A version of Kalton’s theorem for the space of regular operators

• Autores: Foivos Xanthos
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 66, Fasc. 1, 2015, págs. 55-62
• Idioma: inglés
• DOI: 10.1007/s13348-013-0101-8
• Texto completo no disponible (Saber más ...)
• Resumen

  • In this note we extend some recent results in the space of regular operators [appeared in Bu and Wong (Indag Math 23:199–213, 2012), Bu et al. (Collect Math 62:131–137, 2011), and Li et al. (Taiwan J Math 16:207–215, 2012)]. Our main result is the following Banach lattice version of a classical result of Kalton: Let E be an atomic Banach lattice with an order continuous norm and F a Banach lattice. Then the following are equivalent: (i) L^r(E,F) contains no copy of \ell _\infty, (ii) L^r(E,F) contains no copy of c_0, (iii) K^r(E,F) contains no copy of c_0, (iv) K^r(E,F) is a (projection) band in L^r(E,F), (v) K^r(E,F)=L^r(E,F)

• Referencias bibliográficas
  • Albiac, F., Kalton, N.J.: Topics in Banach space theory, Graduate Texts in Mathematics, vol. 233. Springer, New York (2006)
  • Aliprantis, C.D., Burkinshaw, O.: Positive Operators. Springer, Dordrecht (1985)
  • Aqzzouz, B., Elbour, A., Wickstead, A.W.: Positive almost Dunford–Pettis operators and their duality. Positivity 15(2), 185–197 (2011)
  • Bu, Q., Buskes, G., Popov, A.I., Tcaciuc, A., Troitsky, V.G.: The 2-concavification of a Banach lattice equals the diagonal of the Fremlin...
  • Bu, Q., Wong, N.-C.: Some geometric properties inherited by the positive tensor products of atomic Banach latices. Indag. Math. 23, 199–213...
  • Bu, Q., Li, Y., Xue, X.: Some properties of the space of regular operators on atomic Banach lattices. Collect. Math. 62(2), 131–137 (2011)
  • Bu, Q., Buskes, G.: Schauder decompositions and the Fremlin projective tensor product of Banach lattices. J. Math. Anal. Appl. 355(1), 335–351...
  • Bu, Q., Michelle, C., Ji, D.: Reflexivity and the Grothendieck property for positive tensor products of Banach lattices. II. Quaest. Math....
  • Bu, Q., Buskes, G., Lai, W.-K.: The Radon–Nikodym property for tensor products of Banach lattices. II. Positivity 12(1), 45–54 (2008)
  • Chen, Z.L.: On the order continuity of the regular norm. Positivity IV-theory and applications, pp. 45–51. Tech. Univ. Dresden, Dresden (2006)
  • Chen, Z.L., Wickstead, A.W.: The order properties of r-compact operators on Banach lattices. Acta Math. Sin. (Engl. Ser.) 23(3), 457–466 (2007)
  • Chen, Z.L., Wickstead, A.W.: Some applications of Rademacher sequences in Banach lattices. Positivity 2(2), 171–191 (1998)
  • Diestel, J., Uhl, Jr. J.J.: Vector Measures, vol. 15. American Mathematical Society Mathematical Surveys, Providence (1977)
  • Feder, M.: On subspaces of spaces with an unconditional basis and spaces of operators. Ill. J. Math. 24(2), 196–205 (1980)
  • Fremlin, D.H.: Tensor products of Banach lattices. Math. Ann. 211, 87–106 (1974)
  • Kalton, N.J.: Space of compact operators. Math. Ann. 208, 267–278 (1974)
  • Li, Y., Ji, D., Bu, Q.: Copies of c_0 and l_\infty into a regular operator space. Taiwan. J. Math. 16(1), 207–215 (2012)
  • Megginson, R.E.: An Introduction to Banach Space Theory. Springer, New York (1998)
  • Meyer-Nieberg, P.: Banach Lattices. Universitext, Springer, Berlin (1991)
  • Polyrakis, I.A., Xanthos, F.: Cone characterization of Grothendieck spaces and Banach spaces containing c_0. Positivity 15, 677–693 (2011)
  • Schaefer, H.H.: Aspects of Banach lattices. Studies in functional analysis, MAA Stud. Math., vol. 21. Math. Assoc. America, Washington, D.C.,...
  • Schep, A.R.: Cone isomorphisms and almost surjective operators. Indag. Math. (2014, to appear)
  • Wickstead, A.W.: Regular operators between Banach lattices. Positivity, Trends Math., Birkhauser, Basel, pp. 255–279 (2007)
  • Wickstead, A.W.: Converses for the Dodds–Fremlin and Kalton–Saab theorems. Math. Proc. Camb. Philos. Soc. 120(1), 175–179 (1996)
  • Wnuk, W.: On the dual positive Schur property in Banach lattices. Positivity (2014, to appear)