On factorization of operators via the spaces \(l^{p*}\)
A condition is given for a pair of Banach spaces \(X\) and \(Y\) such that \((*)\) each operator \(T:X\to Y\) whose second adjoint \(T^{**}\) is compactly factorable via the space \(l^p\), \(1\leq p\leq +\infty\), itself is compactly factorable via \(l^p\). This condition is that either \(X^*\) or \(Y^{***}\) possesses the Grothendieck approximation property. The author further shows that for \(p=1\) this condition is essential, namely: there exist Banach spaces (with \(Y^{**}\) possessing a boundedly complete basis), for which the assertion \((*)\) is incorrect for the case of \(l^1\)-factorable operators. This implies that one cannot replace the above condition with the weakened condition that ``one of the spaces \(X\) or \(Y^{**}\) has a (restricted) approximation property''.
scientific article; zbMATH DE number 2034821