On some inequalities for numerical radius of operators in Hilbert spaces

Silvestru Sever Dragomir

Abstract


By the use of inequalities for nonnegative Hermitian forms some new inequalities for numerical radius of bounded linear operators in complex Hilbert spaces are established.


Keywords


Schwarz inequality, Buzano inequality, Numerical radius, Operator norm, Operator inequalities.

Full Text:

PDF

References


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

S. S. Dragomir, Some refinements of Schwartz inequality, Simpozionul de Matematici ̧si Aplica ̧tii, Timi ̧soara, Romania, 1-2 Noiembrie 1985, 13–16.

S. S. Dragomir, Gru ̈ss inequality in inner product spaces, The Australian Math Soc. Gazette 26 (1999), No. 2, 66–70.

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

S. S. Dragomir, Some Gru ̈ss type inequalities in inner product spaces, J. Inequal. Pure & Appl. Math. 4 (2) (2003), Article 42. (Online http://jipam.vu.edu.au/article.php?sid=280).

S. S. Dragomir, Reverses of Schwarz, triangle and Bessel inequalities in inner product spaces, J. Inequal. Pure & Appl. Math. 5(3) (2004), Article 76. (Online : http://jipam.vu.edu.au/article.php?sid=432).

S. S. Dragomir, New reverses of Schwarz, triangle and Bessel inequalities in inner product spaces, Austral. J. Math. Anal. & Applics. 1(1) (2004), Article 1. (Online: http://ajmaa.org/cgi-bin/paper.pl?string=nrstbiips.tex ).

S. S. Dragomir, On Bessel and Gru ̈ss inequalities for orthornormal families in inner product spaces, Bull. Austral. Math. Soc. 69(2) (2004), 327–340.

S. S. Dragomir, Advances in Inequalities of the Schwarz, Gru ̈ss and Bessel Type in Inner Product Spaces, Nova Science Publishers Inc, New York, 2005, x+249 p.

S. S. Dragomir, Reverses of the Schwarz inequality in inner product spaces gen- eralising a Klamkin-McLenaghan result, Bull. Austral. Math. Soc. 73 (1) (2006), 69–78.

S. S. Dragomir, Advances in Inequalities of the Schwarz, Triangle and Heisen- berg Type in Inner Product Spaces. Nova Science Publishers, Inc., New York, 2007. xii+243 pp. ISBN: 978-1-59454-903-8; 1-59454-903-6 (Preprint http://rgmia.org/monographs/advancees2.htm)

S. S. Dragomir, Inequalities for the norm and the numerical radius of linear operators in Hilbert spaces. Demonstratio Math. 40 (2007), no. 2, 411–417.

S. S. Dragomir, Some inequalities for the norm and the numerical radius of linear operators in Hilbert spaces, Tamkang J. Math. 39 (2008), no. 1, 1–7.

S. S. Dragomir, Some new Gru ̈ss’ type inequalities for functions of selfadjoint operators in Hilbert spaces, RGMIA Res. Rep. Coll. 11(e) (2008), Art. 12.

S. S. Dragomir, Inequalities for the Cˇebyˇsev functional of two functions of self-adjoint operators in Hilbert spaces, RGMIA Res. Rep. Coll. 11(e) (2008), Art. 17.

S. S. Dragomir, Some inequalities for the Cˇebyˇsev functional of two functions of selfadjoint operators in Hilbert spaces, RGMIA Res. Rep. Coll. 11(e) (2008), Art. 8.

S. S. Dragomir, Inequalities for the Cˇebyˇsev functional of two functions of self- adjoint operators in Hilbert spaces, Aust. J. Math. Anal. & Appl. 6 (2009), Issue 1, Article 7, pp. 1–58.

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 (2009), no. 9-10, 757–767.

S. S. Dragomir, Operator Inequalities of the Jensen, Cˇebyˇsev and Gru ̈ss Type. Springer Briefs in Mathematics. Springer, New York, 2012. xii+121 pp. ISBN: 978-1-4614-1520-6.

S. S. Dragomir, Operator Inequalities of Ostrowski and Trapezoidal Type. Springer Briefs in Mathematics. Springer, New York, 2012. x+112 pp. ISBN: 978-1-4614-1778-1.

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.

S. S. Dragomir, M. V. Boldea and C. Bu ̧se, Norm inequalities of Cˇebyˇsev type for power series in Banach algebras, Preprint RGMIA Res. Rep. Coll. 16 (2013), Art. 73.

S. S. Dragomir and B. Mond, On the superadditivity and monotonicity of Schwarz’s inequality in inner product spaces, Contributions, Macedonian Acad. of Sci and Arts 15 (2) (1994), 5–22.

S. S. Dragomir and B. Mond, Some inequalities for Fourier coefficients in inner product spaces, Periodica Math. Hungarica 32 (3) (1995), 167–172.

S. S. Dragomir, J. Peˇcari ́c and J. S ́andor, The Chebyshev inequality in pre-Hilbertian spaces. II. Proceedings of the Third Symposium of Mathematics and its Applications (Timi ̧soara, 1989), 75–78, Rom. Acad., Timi ̧soara, 1990. MR1266442 (94m:46033)

S. S. Dragomir and J. S ́andor, The Chebyshev inequality in pre-Hilbertian spaces. I. Proceedings of the Second Symposium of Mathematics and its Applications (Timi ̧soara, 1987), 61–64, Res. Centre, Acad. SR Romania, Timi ̧soara, 1988. MR1006000 (90k:46048).

K. E. Gustafson and D. K. M. Rao, Numerical Range, Springer-Verlag, New York, Inc., 1997.

S. Kurepa, Note on inequalities associated with Hermitian functionals, Glasnik Matematˇcki 3 (x23) (1968), 196–205.




DOI: http://dx.doi.org/10.11568/kjm.2017.25.2.247

Refbacks

  • There are currently no refbacks.


ISSN: 1976-8605 (Print), 2288-1433 (Online)

Copyright(c) 2013 By The Kangwon-Kyungki Mathematical Society, Department of Mathematics, Kangwon National University Chuncheon 21341, Korea Fax: +82-33-259-5662 E-mail: kkms@kangwon.ac.kr