A note on absolutely p-summing operators (Q1344896)

Let \(X\), \(Y\) be Banach spaces. As usual \(\Pi_ p(X, Y)\) will stand for the Banach space of all absolutely \(p\)-summing operators from \(X\) into \(Y\) \((1\leq p< +\infty)\). The set of all \(T\in {\mathcal L}(X, Y)\) so that \(T^*\in \Pi_ p(Y^*, X^*)\) is denoted by \(\Pi^ d_ p(X, Y)\). A well-known result of Kwapien, on operators factorizable through \(L_ p\)- spaces, states that \(Y\) is isomorphic to a subspace of an \(L_ p(\mu)\)- space \((1< p< +\infty)\) if and only if \(\Pi^ d_ p(X, Y)\subset \Pi_ p(X, Y)\) for every Banach space \(X\). Among other results, we prove that \(Y\) is isomorphic to a subspace of an \(L_ p(\mu)\)-space \((1\leq p< +\infty)\) if and only if \(\Pi^ d_ p(X, Y)\subset \Pi_ p(X, Y)\) for some \({\mathcal D}_ q\)-space \(X\), where \(q\) is the conjugate number for \(p\).