Weakly equicompact sets of operators defined on Banach spaces

Abstract.

Let X and Y be Banach spaces. We say that a set $M \subset \mathcal{W}(X,Y)$ (the space of all weakly compact operators from X into Y) is weakly equicompact if, for every bounded sequence (x _n ) in X, there exists a subsequence (x _k(n)) so that (Tx _k(n)) is weakly uniformly convergent for T ∈ M. We study some properties of weakly equicompact sets and, among other results, we prove: 1) if $M \subset \mathcal{W}(X,Y)$ is collectively weakly compact, then M ^* is weakly equicompact iff M ^** x ^**={T ^** x ^** : T ∈ M} is relatively compact in Y for every x ^** ∈X ^**; 2) weakly equicompact sets are precompact in $\mathcal{L}(X,Y)$ for the topology of uniform convergence on the weakly null sequences in X.