Khinchin's inequality, Dunford--Pettis and compact operators on the space $\pmb{C([0,1],X)}$

Abstract

We prove that if $X,Y$ are Banach spaces, $\Omega$ a compact Hausdorff space and $U\hbox{\rm :} C(\Omega,X)\to Y$ is a bounded linear operator, and if $U$ is a Dunford--Pettis operator the range of the representing measure $G(\Sigma) \subseteq DP(X,Y)$ is an uniformly Dunford--Pettis family of operators and $\|G\|$ is continuous at $\emptyset$. As applications of this result we give necessary and/or sufficient conditions that some bounded linear operators on the space $C([0,1],X)$ with values in $c_{0}$ or $l_{p}$, ($1\leq p<\infty$) be Dunford--Pettis and/or compact operators, in which, Khinchin's inequality plays an important role.