p-summing operators on injective tensor products of spaces

Abstract

Let $X,Y$ and $Z$ be Banach spaces, and let ${\prod }_p(Y,Z)(1\le p<\infty )$ 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 $T:X{\hat{\otimes }}_{\epsilon }Y⟶Z$ 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 $N\in \mathbb{N}$, 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 ${\ell }_{\infty }^N$. Finally, we show that there is a compact Hausdorff space $K$ and a bounded linear operator $T:C(K){\hat{\otimes }}_{\epsilon }{\ell }_1⟶{\ell }_2$ for which $T^#:\ C(K)\longrightarrow \prod_1(\ell_1, \ell_2)$ is not 2-summing.