p-summing operators on injective tensor products of spaces
| dc.creator | Montgomery-Smith, Stephen J. | |
| dc.creator | Saab, Paulette | |
| dc.date | 1990-07-23 | |
| dc.date | 1999-12-04 | |
| dc.date.accessioned | 2026-07-07T09:04:40Z | |
| dc.date.available | 2026-07-07T09:04:40Z | |
| dc.description | 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 $T: X\hat \otimes_εY\longrightarrow Z$ 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 $C[0,1]\hat \otimes_ε\ell_1$ whose associated operator $T^#$ is 2-summing, but for all $N\in \N$, there exists an $N$-dimensional subspace $U$ of $C[0,1]\hat \otimes_ε\ell_1$ 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 $T:\ C(K)\hat \otimes_ε\ell_1\longrightarrow \ell_2$ for which $T^#:\ C(K)\longrightarrow \prod_1(\ell_1, \ell_2)$ is not 2-summing. | |
| dc.identifier | https://arxiv.org/abs/math/9201215 | |
| dc.identifier | http://arxiv.org/abs/math/9201215 | |
| dc.identifier | B. Royal Soc. Edin. 120A, (1992), 283-296 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/149461 | |
| dc.subject | Functional Analysis | |
| dc.subject | 46B99 | |
| dc.title | p-summing operators on injective tensor products of spaces | |
| dc.type | text |