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Cauchy-Schwarz inequality in semi-inner product \(C^{\ast}\)-modules via polar decomposition. (English) Zbl 1257.46026

Summary: By virtue of the operator geometric mean and the polar decomposition, we present a new Cauchy-Schwarz inequality in the framework of semi-inner product \(C^{\ast }\)-modules over unital \(C^{\ast }\)-algebras and discuss the equality case. We also give several additive and multiplicative type reverses of it. As an application, we present a Kantorovich type inequality on a Hilbert \(C^{\ast }\)-module.

MSC:

46L08 \(C^*\)-modules
47A63 Linear operator inequalities
47A64 Operator means involving linear operators, shorted linear operators, etc.
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[1] Dragomir, S. S., A survey on Cauchy-Bunyakovsky-Schwarz type discrete inequalities, J. Inequal. Pure Appl. Math., 4, 3, 142 (2003), Article 63 · Zbl 1055.26013
[2] Lance, E. C., (Hilbert \(C^\ast \)-Modules. Hilbert \(C^\ast \)-Modules, London Math. Soc. Lecture Note Series, vol. 210 (1995), Cambridge Univ. Press)
[3] Ilisević, D.; Varošanec, S., On the Cauchy-Schwarz inequality and its reverse in semi-inner product \(C^\ast \)-modules, Banach J. Math. Anal., 1, 1, 78-84 (2007) · Zbl 1134.46036
[4] Fujii, J. I.; Fujii, M.; Moslehian, M. S.; Pečarić, J.; Seo, Y., Reverse Cauchy-Schwarz type inequalities in pre-inner product \(C^\ast \)-modules, Hokkaido Math. J., 40, 1-17 (2011)
[5] Dragomir, S. S., A counterpart of Schwarz inequality in inner product spaces, RGMIA Res. Rep. Coll., 6, Suppl. (2003), Article 18 · Zbl 1063.26013
[6] Fujii, M.; Izumino, S.; Nakamoto, R.; Seo, Y., Operator inequalities related to Cauchy-Schwarz and Hölder-McCarthy inequalities, Nihonkai Math. J., 8, 117-122 (1997) · Zbl 0997.47505
[7] Wada, S., On some refinement of the Cauchy-Schwarz inequality, Linear Algebra Appl., 420, 2-3, 433-440 (2007) · Zbl 1121.47010
[8] Niculescu, C. P., Converses of the Cauchy-Schwarz inequality in the \(C^\ast \)-framework, An. Univ. Craiova Ser. Mat. Inform., 26, 22-28 (1999) · Zbl 1014.46027
[9] Joiţa, M., On the Cauchy-Schwarz inequality in \(C^\ast \)-algebras, Math. Rep. (Bucur.), 3(53), 3, 243-246 (2001) · Zbl 1068.46031
[10] Moslehian, M. S.; Persson, L.-E., Reverse Cauchy-Schwarz inequalities for positive \(C^\ast \)-valued sesquilinear forms, Math. Inequal. Appl., 4, 12, 701-709 (2009) · Zbl 1188.46037
[11] Arambasić, Lj.; Bakić, D.; Moslehian, M. S., A treatment of the Cauchy-Schwarz inequality in \(C^\ast \)-modules, J. Math. Anal. Appl., 381, 546-556 (2011) · Zbl 1225.46045
[12] Fujii, J. I., Operator-valued inner product and operator inequalities, Banach J. Math. Anal., 2, 2, 59-67 (2008) · Zbl 1151.47024
[13] Elezović, N.; Marangunić, Lj.; Pečarić, J., Unified treatment of complemented Schwarz and Grüss inequalities in inner product spaces, Math. Inequal. Appl., 8, 2, 223-231 (2005) · Zbl 1078.26014
[14] Ghazanfari, A. G.; Dragomir, S. S., Bessel and Grüss type inequalities in inner product modules over Banach \(\ast \)-algebras, J. Inequal. Appl., 16 (2011), Art ID. 562923 · Zbl 1223.46054
[15] Pusz, W.; Woronowicz, S. L., Functional calculus for sesquilinear forms and the purification map, Rep. Math. Phys., 8, 159-170 (1975) · Zbl 0327.46032
[16] T. Ando, Topics on Operator Inequalities, Lecture Notes (mimeographed), Hokkaido Univ., Sapporo, 1978.; T. Ando, Topics on Operator Inequalities, Lecture Notes (mimeographed), Hokkaido Univ., Sapporo, 1978.
[17] Kubo, F.; Ando, T., Means of positive linear operators, Math. Ann., 246, 205-224 (1980) · Zbl 0412.47013
[18] Furuta, T.; Mićić Hot, J.; Pečarić, J.; Seo, Y., (Mond-Pečarić Method in Operator Inequalities. Mond-Pečarić Method in Operator Inequalities, Monographs in Inequalities, 1 (2005), Element: Element Zagreb) · Zbl 1135.47012
[19] Kantorović, L. V., Functional analysis and applied mathematics, Uspehi Matem. Nauk (N.S.), 3, 6, 89-185 (1948), 28 (in Russian) · Zbl 0034.21203
[20] Yamazaki, T., An extension of Kantorovich inequality to \(n\)-operators via the geometric mean by Ando-Li-Mathias, Linear Algebra Appl., 416, 2-3, 688-695 (2006) · Zbl 1126.47016
[21] Matharu, J. S.; Aujla, J. S., Hadamard product versions of the Chebyshev and Kantorovich inequalities, JIPAM. J. Inequal. Pure Appl. Math., 10, 2, 6 (2009), Article 51 · Zbl 1170.15009
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