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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

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


Author: C. Piñeiro
Journal: Proc. Amer. Math. Soc. 124 (1996), 2013-2020
MSC (1991): Primary 46G10; Secondary 47B10
DOI: https://doi.org/10.1090/S0002-9939-96-03242-X
MathSciNet review: 1307557
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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.


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Additional Information

C. Piñeiro
Affiliation: Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Sevilla, Aptdo. 1160, Sevilla, 41080, Spain
Address at time of publication: Departamento de Matemáticas, Escuela Politécnica Superior, Universidad de Huelva, 21810 La Rábida, Huelva, Spain

DOI: https://doi.org/10.1090/S0002-9939-96-03242-X
Received by editor(s): September 12, 1994
Received by editor(s) in revised form: December 2, 1994
Additional Notes: This research has been partially supported by the D.G.I.C.Y.T., PB 90-893
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1996 American Mathematical Society