## Completely Continuous operators

### Volume 126 / 2012

#### Abstract

A Banach space $X$ has the Dunford–Pettis property (DPP) provided that every weakly compact operator $T$ from $X$ to any Banach space $Y$ is completely continuous (or a Dunford–Pettis operator). It is known that $X$ has the DPP if and only if every weakly null sequence in $X$ is a Dunford–Pettis subset of $X$. In this paper we give equivalent characterizations of Banach spaces $X$ such that every weakly Cauchy sequence in $X$ is a limited subset of $X$. We prove that every operator $T:X\to c_0$ is completely continuous if and only if every bounded weakly precompact subset of $X$ is a limited set.

We show that in some cases, the projective and the injective tensor products of two spaces contain weakly precompact sets which are not limited. As a consequence, we deduce that for any infinite compact Hausdorff spaces $K_1$ and $K_2$, $C(K_1)\otimes _\pi C(K_2)$ and $C(K_1)\otimes _\epsilon C(K_2)$ contain weakly precompact sets which are not limited.