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On -summable sequences in the range of a vector measure

Authors:	Cá ndido Piñ eiro

Institution:	Departamento de Matemáticas, Escuela Politécnica Superior, Universidad de Huelva, 21810 La Rábida, Huelva, Spain

Abstract:	Let . Among other results, we prove that a Banach space has the property that every sequence $(x_{n})\in \ell _{u}^{p}(X)$ lies inside the range of an -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 the conjugate number for . We also prove that, if is an infinite-dimensional ${\mathcal {L}}_{p}$ -space for $1 \leq p < 2$ , then can't have the above property for any .

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