On $p$-summable sequences in the range of a vector measure

Abstract:

Let $p > 2$. Among other results, we prove that a Banach space $X$ has the property that every sequence $(x_{n})\in \ell _{u}^{p}(X)$ lies inside the range of an $X$-valued measure if and only if, for all sequences $(x_{n}^{\ast })$ in $X^{\ast }$ satisfying that the operator $x\in X\rightarrow (\langle x, x_{n}^{\ast }\rangle )\in \ell _{1}$ is 1-summing, the operator $x\in X\rightarrow (\langle x, x_{n}^{\ast }\rangle )\in \ell _{q}$ is nuclear, being $q$ the conjugate number for $p$. We also prove that, if $X$ is an infinite-dimensional ${\mathcal {L}}_{p}$-space for $1 \leq p < 2$, then $X$ can’t have the above property for any $s > 2$.