The ( ð · ) Property in Banach Spaces

A Banach space ð ¸ is said to have ( D ) property if every bounded linear operator 𠑇 ∶ ð ¹ â†’ ð ¸ âˆ— is weakly compact for every Banach space ð ¹ whose dual does not contain an isomorphic copy of ð ‘™ ∞ . 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 ð ¿ 1 ( ð ‘£ ) of real functions, which are integrable with respect to a measure ð ‘£ with values in a Banach space ð ‘‹ , has ( D ) property. We give some other results concerning Banach spaces with ( D ) property.