# Article

Full entry | PDF   (0.2 MB)
Keywords:
$p$-summing linear operators; copies of $l_{p}^{n}$'s uniformly; local structure of a Banach space; multiplication operator; average
Summary:
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 $\lambda \sqrt {2}$-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.
References:
[1] Costara, C., Popa, D.: 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] Defant, A., Floret, K.: 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] Diestel, J., Jarchow, H., Tonge, A.: 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] Lima, Å., Lima, V., Oja, E.: 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] Pietsch, A.: Operator Ideals. Mathematische Monographien 16, VEB Deutscher der Wissenschaften, Berlin (1978). MR 0519680 | Zbl 0399.47039
[7] Popa, D.: 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] Popa, D.: 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] Popa, D.: 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] Sofi, M. A.: 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

Partner of