On p-Compact Sets in Classical Banach Spaces

Given p ≥ 1, we denote by Cp the class of all Banach spaces X satisfying the equality Kp(Y,X) = Πdp(Y,X) for every Banach space Y , Kp (respectively, Πdp ) being the operator ideal of p-compact operators (respectively, of operators with p-summing adjoint). If X belongs to Cp, a bounded set A ⊂ X is relatively p-compact if and only if the evaluation map U∗ A : X∗ −→ ∞(A) is p-summing. We obtain p-compactness criteria valid for Banach spaces in Cp

An approximation property with respect to an operator ideal

Given an operator ideal A, we say that a Banach space X has the approximation property with respect to A if T belongs to {S ◦T : S ∈F(X)} τc for every Banach space Y and every T ∈A(Y,X), τc being the topology of uniform convergence on compact sets. We present several characterizations of this type of approximation property. It is shown that some of the existing approximation properties in the literature may be included in this setting

Duality of measures of non-A-compactness

Let A be a Banach operator ideal. Based on the notion of A-compactness in a Banach space due to Carl and Stephani, we deal with the notion of measure of non-A-compactness of an operator. We consider a map χA (respectively, nA) acting on the operators of the surjective (respectively, injective) hull of A such that χA(T) = 0 (respectively, nA(T) = 0) if and only if the operator T is A-compact (respectively, injectively A-compact). Under certain conditions on the ideal A, we prove an equivalence inequality involving χA(T∗) and nAd(T). This inequality provides an extension of a previous result stating that an operator is quasi p-nuclear if and only if its adjoint is p-compact in the sense of Sinha and Karn

Operators whose adjoints are quasi p-nuclear

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 ⊆{Pn αnxn : (αn) ∈ B`p0}. 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

Some properties and applications of equicompact sets of operators

Let X and Y be Banach spaces. A subset M of K(X,Y ) (the vector space of all compact operators from X into Y endowed with the operator norm) is said to be equicompact if every bounded sequence (xn) in X has a subsequence (xk(n))n such that (Txk(n))n is uniformly convergent for T ∈ M. We study the relationship between this concept and the notion of uniformly completely continuous set and give some applications. Among other results, we obtain a generalization of the classical Ascoli theorem and a compactness criterion in Mc(F,X), the Banach space of all (finitely additive) vector measures (with compact range) from a field F of sets into X endowed with the semivariation norm

On p-compact sets in classical Banach spaces

Given p ≥ 1, we denote by Cp the class of all Banach spaces X satisfying the equality Kp(Y,X) = Πdp(Y,X) for every Banach space Y , Kp (respectively, Πdp ) being the operator ideal of p-compact operators (respectively, of operators with p-summing adjoint). If X belongs to Cp, a bounded set A ⊂ X is relatively p-compact if and only if the evaluation map U∗ A : X∗ −→ ∞(A) is p-summing. We obtain p-compactness criteria valid for Banach spaces in Cp

Conjuntos de operadores que suman de manera uniforme en espacios de funciones continuas

Let X and Y be Banach spaces. A set ℳ of 1-summing operators from X into Y is said to be uniformly summing if the following holds: given a weakly 1-summing sequence (xn) in X, the series∑ n‖ Txn‖ is uniformly convergent in T∈ ℳ. We study some general properties and obtain a characterization of these sets when ℳ is a set of operators defined on spaces of continuous functions.Dejar X y Y Ser espacios de Banach. Un conjuntoℳ de operadores de 1 suma de X dentro Yse dice que está sumando de manera uniforme si se cumple lo siguiente: dada una secuencia débil de 1 suma (Xnorte) en X, las series ∑norte‖TXnorte‖ es uniformemente convergente en T∈ℳ. Estudiamos algunas propiedades generales y obtenemos una caracterización de estos conjuntos cuandoℳ es un conjunto de operadores definidos en espacios de funciones continuas