ON BANACH SPACES VERIFYING GROTHENDIECK'S THEOREM

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 ${\rm dim}\,X = \infty$.