Absolutely summing multilinear operators via interpolation

We use an interpolative technique from \cite{abps} to introduce the notion of multiple $N$-separately summing operators. Our approach extends and unifies some recent results; for instance we recover the best known estimates of the multilinear Bohnenblust-Hille constants due to F. Bayart, D. Pellegrino and J. Seoane-Sep'ulveda. More precisely, as a consequence of our main result, for $1\le t<2$ and $m\in \mathbb{N}$ we prove that $${\left(\sum _{i_1,\dots ,i_m=1}^{\mathrm{\infty }}{\mid U(e_{i_1},\dots ,e_{i_m})\mid }^{\frac{2tm}{2+(m-1)t}}\right)}^{\frac{2+(m-1)t}{2tm}}\le \left[\prod _{j=2}^m\mathrm{\Gamma }{(2-\frac{2-t}{jt-2t+2})}^{\frac{t(j-2)+2}{2t-2jt}}\right]\parallel U\parallel$$ for all complex $m$-linear forms $U:c_{0}\times \cdot \cdot \cdot \times c_{0}\rightarrow $\mathbb{C}$$.