The (𝐷) Property in Banach Spaces

Abstract

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.