C. Piñeiro

Equicompact sets of operators defined on banach spaces

Let X and Y be Banach spaces. We say that a set M ⊂ X(X, Y) (K(X, Y) denotes the space of all compact operators from X into Y) is equicompact if there exists a null sequence (x* n ) n in X* such that ||Tx|| ≤ sup n |x* n (x)| for all x ∈ X and all T ∈ M. It is easy to show that collectively compactness and equicompactness are dual concepts in the following sense: M is equicompact iff M* = {T*: T ∈ M} is collectively compact. We study some properties of equicompact sets and, among other results, we prove: 1) a set M C K(X,Y) is equicompact iff each bounded sequence (x n ) n in X has a subsequence (x k(n) ) n such that (Tx k(n) ) n is a converging sequence uniformly for T ∈ M; 2) if Y does not have finite cotype and M C X(X, Y) is a maximal equicompact set, then, given e > 0 and a finite set {x 1 ,...,x n } in X, there is an operator S ∈ M such that ||Tx i || < (1 + e)||Sx i || for i = 1,...,n and all T ∈ M.

p-CONVERGENT SEQUENCES AND BANACH SPACES IN WHICH p-COMPACT SETS ARE q-COMPACT

We introduce and investigate the notion of p-convergence in a Banach space. Among others, a Grothendieck-like result is obtained; namely, a subset of a Banach space is relatively p-compact if and only if it is contained in the closed convex hull of a p-null sequence. We give a description of the topological dual of the space of all p-null sequences which is used to characterize the Banach spaces enjoying the property that every relatively p-compact subset is relatively q-compact (1 < q < p). As an application, Banach spaces satisfying that every relatively p-compact set lies inside the range of a vector measure of bounded variation are characterized.

A note on sequences lying in the range of a vector measure valued in the bidual

Let X be a Banach space. It is unknown if every subset A of X lying in the range of an X**-valued measure is actually contained in the range of an X-valued measure. In this paper we solve this problem in the case when we consider only vector measures of bounded variation.

UNIFORMLY SUMMING SETS OF OPERATORS ON SPACES OF CONTINUOUS FUNCTIONS

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 ( x n ) in X , the series ∑ n ‖ T x n ‖ 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.