On the compact non-nuclear operator problem (Q1263056)

The problem reads: Let X, Y be two infinite dimensional Banach spaces.: Does there always exist a compact non-nuclear operator f: \(X\to Y?\) This question is settled in the negative by showing that any compact operator from Pisier's space P into its dual \(P^*\) is nuclear. The result uses standard properties of Pisier's space and an interesting approximation result.