\(p\)-convergent operators and the \(p\)-Schur property (Q2300127)

Let \(X,Y\) be Banach spaces. This interesting paper deals with the question of certain subclasses of \({\mathcal{K}}(X,Y)\) being equal to \({\mathcal{K}}(X,Y)\) for all spaces \(Y\) or equivalently for a test space \(Y\).\par For \(1 \leq p < \infty\) let \(C_p(X,Y)\) denote the class of operators which map weakly \(p\)-summable sequences to norm null sequences. It is shown that when \(C_p(X,Y) = {\mathcal{K}}(X,Y)\) for \(Y = \ell^{\infty}\), then these spaces are equal for all Banach spaces \(Y\).\par For \(1 \leq p <q < \infty\), a similar result holds for the inclusion \(C_p(X,Y) \subset C_q(X,Y)\).