Vector measure range duality and factorizations of \((D,p)\)-summing operators from Banach function spaces (Q1882732)

For a dual pair \(D=(\lambda, \lambda')\) of complete countably additive vector measures with values in a Banach space \(X\) and in its dual \(X'\), respectively (see the definitions in [\textit{K. Zheltukhin} and the reviewer, Quaest. Math. 23, No. 1, 77--86 (2000; Zbl 0970.46028)]), it is proved that the space \(L_1(\lambda')\) of scalar functions that are integrable with respect to \(\lambda'\) forms a subspace of the dual to \(L_1(\lambda)\) in the natural duality if and only if the integration operator is an isomorphism of \(L_1(\lambda)\) and \(X\). New concepts of vector measure range duality and of \((D,p)\)-summing operators, where \(D\) is a range dual pair of measures are introduced. Analogues of the Grothendieck-Pietsch domination theorem and of the corresponding factorization are proved in the \((D,p)\)-summing operators setting.