Piñeiro, C. A note on absolutely p-summing operators. (English) Zbl 0814.47024 Collect. Math. 45, No. 2, 133-136 (1994). 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 \(p\). Reviewer: C.Piñeiro (Sevilla) MSC: 47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.) 47A68 Factorization theory (including Wiener-Hopf and spectral factorizations) of linear operators 46B28 Spaces of operators; tensor products; approximation properties Keywords:subspace of an \(L_ p(\mu)\)-space; Banach space of all absolutely \(p\)- summing operators; operators factorizable through \(L_ p\)-spaces PDFBibTeX XMLCite \textit{C. Piñeiro}, Collect. Math. 45, No. 2, 133--136 (1994; Zbl 0814.47024) Full Text: EuDML