{
"id": "math/9201215",
"version": "v2",
"published": "1990-07-23T20:22:00.000Z",
"updated": "1999-12-04T05:08:35.000Z",
"title": "p-summing operators on injective tensor products of spaces",
"authors": [
"Stephen J. Montgomery-Smith",
"Paulette Saab"
],
"journal": "B. Royal Soc. Edin. 120A, (1992), 283-296",
"categories": [
"math.FA"
],
"abstract": "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.",
"revisions": [
{
"version": "v2",
"updated": "1999-12-04T05:08:35.000Z"
}
],
"analyses": {
"subjects": [
"46B99"
],
"keywords": [
"injective tensor products",
"p-summing operators",
"compact hausdorff space",
"identity operator",
"bounded linear operator"
],
"tags": [
"journal article"
],
"note": {
"typesetting": "TeX",
"pages": 0,
"language": "en",
"license": "arXiv",
"status": "editable"
}
}
}