Density of finite rank operators in the Banach space of p-compact operators

Abstract

A Banach space X is said to have the k p -approximation property (k p -AP) if for every Banach space Y , the space F (Y , X) of finite rank operators is dense in the space K p (Y , X) of p-compact operators endowed with its natural ideal norm k p . In this paper we study this notion that has been previously treated by Sinha and Karn (2002) in [15]. As application, the k p -AP of dual Banach spaces is characterized via density of finite rank operators in the space of quasi p-nuclear operators for the p-summing norm. This allows to obtain a relation between the k p -AP and Saphar's approximation property. As another application, the k p -AP is characterized in terms of a trace condition. Finally, we relate the k p -AP to the (p, p)-approximation property introduced in Sinha and Karn (2002) [15] for subspaces of L p (μ)-spaces.

$p$-Convergent sequences and Banach spaces in which $p$-compact sets are $q$-compact

“…Since φ is p-compact, φ * is quasi p-nuclear [3,Corollary 3.4] and, in particular, p-summing. So, according to Proposition 2.4, the sequence φ(γ n ) is p-null in X.…”

“…In [3], it is proved that, for every q ≥ 1 and every infinite dimensional Banach space, there exist relatively compact subsets that are not relatively q-compact [3,Theorem 3.14]. The objective of this section is to find out if this result is also true when we replace compact (= ∞-compact) with p-compact, p > q.…”

“…In [3,Theorem 3.14], Serrano and the authors have proved that every infinite dimensional Banach space has relatively compact sets failing to be q-compact for every 1 ≤ q < ∞. Section 3 is devoted to find out if this result is also true when we replace compact (= ∞-compact) with p-compact, p > q.…”

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

“…Reinforced by the fact that there are Banach spaces which lack the approximation property (the first example given by Enflo [13]), important variants of this property have emerged and were intensively studied, see [4,11,18,21] and references therein. In particular, there is a recent inclination to study approximation properties related to (Banach) operator ideals, as it can be seen for instance in [1,5,7,9,14,16,17,19,22,23,29,30].…”

“…With the particular case of N p , on the one hand, we cover the notion of p-approximation property introduced by Sinha and Karn [29] and studied by many authors in the last years, see for instance [1,5,7,16]. On the other hand, we cover the κ p -approximation property defined by Delgado, Piñeiro and Serrano [9] and studied later in [14,16,23]. Also, we address the K A -uniform approximation property in terms of a modified -product of Schwartz.…”