Annals of Functional Analysis

Advances in Operator Cauchy--Schwarz inequalities and their reverses

Abstract

The Cauchy-Schwarz (C-S) inequality is one of the most famous inequalities in mathematics. In this survey article, we first give a brief history of the inequality. Afterward, we present the C-S inequality for inner product spaces. Focusing on operator inequalities, we then review some significant recent developments of the C-S inequality and its reverses for Hilbert space operators and elements of Hilbert $C^*$-modules. In particular, we pay special attention to an operator Wielandt inequality.

Article information

Source
Ann. Funct. Anal., Volume 6, Number 3 (2015), 275-295.

Dates
First available in Project Euclid: 17 April 2015

https://projecteuclid.org/euclid.afa/1429286046

Digital Object Identifier
doi:10.15352/afa/06-3-20

Mathematical Reviews number (MathSciNet)
MR3336919

Zentralblatt MATH identifier
1312.47022

Citation

Aldaz, J. M.; Barza, S.; Fujii, M.; Moslehian, M. S. Advances in Operator Cauchy--Schwarz inequalities and their reverses. Ann. Funct. Anal. 6 (2015), no. 3, 275--295. doi:10.15352/afa/06-3-20. https://projecteuclid.org/euclid.afa/1429286046

References

• J.M. Aldaz, A stability version of Hölder's inequality, J. Math. Anal. Appl. 343 (2008), no. 2, 842–852.
• J.M. Aldaz, Strengthened Cauchy-Schwarz and Hölder inequalities, J. Inequal. Pure Appl. Math. 10 (2009), no. 4, Article 116, 6 pp.
• H. Alzer, On the Cauchy-Schwarz inequality, J. Math. Anal. Appl. 234 (1999), no. 1, 6–14.
• T. Ando, Topics on operator inequality, Hokkaido Univ. Lecture Note, 1978.
• Lj. Arambasić, D. Bakić and M.S. Moslehian, A treatment of the Cauchy–Schwarz inequality in $C^*$-modules, J. Math. Anal. Appl. 381 (2011) 546–556.
• J.S. Aujla and H.L. Vasudeva, Inequalities involving Hadamard product and operator means, Math. Japon. 42 (1995), no. 2, 265–272.
• F.L. Bauer and A.S. Householder, Some inequalities involving the euclidean condition of a matrix, Numer. Math. 2 (1960), 308–311
• R. Bhatia and C. Davis, A Cauchy-Schwartz inequality for operators with applications Linear Algebra Appl. 223/224 (1995), 119–129.
• R. Bhatia and C. Davis, More operator versions of the Schwarz inequality, Comm. Math. Phys. 215 (2000), no. 2, 239–244.
• V. Bouniakowsky, Sur quelques inegalités concernant les intégrales aux différences finies, Mem. Acad. Sci. St. Petersbourg 7 (1859), no. 1, p. 9.
• M.L. Buzano, Generalizzatione della diseguagliazza di Cauchy?Schwarz, Rend. Sem. Mat. Univ. Politech. Torino 31 (1971/73) 405-?409.
• A. Cauchy, Oeuvres 2, III (1821), p. 373.
• P. Cerone, Y.J. Cho, S.S. Dragomir and S.S. Kim, Refinements of some reverses of Schwarz's inequality in $2$-inner product spaces and applications for integrals, J. Indones. Math. Soc. 12 (2006), no. 2, 185–199.
• M.-D. Choi, A Schwartz inequality for positive linear maps on $C^*$-algebras, Illinois J. Math. 18 (1974), 565–574.
• J.B. Diaz and F.T. Metcalf, Stronger forms of a class of inequalities of G. Pólya-G. Szegö and L.V. Kantorovich, Bull. Amer. Math. Soc. 69 (1963), 415–418.
• S.S. Dragomir, A survey on Cauchy–Bunyakovsky–Schwarz type discrete inequalities, J. Inequal. Pure Appl. Math. 4 (2003), no. 3, Article 63, 142 pp.
• S.S. Dragomir, Advances in inequalities of the Schwarz, triangle and Heisenberg type in inner product spaces, Nova Science Publishers, Inc., New York, 2007.
• N. Elezović, Lj. Marangunić and J.E. Pečarić, Unified treatment of complemented Schwarz and Grüss inequalities in inner product spaces, Math. Inequal. Appl. 8 (2005), no.2, 223–231.
• J.I. Fujii, Operator-valued inner product and operator inequalities, Banach J. Math. Anal. 2 (2008), no. 2, 59–67.
• J.I. Fujii, M. Fujii, M.S. Moslehian and Y. Seo, Cauchy-Schwarz inequality in semi-inner product $C^*$-modules via polar decomposition, J. Math. Anal. Appl. 394 (2012), no. 2, 835–840.
• M. Fujii, Y. Katayama and R. Nakamoto, Generalizations of the Wielandt theorem, Math. Japon. 49 (1999) 217-222.
• M. Fujii, S. Izumino, R. Nakamoto and Y. Seo, Operator inequalities related to Cauchy–Schwarz and Holder-McCarthy inequalities, Nihonkai Math. J. 8 (1997) 117–122.
• M. Fujii, J. Mićić Hot, J. Pečarić and Y. Seo, Recent Developments of Mond-Pečarić Method in Operator Inequalities, Inequalities for bounded selfadjoint operators on a Hilbert space, II, Element, Zagreb, 2012.
• M. Fujii and Y. Seo, Wielandt type extensions of the Heinz-Kato-Furuta inequality, Operator Theory: Adv. Appl. 127 (2001), 267–277.
• T. Furuta, An extension of the Heinz-Kato theorem, Proc. Amer. Math. Soc. 120 (1994), 785–787.
• T. Furuta, J. Mićić Hot, J. Pečarić and Y. Seo, Mond–Pečarić Method in Operator Inequalities, Element, Zagreb, 2005.
• R.A. Horn and C.R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 1985.
• D. Ilišević and S. Varošanec, On the Cauchy–Schwarz inequality and its reverse in semi-inner product $C^*$-modules, Banach J. Math. Anal. 1 (2007), 78–84.
• D.R. Jocić, Cauchy–Schwarz and means inequalities for elementary operators into norm ideals, Proc. Amer. Math. Soc. 126 (1998), no. 9, 2705–2711.
• M. Joiţa, On the Cauchy–Schwarz inequality in $C^*$-algebras, Math. Rep. (Bucur.) 3(53) (2001), no. 3, 243–246.
• R.V. Kadison, A generalized Schwarz inequality and algebraic invariants for operator algebras, Ann. of Math. (2) 56 (1952), 494–503.
• F. Kittaneh, Some norm inequalities for operators, Canad. Math. Bull. 42 (1999), no. 1, 87–96.
• M.S. Klamkin and R.G. Mclenaghan, An ellipse inequality, Math. Mag. 50 (1977), 261–263.
• E.C. Lance, Hilbert $C^{*}$-modules, A Toolkit for Operator Algebraists, London Math. Soc. Lecture Series, 210. Cambridge University Press, Cambridge, 1995.
• E.-Y. Lee, A matrix reverse Cauchy–Schwarz inequality, Linear Algebra Appl. 430 (2009), no. 2-3, 805–810.
• J. Ma, An identity in real inner product space, J. Inequal. Pure Appl. Math. 8 (2007), no. 2, Art. 48, 4 pp.
• R. Mathias, A note on: "More operator versions of the Schwarz inequality" [Comm. Math. Phys. 215 (2000), no. 2, 239-244; by R. Bhatia and C. Davis], Positivity 8 (2004), no. 1, 85–87.
• M.S. Moslehian, R. Nakamoto and Y. Seo, A Diaz–Metcalf type inequality for positive linear maps and its applications, Electron. J. Linear Algebra 22 (2011), 179–190.
• M.S. Moslehian and L.-E. Persson, Reverse Cauchy–Schwarz inequalities for positive $C^*$-valued sesquilinear forms, Math. Inequal. Appl. 12 (2009), no. 4, 701–709.
• R.B. Nelsen, Proof without Words: The Cauchy-Schwarz Inequality, Math. Mag. 67 (1994), no. 1, 20.
• C.P. Niculescu, Converses of the Cauchy–Schwarz inequality in the $C^*$-framework, An. Univ. Craiova Ser. Mat. Inform. 26 (1999), 22–28.
• G. Pólya and G. Szegö, Aufgaben und Lehrsätze aus der Analysis, Vol. 1, Berlin 1925, pp. 57 and 213–214.
• O. Shisha and B. Mond, Bounds on differences of means, Inequalities I, New York-London, 1967, 293–308.
• J.M. Steele, The Cauchy–Schwarz master class. An introduction to the art of mathematical inequalities, MAA Problem Books Series. Mathematical Association of America, Washington, DC; Cambridge University Press, Cambridge, 2004.
• W.F. Stinespring, Positive functions on $C^*$-algebras, Proc. Amer. Math. Sci. 6 (1955), 211–216.
• S.S. Wagner, Notices Amer. Math. Soc. 12 (1965), 220.
• G.S. Watson, Serial correlation in regression analysis I, Biometrika 42 (1955), 327–342.
• D.B. Zagier, An inequality converse to that of Cauchy, Indag. Math. (N.S.) 39 (1977), no. 4, 349–351.