On Persson's theorem concerning p-nuclear operators (Q1084313)

Let E, F be Banach spaces, p a real number such that \(1<p<\infty\) and \(1/p+1/p'=1\). We say that a linear operator T from E into F is of type \(N^ p\) if it is factorized by the bounded linear operators \(V:E\to \ell^{p'}\), \(D:\ell^{p'}\to \ell^ 1\) and \(W:\ell^ 1\to F\), where \(D(\alpha_ n)\) is a diagonal operator with \(\sum_{n}| \alpha_ n|^ p<\infty\). The aim of this paper is to prove the following: Theorem. (1) A Banach space E is isomorphic to a quotient of some \(L^ p\) if and only if for each Banach space F, every operator of type \(N^ p\) from F into E is p-nuclear. (2) A Banach space E is isomorphic to a subspace of some \(L^{p'}\) if and only if for each Banach space F, every p-nuclear operator from E into F is of type \(N^ p\).