Hypo-q-Norms on a Cartesian Product of Algebras of Operators on Banach Spaces

Silvestru Sever Dragomir 1
  • 1 Mathematics, College of Engineering & Science, Victoria University, DST-NRF Centre of Excellence in the Mathematical and Statistical Sciences, School of Computer Science & Applied Mathematics, University of the Witwatersrand, PO Box 14428, Melbourne City, Australia


In this paper we consider the hypo-q-operator norm and hypo-q-numerical radius on a Cartesian product of algebras of bounded linear operators on Banach spaces. A representation of these norms in terms of semi-inner products, the equivalence with the q-norms on a Cartesian product and some reverse inequalities obtained via the scalar reverses of Cauchy-Buniakowski-Schwarz inequality are also given.

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1] S.S. Dragomir, A survey on Cauchy-Bunyakovsky-Schwarz type discrete inequalities, J. Inequal. Pure Appl. Math. 4 (2003), no. 3, Art. 63, 142 pp. Available at https://www.emis.de/journals/JIPAM/article301.html?sid=301.

  • [2] S.S. Dragomir, A counterpart of Schwarz’s inequality in inner product spaces, East Asian Math. J. 20 (2004), no. 1, 1–10. Preprint RGMIA Res. Rep. Coll. 6 (2003), Supplement, Art. 18. Available at http://rgmia.org/papers/v6e/CSIIPS.pdf.

  • [3] S.S. Dragomir, Semi-Inner Products and Applications, Nova Science Publishers, Inc., Hauppauge, NY, 2004.

  • [4] S.S. Dragomir, Advances in Inequalities of the Schwarz, Grüss and Bessel Type in Inner Product Spaces, Nova Science Publishers, Inc., Hauppauge, NY, 2005.

  • [5] S.S. Dragomir, Reverses of the Schwarz inequality generalising a Klamkin-McLenaghan result, Bull. Austral. Math. Soc. 73 (2006), no. 1, 69–78.

  • [6] S.S. Dragomir, The hypo-Euclidean norm of an n-tuple of vectors in inner product spaces and applications, J. Inequal. Pure Appl. Math. 8 (2007), no. 2, Art. 52, 22 pp. Available at https://www.emis.de/journals/JIPAM/article854.html?sid=854.

  • [7] S.S. Dragomir, Hypo-q-norms on a Cartesian product of normed linear spaces, Preprint RGMIA Res. Rep. Coll. 20 (2017), Art. 153. Available at http://rgmia.org/papers/v20/v20a153.pdf.

  • [8] S.S. Dragomir, Inequalities for hypo-q-norms on a Cartesian product of inner product spaces, Preprint RGMIA Res. Rep. Coll. 20 (2017), Art. 168. Available at http://rgmia.org/papers/v20/v20a168.pdf.

  • [9] J.R. Giles, Classes of semi-inner-product spaces, Trans. Amer. Math. Soc. 129 (1967), 436–446.

  • [10] B.W. Glickfeld, On an inequality of Banach algebra geometry and semi-inner product space theory, Illinois J. Math. 14 (1970), 76–81.

  • [11] G. Lumer, Semi-inner-product spaces, Trans. Amer. Math. Soc. 100 (1961), 29–43.

  • [12] P.M. Miličić, Sur le semi-produit scalaire dans quelques espaces vectoriels normés, Mat. Vesnik 8(23) (1971), 181–185.

  • [13] P.M. Miličić, Une généralisation naturelle du produit scalaire dans un espace normé et son utilisation, Publ. Inst. Math. (Beograd) (N.S.) 42(56) (1987), 63–70.

  • [14] P.M. Miličić, La fonctionelle g et quelques problèmes des meilleures approximations dans des espaces normés, Publ. Inst. Math. (Beograd) (N.S.) 48(62) (1990), 110–118.

  • [15] M.S. Moslehian, M. Sattari and K. Shebrawi, Extensions of Euclidean operator radius inequalities, Math. Scand. 120 (2017), no. 1, 129–144.

  • [16] B. Nath, On a generalization of semi-inner product spaces, Math. J. Okayama Univ. 15 (1971), no. 1, 1–6.

  • [17] P.L. Papini, Un’osservazione sui prodotti semi-scalari negli spazi di Banach, Boll. Un. Mat. Ital. (4) 2 (1969), 686–689.

  • [18] I. Roşca, Semi-produit scalaire et représentations de type de Riesz pour les fonctionelles linéaires et bornées sur les espaces normés, C.R. Acad. Sci. Paris Sér. A–B 283 (1976), no. 3, Ai, A79–A81.

  • [19] O. Shisha and B. Mond, Bounds on differences of means, in: O. Shisha (ed.), Inequalities, Academic Press, Inc., New York, 1967, pp. 293–308.


Journal + Issues