Operators whose adjoints are quasi p-nuclear

Abstract

For p ≥ 1, a set K in a Banach space X is said to be relatively p-compact if there exists a p-summable sequence (xn) in X with K ⊆ { P n αnxn : (αn) ∈ B p }. We prove that an operator T : X → Y is p-compact (i.e., T maps bounded sets to relatively p-compact sets) iff T * is quasi p-nuclear. Further, we characterize p-summing operators as those operators whose adjoints map relatively compact sets to relatively p-compact sets.

On p-compact sets in classical Banach spaces

“…Anyway, we have the following way to generate 2-compact sets in c 0 : if A ⊂ 2 is relatively compact, then A is relatively 2-compact as a subset of c 0 . In fact, the identity map from 2 to c 0 has 1-summing (hence, 2-summing) adjoint, so that operator maps relatively compact sets in 2 to relatively 2-compact sets in c 0 [8,Theorem 3.14]. This example inspires the following lemma:…”

“…A bounded subset A of a Banach space X is relatively p-compact if and only if the corresponding evaluation map U * [8,Proposition 3.5]. Nevertheless, for a wide class of Banach spaces, the relative p-compactness of a set is characterized just by the p-summability of its evaluation map.…”

“…Conversely, suppose A ⊂ X is such that there exist a relatively compact set K ⊂ 2 and an operator φ : 2 −→ X verifying A ⊂ φ(K). According to [9, Theorem 3.1], φ * is 2-summing, so φ map relatively compact sets in 2 to relatively 2-compact sets in X [8,Theorem 3.14].…”

“…A set A ⊂ X is said to be relatively p-compact if there exists (x n ) ∈ p (X) ((x n ) ∈ c 0 (X) if p = ∞) such that A ⊂ p-co (x n ). This nice notion has provoked the interest of several authors (see, for instance, [2], [6], [8] and [14]), whose contributions have made possible a deeper acknowledge of p-compactness in arbitrary Banach spaces. Anyway, there is no much information or examples of relative p-compact sets in concrete Banach spaces.…”

“…In [8], it is proved that a bounded subset A of an arbitrary Banach space X is relatively p-compact if and only if the corresponding evaluation map U * A : x * ∈ X * −→ ( x * , a ) a∈A ∈ ∞ (A) is p-nuclear ([8, Proposition 3.5]). However, for a wide class, say C p , of Banach spaces, the relatively p-compactness of any bounded set A occurs whenever U * A is just p-summing.…”