Weakly equicompact sets of operators defined on Banach spaces

Let X and Y be Banach spaces. We say that a set (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 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 for the topology of uniform convergence on the weakly null sequences in X. Received: 14 February 2005; revised: 1 June 2005