### 2-summing multiplication operators

Let 1 ≤ p < ∞, $={\left(X\u2099\right)}_{n\in \mathbb{N}}$ be a sequence of Banach spaces and ${l}_{p}\left(\right)$ the coresponding vector valued sequence space. Let $={\left(X\u2099\right)}_{n\in \mathbb{N}}$, $={\left(Y\u2099\right)}_{n\in \mathbb{N}}$ be two sequences of Banach spaces, $={\left(V\u2099\right)}_{n\in \mathbb{N}}$, Vₙ: Xₙ → Yₙ, a sequence of bounded linear operators and 1 ≤ p,q < ∞. We define the multiplication operator ${M}_{}:{l}_{p}\left(\right)\to {l}_{q}\left(\right)$ by ${M}_{}\left({\left(x\u2099\right)}_{n\in \mathbb{N}}\right):={\left(V\u2099\left(x\u2099\right)\right)}_{n\in \mathbb{N}}$. We give necessary and sufficient conditions for ${M}_{}$ to be 2-summing when (p,q) is one of the couples (1,2), (2,1), (2,2), (1,1), (p,1), (p,2), (2,p), (1,p), (p,q); in the last case 1 < p < 2, 1 < q < ∞.