Journal of the
Korean Mathematical Society JKMS

We give the necessary condition for an operator defined on a cartesian product of $c_{0}\left( \mathcal{X}\right) $ spaces\ to be summing or dominated and we show that for the multiplication operators this condition is also sufficient. By using these results, we show that $\Pi _{s}\left( c_{0},\ldots,c_{0};c_{0}\right) $ contains a copy of $l_{s}\left( l_{2}^{m}\mid m\in \mathbb{N}\right) $ for $s>2$ or a copy of $l_{s}\left( l_{1}^{m}\mid m\in \mathbb{N}\right) $, for any $1\leq s<\infty $. Also, $\Delta _{s_{1},\ldots,s_{n}}\left( c_{0},\ldots,c_{0};c_{0}\right) $ contains a copy of $ l_{v_{n}\left( s_{1},\ldots,s_{n}\right) }$ if $v_{n}\left( s_{1},\ldots,s_{n}\right) \leq 2$ or a copy of $l_{v_{n}\left( s_{1},\ldots,s_{n}\right) }\left( l_{2}^{m}\mid m\in \mathbb{N} \right) $ if $2

Keywords: $\left( s,s_{1},\ldots s_{n}\right) $-summing operator, $\left(s_{1},\ldots s_{n}\right) $-dominated operator

MSC numbers: Primary 47B10, 47L20; Secondary 46B45