Operator ideal properties of the integration map of a vector measure (Q2252918)

Let \(X\) be a Banach space and \(m\) an \(X\)-valued vector measure. Let \(L^1(m)\) denote the space of \(m\)-integrable functions equipped with the lattice norm. In this interesting paper, the authors study the operator \(f \rightarrow \int f\, dm\). They show that if it is compact, then \(L^1(m) = L^1(|m|)\). This is also the case if the operator is completely continuous, provided \(X\) contains no isomorphic copy of \(\ell^1\) and has an unconditional basis.