Factorization of weakly compact operators between Banach spaces and Fréchet or (LB)-spaces (Q2867598)

The authors prove the following results: (1)~There are a Banach space \(X\), a complete (LB)-space \(E\) and a weakly compact operator \(T\in L(X,E)\) which does not factorize through a reflexive Banach space. (2)~A reflexive operator from a locally convex space into a Banach space factorizes through a reflexive Banach space if and only if it is weakly compact. (3)~There exist a Fréchet-Montel space \(F\) and a continuous surjection \(T:F\to X\) onto a Banach space \(X\) which does not factorize through a reflexive Banach space.