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

Abstract

We prove that if $X,Y$ are Banach spaces, $\Omega$ a compact Hausdorff space and $U:C(\Omega ,X)\to Y$ 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 $\Vert G\Vert$ is continuous at $\varnothing$. 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 $l_p$, ($1\le p<\infty$) be Dunford--Pettis and/or compact operators, in which, Khinchin's inequality plays an important role.

18 pages