Bounded approximation properties via integral and nuclear operators

Abstract

Abstract. Let X be a Banach space and let A be a Banach operator ideal. We say that X has the λ-bounded approximation property for A (λ-BAP for A) if for every Banach space Y and every operator T ∈ A(X, Y ), there exists a net (S α ) of finite rank operators on X such that S α → I X uniformly on compact subsets of X andWe prove that the (classical) λ-BAP is precisely the λ-BAP for the ideal I of integral operators, or equivalently, for the ideal SI of strictly integral operators. We also prove that the weak λ-BAP is precisely the λ-BAP for the ideal N of nuclear operators.

Inner and outer inequalities with applications to approximation properties

“…As proved in [21] (and clear from Theorems 1.1 and 3.6), this conjecture means precisely that the weak λ-BAP is different from the λ-BAP. Results of [16], [21], [23], [29], and of the present paper seem to support this conjecture. For p = 1, condition (c) of Theorem 4.3 would be exactly the same as condition (c) of Theorem 3.6.…”

“…Condition (b) of Theorem 3.6 was very recently applied to prove main results in [16]. This condition appears to be of essence of the weak BAP: it opens a way towards remarkably simple proofs of main results concerning the weak BAP, using, of course, some basics on classical tensor products (as, e.g., (a)⇔(c) in Proposition 3.5, (a)⇔(b) in Theorem 1.1, or the coincidence of N (X, Y * ) and I(X, Y * ), with the equality of norms, whenever X * or Y * has the Radon-Nikodým property).…”

Ideal topologies and corresponding approximation properties

“…The consideration of problems on the approximation by operators belonging to a given operator ideal was a question of time. Indeed, a number of approximation properties (APs) with respect to operator ideals-and other ones that are somehow related to operator ideals-have been studied in the last three decades, see, e.g., [6,11,13,15,19,20,29,34,35,37,38,39,40,41,49,50,57,58,59,62,64]. The reader is also referred to the surveys [51,52] and to the references therein.…”

Bounded approximation properties in terms of $C[0,1]$

“…Recently, an approximation property which is bounded for A was introduced and studied in [11] as follows. We say that X has the λ-bounded approximation property for A (λ-BAP for A ) if for every Banach space Y and every operator T ∈ A (X, Y ) there exists a net (S α ) ⊂ F (X) such that S α → I X uniformly on compact subsets of X and lim sup…”