Compactness of the integration operator associated with a vector measure (Q2773433)

Let \(X\) be a complex Banach space, let \(\Sigma\) be a \(\sigma\)-algebra of subsets of a set \(\Omega\), and let \(m\colon \Sigma\to X\) be a vector measure. The integral operator \(I_m\colon L^1(m)\to X\) is given by the Bochner integral \(I_m(f):=\smallint_{\Omega}f dm\), \(f\in L^1(m)\). For a vector measure \(m\) with finite variation, the authors proved that \(I_m\) is compact if and only if \(m\) has a Radon-Nikodým derivative \(G:=dm/d|m|\) with respect to the variation \(|m|\) of \(m\), and the function \(G\) has \(|m|\)-essentially compact range in \(X\). Moreover, in every infinite-dimensional Banach space there exist nontrivial vector measures \(m\) with finite variation such that \(I_m\) is compact, and other \(m\) still with finite variation such that \(I_m\) is not compact. If \(m\) has infinite variation, then \(I_m\) is never compact.