Operators into \(L^ 1\) of a vector measure and applications to Banach lattices (Q811509)

Let \(\nu\) be a vector measure with values in a Banach space \(X\), and let \(L^ 1(\nu)\) be the space of real functions that are integrable with respect to \(\nu\). To every bounded linear operator from a Banach space \(Y\) into \(L^ 1(\nu)\) we associate a vector measure with values in \({\mathcal L}(Y,X)\). We study how the properties of the measure depend on the operator. As an application we prove that if \(X\) is an order continuous atomic Banach lattice, every operator from \(Y\) into \(X\) is compact if and only if \({\mathcal L}(Y,X)\) does not contain an isomorphic copy of \(\ell_ \infty\).