Summary: We give the necessary condition for an operator defined on a Cartesian product of \(c_{0}(\chi)\) 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 \(\Pi_{s}(c_{0},\ldots,c_{0};c_{0})\) contains a copy of \(l_{s}(l_{2}^{m}\mid m\in \mathbb{N})\) for \(s>2\) or a copy of \(l_{s}(l_{1}^{m}\mid m\in \mathbb{N})\), for any \(1\leq s<\infty\). Also, \(\Delta_{s_{1},\ldots,s_{n}}(c_{0},\ldots,c_{0};c_{0}\) contains a copy of \(l_{v_{n}(s_{1},\ldots,s_{n})}\) if \(v_{n}(s_{1},\ldots,s_{n})\leq 2\) or a copy of \(l_{v_{n}}(s_{1},\ldots,s_{n})(l_{2}^{m}\mid m \in \mathbb{N})\) if \(2<v_{n}(s_{1},\ldots,s_{n})\), where \(\frac{1}{v_{n}(s_{1},\ldots,s_{n})}=\frac{1}{s_{1}}+\cdots +\frac{1}{s_{n}}\). We find also the necessary and sufficient conditions for bilinear operators induced by some method of summability to be \(1\)-summing or \(2\) -dominated.