Grothendieckʼs nuclear operator theorem revisited with an application to p-null sequences

Let X and Y be Banach spaces and let α be a tensor norm. The principal result is the following theorem. If either X⁎⁎⁎ or Y has the approximation property, then each α-nuclear operator T:X⁎→Y such that T⁎(Y⁎)⊂X can be approximated in the α-nuclear norm by finite-rank operators of type X⊗Y. In the special case of (Grothendieck) nuclear operators, the theorem provides a strengthening for the classical theorem on the nuclearity of operators with a nuclear adjoint. The hypotheses about the approximation property are essential. The main application yields an affirmative answer to [C. Piñeiro, J.M. Delgado, p-Convergent sequences and Banach spaces in which p-compact sets are q-compact, Proc. Amer. Math. Soc. 139 (2011) 957–967]: for p⩾1, a sequence (xn)⊂X is p-null if and only if and (xn) is relatively p-compact in X.