## Illinois Journal of Mathematics

### Vector measures and nuclear operators

M. A. Sofi

#### Abstract

Among other results we prove that for a Banach space $X$ and $1 < p < \infty$, all $p$-unconditionally Cauchy sequences in $X$ lie inside the range of a $Y$-valued measure of bounded variation for some Banach space $Y$ containing $X$ if and only if each $\ell_1$-valued $2$-summing map on $X$ induces a nuclear map on $X$ valued in $\ell_q$, $q$ being conjugate to $p$. We also characterise Banach spaces $X$ with the property that all $\ell_2$-valued absolutely summing maps on $X$ are already nuclear as those for which $X^\ast$ has the (GT) and (GL) properties.

#### Article information

Source
Illinois J. Math., Volume 49, Number 2 (2005), 369-383.

Dates
First available in Project Euclid: 13 November 2009

https://projecteuclid.org/euclid.ijm/1258138023

Digital Object Identifier
doi:10.1215/ijm/1258138023

Mathematical Reviews number (MathSciNet)
MR2163940

Zentralblatt MATH identifier
1083.46025

#### Citation

Sofi, M. A. Vector measures and nuclear operators. Illinois J. Math. 49 (2005), no. 2, 369--383. doi:10.1215/ijm/1258138023. https://projecteuclid.org/euclid.ijm/1258138023