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ON BANACH SPACES VERIFYING GROTHENDIECK'S THEOREM

Published online by Cambridge University Press:  08 October 2003

QINGYING BU
Affiliation:
Department of Mathematics, University of Mississippi, University, MS 38677, USA, qbu@olemiss.edu
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Abstract

The projective tensor product $\ell_2\,{\hat{\otimes}}\,X$ of $\ell_2$ with any Banach space $X$ sits inside the space ${\rm Rad}(X)$ of all almost unconditionally summable sequences in $X$. If $X$ is of cotype 2 and $u: X \longrightarrow Y$ is 2-summing, then $u$ takes ${\rm Rad}(X)$ into $\ell_2\,{\hat{\otimes}}\,Y$. Consequently, if $X$ is of cotype 2, then every operator from $X$ to $\ell_2$ is 1-summing if and only if $\ell_1\,{\check{\otimes}}\,X \subseteq \ell_2\,{\hat{\otimes}}\,X$. In this case, each 2-summing operator from $\ell_2$ to $X$ is nuclear, and $X$ does not have non-trivial type provided that ${\rm dim}\,X = \infty$.

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Notes and Papers
Copyright
© The London Mathematical Society 2003

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