Compact integration operators for Fréchet-space-valued measures.
Let \(X\) be a Fréchet space with an increasing sequence of continuous seminorms \(p_1\leq p_2\leq\cdots\) determining the topology of \(X\), and for each \(k\geq 1\), let \(X_k\) be the Banach space obtained from the quotient normed space \(X/p_k^{-1}(\{0\})\) by completion. Let \(\Sigma\) be a \(\sigma\)-algebra of subsets of a non-empty set \(\Omega\), and let \(m:\Sigma\to X\) be a vector measure. Also, for each \(k\geq 1\), let \(\Pi_k\) be the canonical quotient map of \(X\) onto \(X_k\) and let \(m_k=\Pi_k{\circ}m\) be the Banach space valued measure with the variation measure \(| m_k| \). The integration operator \(I_m:L^1(m)\to X\) is defined by \(I_mf:=\smallint_{\Omega}f\,dm\) for \(f\in L^1(m)\). The authors prove that \(I_m:L^1(m)\to X\) is compact if and only if there exists a positive integer \(k\) and a Bochner \(| m_k| \)-integrable function \(G:\Omega\to X\) such that (i) \(L^1(m)=L^1(m_k)\); (ii) \(m(E)=(B)\smallint_EG\,d| m_k| \) for \(E\in\Sigma\) and (iii) \(G\) has \(| m_k| \)-essentially relatively compact range in \(X\). In this case, \(m\) necessarily has finite variation and \(I_mf=(B)\smallint_{\Omega}fG\,d| m_k|\) for \(f\in L^1(m)\).
scientific article; zbMATH DE number 1893641
scientific article