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

[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

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

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