The (D) Property in Banach Spaces

A Banach space E is said to have (D) property if every bounded linear operator T : F → E* is weakly compact for every Banach space F whose dual does not contain an isomorphic copy of l∞. Studying this property in connection with other geometric properties, we show that every Banach space whose dual has (V∗) property of Pełczyński (and hence every Banach space with (V) property) has (D) property. We show that the space L1(v) of real functions, which are integrable with respect to a measure v with values in a Banach space X, has (D) property. We give some other results concerning Banach spaces with (D) property.