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


Author: Cándido Piñeiro
Journal: Proc. Amer. Math. Soc. 125 (1997), 2073-2082
MSC (1991): Primary 46G10; Secondary 47B10
DOI: https://doi.org/10.1090/S0002-9939-97-03817-3
MathSciNet review: 1377003
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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 $s > 2$.


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Additional Information

Cándido Piñeiro
Email: candido@colon.uhu.es

DOI: https://doi.org/10.1090/S0002-9939-97-03817-3
Received by editor(s): November 30, 1995
Received by editor(s) in revised form: January 31, 1996
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1997 American Mathematical Society

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