerical radius in operator theory, J. Reine Angew. Math.237 (1969), 166–174.10.1515/crll.1969.237.166
Open DOI Search in Google Scholar
[30] E. R. Lorch, The Cauchy-Schwarz inequality and self-adjoint spaces. Ann. of Math. (2) 46, (1945), 468–473.10.2307/1969163
Open DOI Search in Google Scholar
[31] C. Lupu, D. Schwarz, Another look at some new Cauchy-Schwarz type inner product inequalities. Appl. Math. Comput.231 (2014), 463–477.10.1016/j.amc.2013.11.047
Open DOI Search in Google Scholar
[32] M. Marcus, The Cauchy-Schwarz inequality in the exterior algebra. Quart. J. Math. Oxford Ser. (2) 17 (1966), 61–63.10.1093/qmath/17.1.61
Open DOI Search in Google Scholar
[33] P. R. Mercer, A refined Cauchy-Schwarz inequality. Internat. J. Math. Ed. Sci. Tech.38 (6) (2007), 839–842.10.1080/00207390701359370
Open DOI Search in Google Scholar
[34] F. T. Metcalf, A Bessel-Schwarz inequality for Gramians and related bounds for determinants. Ann. Mat. Pura Appl. (4) 68 (1965), 201–232.
Search in Google Scholar
[35] V. Müller, The numerical radius of a commuting product, Michigan Math. J.39 (1988), 255–260.10.1307/mmj/1029003752
Open DOI Search in Google Scholar
[36] K. Okubo, T. Ando, Operator radii of commuting products, Proc. Amer. Math. Soc.56 (1976), 203–210.10.1090/S0002-9939-1976-0405132-6
Open DOI Search in Google Scholar
[37] T. Precupanu, On a generalization of Cauchy-Buniakowski-Schwarz inequality. An. Şti. Univ. “Al. I. Cuza” Iaşi Secţ. I a Mat. (N.S.) 22 (2) (1976), 173–175.
Search in Google Scholar
[38] K. Trenčevski, R. Malčeski, On a generalized n-inner product and the corresponding Cauchy-Schwarz inequality. J. Inequal. Pure Appl. Math.7 (2) (2006), Article 53, 10 pp.
Search in Google Scholar
[39] G.-B. Wang, J.-P. Ma, Some results on reverses of Cauchy-Schwarz inequality in inner product spaces. Northeast. Math. J.21 (2) (2005), 207–211.
Search in Google Scholar

Articles in this issue

DOI: https://doi.org/10.1515/ausm-2016-0005 | Journal eISSN: 2066-7752

Journal RSS Feed

Language: English

Page range: 75 - 92

Submitted on: Sep 22, 2015

Published online: Aug 2, 2016

Published in print: Aug 1, 2016

Published by: Sapientia Hungarian University of Transylvania

In partnership with: Paradigm Publishing Services

Keywords:

inner product spaces,

Schwarz’s inequality,

Buzano’s inequality,

projection,

selfadjoint operators,

unitary operators,

operator norm,

numerical radius

MSC:

46C05,

26D15,

26D1

Related subjects:

Mathematics,

General mathematics

© 2016 Sever S. Dragomir, published by Sapientia Hungarian University of Transylvania
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Volume 8 (2016): Issue 1 (August 2016)