The ( D ) Property in Banach Spaces

Abstract

A Banach space $E$ is said to have (D) property if every bounded linear operator $T:F\to E^{\mathrm{\{\backslash ast\}}}$ is weakly compact for every Banach space $F$ whose dual does not contain an isomorphic copy of $l_{\infty }$. 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 $L^1\left(v\right)$ 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.