Weakly equicompact sets of operators defined on Banach spaces

Let X and Y be Banach spaces. We say that a set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M \subset \mathcal{W}(X,Y)$$\end{document} (the space of all weakly compact operators from X into Y) is weakly equicompact if, for every bounded sequence (xn) in X, there exists a subsequence (xk(n)) so that (Txk(n)) is weakly uniformly convergent for T ∈ M. We study some properties of weakly equicompact sets and, among other results, we prove: 1) if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M \subset \mathcal{W}(X,Y)$$\end{document} 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{L}(X,Y)$$\end{document} for the topology of uniform convergence on the weakly null sequences in X.