Asplund operators and A. E. convergence

. Let E and F be Banach spaces, and U: E --> F a bounded linear operator. The following are equivalent: 1. o2. (a) U is an Asplund operator.3. (b) Let (Xn) be a sequence of Bochner measurable functions with values in E and supn||;Xn|| < [infinity] a.e. If (Xn) converges scalarly to 0 in E, then (UXn) converges to 0 weakly in F almost everywhere.4. (c) If (Xn) is a weak sequential potential of class (B) in E, then (UXn) converges to 0 weakly in F almost everywhere.5. (d) If (Xn) is a strong potential of class (B) in E, then (UXn) converges to 0 weakly in F almost everywhere.