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

Abstract

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 (x n ) ⊂ X is p-null if and only if lim x n = 0 and (x n ) is relatively p-compact in X.

On null sequences for Banach operator ideals, trace duality and approximation properties

“…Notice that when is minimal, Theorem (c) gives a representation of as a Banach tensor product. Also, this result can be seen as an extension of the characterization given, with a different approach, by Oja for p ‐null sequences and J. M. Kim for unconditionally p ‐null sequences. To be more precise we introduce some definitions.…”

“…By [, Corollary 4.2], unconditionally p ‐null sequences and ‐null sequences coincide. Therefore, once these concepts are described under the framework of the metric Carl–Stephani theory, an application of Theorem (c) for covers the result due to Oja [, Theorem 4.1] for p ‐null sequences, . With , the result by Kim [, Theorem 1.1], , is covered.…”

“…The notions of null sequences and compact sets were shown to be closely related from the nowadays classical result of Grothendieck which characterizes relatively compact sets as those contained in the absolutely convex hull of a norm null sequence of vectors of the space. In the recent years strong forms of compactness have been studied, see for instance , , , , , , , , , . Many of the results obtained can be revisited under the Carl–Stephani theory of ‐compact sets and ‐null sequences , where denotes an arbitrary operator ideal.…”

The Banach ideal of A-compact operators and related approximation properties

“…The class of p-compact sets fits in this framework for the ideal N p of right p-nuclear operators. This fact and the notion of A-null sequences [3] allow us to give another proof of a question posed in [8] and solved by Oja in her recent work [24]. In Section 2 we use m A to endow K A with a norm · K A , under which it is a Banach operator ideal.…”

“…In [8], Delgado and Piñeiro define p-null sequences, p ≥ 1, as follows. In [24] the author describes the space of p-null sequences as a tensor product via the ChevetSaphar tensor norm and, as an application, the result is obtained. Here, we show that the Delgado-Piñeiro-Oja result is an immediate consequence of the next two propositions.…”