A note on \(p\)-nuclear operators (Q1924833)

Let \(X\) be a Banach space and let \(1\leq r< +\infty\). We prove that \(X^*\) is isomorphic to a subspace of an \(L^r (\mu)\)-space if and only if the operator \((\alpha_n)\in \ell_r\to \sum \alpha_n x_n\in X\) is \(s\)-nuclear whenever \(\sum |x_n |^s< +\infty\), \(s\) being the conjugate exponent for \(r\).