Czechoslovak Mathematical Journal, Vol. 67, No. 2, pp. 457-467, 2017


Copies of $l_p^n$'s uniformly in the spaces $\Pi_2( C[ 0,1] ,X) $ and $\Pi_1(C[ 0,1],X) $

Dumitru Popa

Received January 8, 2016.  First published May 4, 2017.

Abstract:  We study the presence of copies of $l_p^n$'s uniformly in the spaces $\Pi_2( C[ 0,1] ,X) $ and $\Pi_1( C[0,1] ,X)$. By using Dvoretzky's theorem we deduce that if $X$ is an infinite-dimensional Banach space, then $\Pi_2( C[ 0,1] ,X) $ contains $\lambda\sqrt2$-uniformly copies of $l_{\infty}^n$'s and $\Pi_1( C[ 0,1] ,X) $ contains $\lambda$-uniformly copies of $l_2^n$'s for all $\lambda>1$. As an application, we show that if $X$ is an infinite-dimensional Banach space then the spaces $\Pi_2( C[ 0,1] ,X) $ and $\Pi_1( C[ 0,1] ,X) $ are distinct, extending the well-known result that the spaces $\Pi_2( C[ 0,1],X) $ and $\mathcal{N}( C[ 0,1] ,X) $ are distinct.
Keywords:  $p$-summing linear operators; copies of $l_p^n$'s uniformly; local structure of a Banach space; multiplication operator; average
Classification MSC:  46B07, 47B10, 47L20, 46B28
DOI:  10.21136/CMJ.2017.0009-16


References:
[1] C. Costara, D. Popa: Exercises in Functional Analysis. Kluwer Texts in the Mathematical Sciences 26, Kluwer Academic Publishers Group, Dordrecht (2003). DOI 10.1007/978-94-017-0223-2 | MR 2027363 | Zbl 1070.46001
  [2] A. Defant, K. Floret: Tensor Norms and Operator Ideals. North-Holland Mathematics Studies 176, North-Holland Publishing, Amsterdam (1993). DOI 10.1016/s0304-0208(08)x7019-7 | MR 1209438 | Zbl 0774.46018
  [3] J. Diestel, H. Jarchow, A. Tonge: Absolutely Summing Operators. Cambridge Studies in Advanced Mathematics 43, Cambridge University Press, Cambridge (1995). DOI 10.1017/CBO9780511526138 | MR 1342297 | Zbl 0855.47016
  [4] J. Diestel, J. J. Uhl, Jr.: Vector Measures. Mathematical Surveys 15, American Mathematical Society, Providence (1977). DOI 10.1090/surv/015 | MR 0453964 | Zbl 0369.46039
  [5] A. Lima, V. Lima, E. Oja: Absolutely summing operators on $C[0,1]$ as a tree space and the bounded approximation property. J. Funct. Anal. 259 (2010), 2886-2901. DOI 10.1016/j.jfa.2010.07.017 | MR 2719278 | Zbl 1207.46019
  [6] A. Pietsch: Operator Ideals. Mathematische Monographien 16, VEB Deutscher der Wissenschaften, Berlin (1978). MR 0519680 | Zbl 0399.47039
  [7] D. Popa: Examples of operators on $C[0,1]$ distinguishing certain operator ideals. Arch. Math. 88 (2007), 349-357. DOI 10.1007/s00013-006-1916-2 | MR 2311842 | Zbl 1124.47013
  [8] D. Popa: Khinchin's inequality, Dunford-Pettis and compact operators on the space $C([0,1],X)$. Proc. Indian Acad. Sci., Math. Sci. 117 (2007), 13-30. DOI 10.1007/s12044-007-0002-4 | MR 2300675 | Zbl 1124.47023
  [9] D. Popa: Averages and compact, absolutely summing and nuclear operators on $C(\Omega)$. J. Korean Math. Soc. 47 (2010), 899-924. DOI 10.4134/JKMS.2010.47.5.899 | MR 2722999 | Zbl 1214.47023
  [10] M. A. Sofi: Factoring operators over Hilbert-Schmidt maps and vector measures. Indag. Math., New Ser. 20 (2009), 273-284. DOI 10.1016/S0019-3577(09)80014-1 | MR 2599817 | Zbl 1193.46005

Access to the full text of journal articles on this site is restricted to the subscribers of Myris Trade. To activate your access, please contact Myris Trade at myris@myris.cz.
Subscribers of Springer need to access the articles on their site, which is https://link.springer.com/journal/10587.

Affiliations:   Dumitru Popa, Department of Mathematics, Ovidius University of Constanţa, Bd. Mamaia 124, 900527 Constanţa, Romania, e-mail: dpopa@univ-ovidius.ro

 
PDF available at: