Trace inequalities of Cassels and Grüss type for operators in Hilbert spaces

Sever S. Dragomir

doi:10.1515/ausm-2017-0006

Abstract

Some trace inequalities of Cassels type for operators in Hilbert spaces are provided. Applications in connection to Grüss inequality and for convex functions of selfadjoint operators are also given.

References

[1] G. A. Anastassiou, Grüss type inequalities for the Stieltjes integral, Nonlinear Funct. Anal. Appl., 12 (2007), no. 4, 583–593.
Search in Google Scholar Back to article
[2] G. A. Anastassiou, Chebyshev-Grüss type and comparison of integral means inequalities for the Stieltjes integral, Panamer. Math. J., 17 (2007), no. 3, 91–109.
Search in Google Scholar Back to article
[3] G. A. Anastassiou, Chebyshev-Grüss type inequalities via Euler type and Fink identities, Math. Comput. Modelling, 45 (2007), no. 9–10, 1189–1200
Search in Google Scholar Back to article
[4] T. Ando, Matrix Young inequalities, Oper. Theory Adv. Appl., 75 (1995), 33–38.
Search in Google Scholar Back to article
[5] R. Bellman, Some inequalities for positive definite matrices, in: E.F. Beckenbach (Ed.), General Inequalities 2, Proceedings of the 2nd International Conference on General Inequalities, Birkhäuser, Basel, 1980, pp. 89–90.10.1007/978-3-0348-6324-7_8
Search in Google Scholar Back to article
[6] E. V. Belmega, M. Jungers, S. Lasaulce, A generalization of a trace inequality for positive definite matrices, Aust. J. Math. Anal. Appl., 7 (2010), no. 2, Art. 26, 5 pp.
Search in Google Scholar Back to article
[7] N. G. de Bruijn, Problem 12, Wisk. Opgaven, 21 (1960), 12–14.
Search in Google Scholar Back to article
[8] P. Cerone, S. S. Dragomir, A refinement of the Grüss inequality and applications. Tamkang J. Math., 38 (2007), no. 1, 37–49.
Search in Google Scholar Back to article
[9] P. Cerone, S. S. Dragomir, New bounds for the Čebyšev functional. Appl. Math. Lett., 18 (2005), no. 6, 603–611.
Search in Google Scholar Back to article
[10] P. Cerone, S. S. Dragomir, Chebychev functional bounds using Ostrowski seminorms, Southeast Asian Bull. Math., 28 (2004), no. 2, 219–228.
Search in Google Scholar Back to article
[11] D. Chang, A matrix trace inequality for products of Hermitian matrices, J. Math. Anal. Appl., 237 (1999) 721–725.
Search in Google Scholar Back to article
[12] L. Chen, C. Wong, Inequalities for singular values and traces, Linear Algebra Appl., 171 (1992), 109–120.10.1016/0024-3795(92)90253-7
Search in Google Scholar Back to article
[13] I. D. Coop, On matrix trace inequalities and related topics for products of Hermitian matrix, J. Math. Anal. Appl., 188 (1994), 999–1001.10.1006/jmaa.1994.1475
Search in Google Scholar Back to article
[14] S. S. Dragomir, Some Grüss type inequalities in inner product spaces, J. Inequal. Pure & Appl. Math., 4(2003), No. 2, Article 42, [ON LINE: http://jipam.vu.edu.au/v4n2/032_03.html].
Search in Google Scholar Back to article
[15] S. S. Dragomir, A Survey on Cauchy-Bunyakovsky-Schwarz Type Discrete Inequalities, RGMIA Monographs, Victoria University, 2002. (ON LINE: http://rgmia.vu.edu.au/monographs/).
Search in Google Scholar Back to article
[16] S. S. Dragomir, A counterpart of Schwarz’s inequality in inner product spaces, East Asian Math. J., 20 (1) (2004), 1-10. Zbl 1094.26017. Preprint RGMIA Res. Rep. Coll., 6(2003), Supplement, Article 18. [ON LINE: http://rgmia.vu.edu.au/v6(E).html].
Search in Google Scholar Back to article
[17] S. S. Dragomir, Grüss inequality in inner product spaces, The Australian Math Soc. Gazette, 26 (1999), No. 2, 66–70.
Search in Google Scholar Back to article
[18] 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
Search in Google Scholar Back to article
[19] S. S. Dragomir, Some discrete inequalities of Grüss type and applications in guessing theory, Honam Math. J., 21 (1) (1999), 145–156.
Search in Google Scholar Back to article
[20] S. S. Dragomir, Some integral inequalities of Grüss type, Indian J. of Pure and Appl. Math.,31 (4) (2000), 397–415.
Search in Google Scholar Back to article
[21] S. S. Dragomir, Advances in Inequalities of the Schwarz, Grüss and Bessel Type in Inner Product Spaces, Nova Science Publishers Inc., New York, 2005, x+249 pp.
Search in Google Scholar Back to article
[22] S. S. Dragomir, G. L. Booth, On a Grüss-Lupaş type inequality and its applications for the estimation of p-moments of guessing mappings, Mathematical Communications,5 (2000), 117–126.
Search in Google Scholar Back to article
[23] S. S. Dragomir, A Grüss type integral inequality for mappings of r-Hölder’s type and applications for trapezoid formula, Tamkang J. of Math., 31 (1) (2000), 43–47.10.5556/j.tkjm.31.2000.413
Search in Google Scholar Back to article
[24] S. S. Dragomir, I. Fedotov, An inequality of Grüss’ type for Riemann-Stieltjes integral and applications for special means, Tamkang J. of Math., 29 (4) (1998), 286–292.10.5556/j.tkjm.29.1998.4257
Search in Google Scholar Back to article
[25] S. S. Dragomir, Čebyšev’s type inequalities for functions of selfadjoint operators in Hilbert spaces, Linear Multilinear Algebra, 58 (2010), no. 7-8, 805–814. Preprint RGMIA Res. Rep. Coll., 11 (2008), Supp. Art. 9. [Online http://rgmia.org/v11(E).php]10.1080/03081080902992104
Search in Google Scholar Back to article
[26] S. S. Dragomir, Grüss’ type inequalities for functions of selfadjoint operators in Hilbert spaces, Ital. J. Pure Appl. Math. No. 28 (2011), 207–224. Preprint RGMIA Res. Rep. Coll.11 (2008), Supp. Art. 11. [Online http://rgmia.org/v11(E).php]
Search in Google Scholar Back to article
[27] S. S. Dragomir, New inequalities of the Kantorovich type for bounded linear operators in Hilbert spaces, Linear Algebra Appl.428 (2008), no. 11–12, 2750–2760.
Search in Google Scholar Back to article
[28] S. S. Dragomir, Some Čebyšev’s type trace inequalities for functions of selfadjoint operators in Hilbert spaces, Preprint RGMIA Res. Rep. Coll.17 (2014), Art. 111. [Online http://rgmia.org/papers/v17/v17a111.pdf].
Search in Google Scholar Back to article
[29] S. S. Dragomir, Some Grüss type inequalities for trace of operators in Hilbert spaces, Preprint RGMIA Res. Rep. Coll.17 (2014), Art. 114. [Online http://rgmia.org/papers/v17/v17a114.pdf].
Search in Google Scholar Back to article
[30] S. S. Dragomir, Reverse Jensen’s type trace inequalities for convex functions of selfadjoint operators in Hilbert spaces, Preprint RGMIA Res. Rep. Coll.17 (2014), Art. 118. [Online http://rgmia.org/papers/v17/v17a118.pdf].
Search in Google Scholar Back to article
[31] A. M. Fink, A treatise on Grüss’ inequality, Analytic and Geometric Inequalities, 93–113, Math. Appl. 478, Kluwer Academic Publ., 1999.10.1007/978-94-011-4577-0_7
Search in Google Scholar Back to article
[32] S. Furuichi, M. Lin, Refinements of the trace inequality of Belmega, Lasaulce and Debbah. Aust. J. Math. Anal. Appl., 7 (2010), no. 2, Art. 23, 4 pp.
Search in Google Scholar Back to article
[33] T. Furuta, J. Mićić Hot, J. Pečarić, Y. Seo, Mond-Pečarić Method in Operator Inequalities. Inequalities for Bounded Selfadjoint Operators on a Hilbert Space, Element, Zagreb, 2005.
Search in Google Scholar Back to article
[34] W. Greub and W. Rheinboldt, On a generalisation of an inequality of L.V. Kantorovich, Proc. Amer. Math. Soc.,10 (1959), 407–415.10.1090/S0002-9939-1959-0105028-3
Search in Google Scholar Back to article
[35] G. Grüss, Über das Maximum des absoluten Betrages von 1b−a∫abf(x)g(x)dx−1(b−a)2∫abf(x)dx∫abg(x)dx${1 \over {{\rm{b}} - {\rm{a}}}}\int_{\rm{a}}^{\rm{b}} {{\rm{f}}({\rm{x}}){\rm{g}}({\rm{x}}){\rm{dx}} - {1 \over {\left( {{\rm{b}} - {\rm{a}}} \right)^2 }}\int_{\rm{a}}^{\rm{b}} {{\rm{f}}({\rm{x}}){\rm{dx}}} \int_{\rm{a}}^{\rm{b}} {{\rm{g}}({\rm{x}}){\rm{dx}}} } $ , Math. Z., 39 (1935), 215–226.10.1007/BF01201355
Back to article
[36] H. D. Lee, On some matrix inequalities, Korean J. Math., 16 (2008), No. 4, 565–571.
Search in Google Scholar Back to article
[37] L. Liu, A trace class operator inequality, J. Math. Anal. Appl., 328 (2007), 1484–1486.10.1016/j.jmaa.2006.04.092
Search in Google Scholar Back to article
[38] Z. Liu, Refinement of an inequality of Grüss type for Riemann-Stieltjes integral, Soochow J. Math.,30 (4) (2004), 483–489.
Search in Google Scholar Back to article
[39] S. Manjegani, Hölder and Young inequalities for the trace of operators, Positivity, 11 (2007), 239–250.10.1007/s11117-006-2054-6
Search in Google Scholar Back to article
[40] A. Matković, J. Pečarić, I. Perić, A variant of Jensen’s inequality of Mercer’s type for operators with applications, Linear Algebra Appl., 418 (2006), no. 2–3, 551–564.
Search in Google Scholar Back to article
[41] D. S. Mitrinović, J. E. Pečarić, A.M. Fink, Classical and New Inequalities in Analysis, Kluwer Academic Publishers, Dordrecht, 1993.10.1007/978-94-017-1043-5
Search in Google Scholar Back to article
[42] H. Neudecker, A matrix trace inequality, J. Math. Anal. Appl., 166 (1992), 302–303.10.1016/0022-247X(92)90344-D
Search in Google Scholar Back to article
[43] J. Pečarić, J. Mićić, Y. Seo, Inequalities between operator means based on the Mond-Pečarić method. Houston J. Math., 30 (2004), no. 1, 191–207
Search in Google Scholar Back to article
[44] G. Pólya, G. Szegö, Aufgaben und Lehrsätze aus der Analysis, Vol. 1, Berlin 1925, pp. 57 and 213–214.10.1007/978-3-662-38381-0_1
Search in Google Scholar Back to article
[45] M. B. Ruskai, Inequalities for traces on von Neumann algebras, Commun. Math. Phys.26 (1972), 280–289.10.1007/BF01645523
Search in Google Scholar Back to article
[46] K. Shebrawi, H. Albadawi, Operator norm inequalities of Minkowski type, J. Inequal. Pure Appl. Math., 9 (1) (2008), 1–10, article 26.10.1155/2009/492154
Search in Google Scholar Back to article
[47] K. Shebrawi, H. Albadawi, Trace inequalities for matrices, Bull. Aust. Math. Soc., 87 (2013), 139–148.10.1017/S0004972712000627
Search in Google Scholar Back to article
[48] O. Shisha and B. Mond, Bounds on differences of means, Inequalities I, New York-London, 1967, 293–308.
Search in Google Scholar Back to article
[49] B. Simon, Trace Ideals and Their Applications, Cambridge University Press, Cambridge, 1979.
Search in Google Scholar Back to article
[50] Z. Ulukök and R. Türkmen, On some matrix trace inequalities. J. Inequal. Appl., 2010, Art. ID 201486, 8 pp.10.1155/2010/201486
Search in Google Scholar Back to article
[51] X. Yang, A matrix trace inequality, J. Math. Anal. Appl., 250 (2000) 372–374.
Search in Google Scholar Back to article
[52] X. M. Yang, X. Q. Yang, K. L. Teo, A matrix trace inequality, J. Math. Anal. Appl., 263 (2001), 327–331.10.1006/jmaa.2001.7613
Search in Google Scholar Back to article
[53] Y. Yang, A matrix trace inequality, J. Math. Anal. Appl., 133 (1988) 573–574.
Search in Google Scholar Back to article

Trace inequalities of Cassels and Grüss type for operators in Hilbert spaces

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