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

Abstract

Let scriptA be a Banach operator ideal and X be a Banach space. We undertake the study of the vector space of scriptA‐null sequences of Carl and Stephani on X, c0,Afalse(Xfalse), from a unified point of view after we introduce a norm which makes it a Banach space. To give accurate results we consider local versions of the different types of accessibility of Banach operator ideals. We show that in the most common situations, when scriptA is right‐accessible for (ℓ1;X), c0,Afalse(Xfalse) behaves much alike c0false(Xfalse). When this is the case we give a geometric tensor product representation of c0,Afalse(Xfalse). On the other hand, we show an example where the representation fails. Also, via a trace duality formula, we characterize the dual space of c0,Afalse(Xfalse). We apply our results to study some problems related with the KA‐approximation property giving a trace condition which is used to solve the remaining case (p=1) of a problem posed by Kim (2015). Namely, we prove that if a dual space has the K1‐approximation property then the space has the Ku1‐approximation property.

Lifting some approximation properties from a dual space $X’$ to the Banach space $X$

“…This question was already answered for particular A-compact sets in several papers, see [2,7,8,9,11] among others. Our objective is to study the above problem for a wide class of A-compact sets.…”

“…In [11,Proposition 1.8], it was shown that A-compact sets are determined by operators in A with domain in 1 . Since Theorems 2.4 and 2.5 involve A-compact sets determined by operators with domain c 0 , we make a brief discussion of this type of compact sets in Section 1.…”

“…Following Carl and Stephani [1], we call a subset K of X relatively A-compact if there exist a Banach space Z, an operator T ∈ A(Z, X) and a compact subset L of Z such that K ⊂ T (L). The measure of relatively A-compact sets was introduced in [11] as follows. Given a relatively A-compact subset K of X, the size of K is given by…”

“…From [3,22.1] we know that A, A min and A max are associated with the same tensor norm. In light of [11], the theory of A-compactness is simpler whenever A is a right-accessible Banach operator ideal. This happens when the identity A min = A • F holds isometrically.…”

“…We now focus on a class of approximation properties determined by A-compact sets. Following [11,Proposition 3.1], a Banach space X has the K A -approximation property (K A -AP for short) if, given ε > 0 and an A-compact set K ⊂ X, there exists a finite rank operator S :…”