Grothendieck’s 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 lim xn = 0 and (xn) is relatively p-compact in X. © 2012 Elsevier Inc. All rights reserved.