Copies of $l_p^n$’s uniformly in the spaces ${\Pi }_2(C[0,1],X)$ and ${\Pi }_1(C[0,1],X)$

@article{Popa2017,
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 $\lambda \sqrt\{2\}$-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.},
author = {Popa, Dumitru},
journal = {Czechoslovak Mathematical Journal},
keywords = {$p$-summing linear operators; copies of $l_\{p\}^\{n\}$’s uniformly; local structure of a Banach space; multiplication operator; average},
language = {eng},
number = {2},
pages = {457-467},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Copies of $l_\{p\}^\{n\}$’s uniformly in the spaces $\Pi _\{2\}( C[ 0,1] ,X) $ and $\Pi _\{1\}(C[ 0,1],X) $},
url = {http://eudml.org/doc/288204},
volume = {67},
year = {2017},
}

TY - JOUR
AU - Popa, Dumitru
TI - Copies of $l_{p}^{n}$’s uniformly in the spaces $\Pi _{2}( C[ 0,1] ,X) $ and $\Pi _{1}(C[ 0,1],X) $
JO - Czechoslovak Mathematical Journal
PY - 2017
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 67
IS - 2
SP - 457
EP - 467
AB - 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.
LA - eng
KW - $p$-summing linear operators; copies of $l_{p}^{n}$’s uniformly; local structure of a Banach space; multiplication operator; average
UR - http://eudml.org/doc/288204
ER -