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 | References | Similar Articles | Additional Information

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

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