  Let $X,Y$ and $Z$ be Banach spaces, and let $\prod_p(Y,Z) (1\leq p<\infty)$
denote the space of $p$-summing operators from $Y$ to $Z$. We show that, if $X$
is a {\it \$}$_\infty$-space, then a bounded linear operator $T: X\hat
\otimes_\epsilon Y\longrightarrow Z$ is 1-summing if and only if a naturally
associated operator $T^#: X\longrightarrow \prod_1(Y,Z)$ is 1-summing. This
result need not be true if $X$ is not a {\it \$}$_\infty$-space. For $p>1$,
several examples are given with $X=C[0,1]$ to show that $T^#$ can be
$p$-summing without $T$ being $p$-summing. Indeed, there is an operator $T$ on
$C[0,1]\hat \otimes_\epsilon \ell_1$ whose associated operator $T^#$ is
2-summing, but for all $N\in \N$, there exists an $N$-dimensional subspace $U$
of $C[0,1]\hat \otimes_\epsilon \ell_1$ such that $T$ restricted to $U$ is
equivalent to the identity operator on $\ell^N_\infty$. Finally, we show that
there is a compact Hausdorff space $K$ and a bounded linear operator $T:\
C(K)\hat \otimes_\epsilon \ell_1\longrightarrow \ell_2$ for which $T^#:\
C(K)\longrightarrow \prod_1(\ell_1, \ell_2)$ is not 2-summing.
