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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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

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