Maximal quasi-normed extension of quasi-normed lattices

Maximal quasi-normed extension of quasi-normed lattices

Автор: Kusraev Anatoly Georgievich, Tasoev Batradz Botazovich

Журнал: Владикавказский математический журнал @vmj-ru

Статья в выпуске: 3 т.19, 2017 года.

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The purpose of this article is to extend the Abramovich's construction of a maximal normed extension of a normed lattice to quasi-Banach setting. It is proved that the maximal quasi-normed extension Xϰ of a Dedekind complete quasi-normed lattice X with the weak σ-Fatou property is a quasi-Banach lattice if and only if X is intervally complete. Moreover, Xϰ has the Fatou and the Levi property provided that X is a Dedekind complete quasi-normed space with the Fatou property. The possibility of applying this construction to the definition of a space of weakly integrable functions withrespect to a measure taking values from a quasi-Banach lattice is also discussed, since the duality based definition does not work in the quasi-Banach setting.

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Quasi-banach lattice, maximal quasi-normed extension, fatou property, levi property vector measure, space of weakly integrable functions

Короткий адрес: https://sciup.org/14318581

IDR: 14318581   |   DOI: 10.23671/VNC.2017.3.7111

Список литературы Maximal quasi-normed extension of quasi-normed lattices

  • Abramovich Yu. A. On maximal normed extension of a semi-ordered normed spaces. Izv. Vyssh. Uchebn. Zaved. Mat., 1970, vol.3, pp. 7-17.
  • Abramovich Y. A. and Aliprantis C. D. An Invitation to Operator Theory. Providence (R.I.): Amer. Math. Soc, 2002. iv+530 p. (Graduate Stud. in Math., vol. 50).
  • Aliprantis C. D. and Burkinshaw O. Positive Operators. London etc.: Acad. Press Inc., 1985. xvi+367 p.
  • Calabuig J. M., Delgado O., Juan M. A., and Sanchez Perez E. A. On the Banach lattice structure of L1w of a vector measure on a δ-ring. Collect. Math. 2014, vol. 65, pp. 67-85.
  • Curbera G. P. and Ricker W. J. Vector measures, integration, applications, Positivity. Basel: Birkhauser, 2007, pp. 127-160 (Trends Math.).
  • Hyers D. H. A note on linear topological spaces. Bull. Amer. Math. Soc., 1938, vol. 44, № 2, pp. 76-80.
  • Juan A. M. and Sanchez-Perez E. A. Maurey-Rosenthal domination for abstract Banach lattices. J. Ineq. and Appl., 2013, vol. 213, pp. 1-12.
  • Kalton N. J. Convexity conditions for non-locally convex lattices. Glasgow Math. J., 1984, vol. 25, pp. 141-142.
  • Kalton N. J. Quasi-Banach spaces. Handbook of the Geometry of Banach Spaces (Eds. W. B. Johnson and J. Lindenstrauss). Amsterdam a.o.: Elsevier, 2003. P. 1118-1130.
  • Kusraev A. G. and Tasoev B. B. Kantorovich-Wright integration and representation of quasi-Banach lattices. Dokl. Math. 2017, vol. 474, pp. 15-18.
  • Kusraev A. G. and Tasoev B. B. Kantorovich-Wright integration and representation of vector lattices//J. Math. Anal. Appl. 2017, vol. 455, pp. 554-568.
  • Kusraev A. G. and Tasoev B. B. Kantorovich-Wright integration and representation of quasi-Banach lattices//J. Math. Anal. Appl. (to appear).
  • Luxemburg W. A. J. and Zaanen A. C. Riesz Spaces. Vol. 1. Amsterdam-London: North-Holland, 1971. 514 pp.
  • Lewis D. R. On integration and summability in vector spaces. Illinois J. Math. 1972, vol. 16, pp. 294-307.
  • Maligranda L. Type, cotype and convexity properties of quasi-Banach spaces. Proc. of the International Symposium on Banach and Function Spaces Kitakyushu. Japan, 2003, pp. 83-120.
  • Masani P. R. and Niemi H. The integration theory of Banach space valued measures and the Tonelli-Fubini theorems. II. Pettis integration. Adv. Math., 1989, vol. 75, pp. 121-167.
  • Meyer-Nieberg P. Banach Lattices. Berlin etc.: Springer, 1991. xvi+395 p.
  • Okada S., Ricker W. J., and Sanchez-Perez E. A. Optimal Domain and Integral Extension of Operators Acting in Function Spaces. Basel: Birkhauser, 2008. (Oper. Theory Adv. Appl., vol. 180).
  • Pietsch A. Operator Ideals. Berlin: Deutsch. Verlag Wiss., 1978; North-Holland, Amsterdam-London-N.Y.-Tokyo, 1980.
  • Sanchez Perez E. A. and Tradacete P. Bartle-Dunford-Schwartz integration for positive vector measures and representation of quasi-Banach lattices. J. Nonlin. and Conv. Anal. 2016, vol. 17, no. 2, pp. 387-402.
  • Veksler A. I. On realization of Archimedean K-lineals. Sib. Math. J., 1962, vol. 3, no. 1, pp. 7-16.
  • Veksler A. I. The concept of a linear lattice which is normal in itself, and certain applications of this concept to the theory of linear and linear normed lattices. Isv. Vuzov. Math., 1966, vol. 4, pp. 13-22.
  • Veksler A. I. Interval completeness and intervally complete normability of KN-lineals. Isv. Vuzov. Math., 1970, vol. 4, pp. 36-46.
  • Vulikh B. Z. Introduction to the Theory of Partially Ordered Spaces. M.: Fizmatgiz, 1961.
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