Characterizations of subspaces, quotients and subspaces of quotients of \(L_ p\) (Q1084312)

Let E be a Banach space, \(1<p<\infty\) and \(1/p+1/p'=1\). Denote by \((e_ n)\) the canonical basis of \(\ell^{p'}\). Then the following theorem is proved: Theorem. (1) E is isomorphic to a quotient of some \(L^ p\) if and only if for each operator \(T:\ell^{p'}\to E\), \(\sum_{n}\| Te_ n\|^ p<\infty\) implies that T is p-integral. (2) E is isomorphic to a subspace of a quotient of some \(L^ p\) if and only if for each operator \(T:\ell^{p'}\to E\), \(\sum_{n}\| Te_ n\|^ p<\infty\) implies that T is p-summing. Similar characterization of subspaces of \(L^ p\) is also given.