Some theory for summability with respect to purely atomic vector measures and $p$-summing operators

Let $E$ and $X$ be Banach spaces, $m\colon 2^\mathbb{N}\to E$ be a vector measure and $(x_k)_{k=1}^\infty$ be a sequence in $X$. Roughly speaking, we say that $(x_k)_{k=1}^\infty$ is $m$-summable if the sequence $(x_k e^\ast(m(\{k\}))_{k=1}^\infty$ is summable (in the classical sense) for every $e^\ast\in E^\ast$. This method of summation can be seen as a generalization of the weighted summability and is closely related to the space $\ell^1(m)$ of integrable functions with respect to a purely atomic vector measure $m$. This relationship will allows us to obtain trivial characterizations of unconditionally $m$-summable sequences and a version of Dvoretzky-Rogers Theorem for Banach function spaces under mild conditions for $m$. We also provide examples of non summable sequences which are $m$-summable and obtain non trivial versions of $\ell^1$-Schur and Orlicz-Pettis Theorems for this kind of summability. We finish the paper by defining $(m,p)$-summing operators and proving that, with a condition of separability, they coincide with the classical $p$-summing operators.