Banach spaces in which every $p$-weakly summable sequence lies in the range of a vector measure

Abstract:

Let $X$ be a Banach space. For $1<p<+\infty$ we prove that the identity map $I_X$ is $(1,1,p)$-summing if and only if the operator $x^*\in X^*\to \sum \langle x_n,x^*\rangle e_n\in l_q$ is nuclear for every unconditionally summable sequence $(x_n)$ in $X$, where $q$ is the conjugate number for $p$. Using this result we find a characterization of Banach spaces $X$ in which every $p$-weakly summable sequence lies inside the range of an $X^{**}$-valued measure (equivalently, every $p$-weakly summable sequence $(x_n)$ in $X$, satisfying that the operator $(\alpha _n)\in l_q\to \sum \alpha _nx_n\in X$ is compact, lies in the range of an $X$-valued measure) with bounded variation. They are those Banach spaces such that the identity operator $I_{X^*}$ is $(1,1,p)$-summing.