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

Dumitru Popa
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Abstract

We prove that if $X,Y$ are Banach spaces, $Ω$ a compact Hausdorff space and $U\hbox{\rm :} C(Ω,X)\to Y$ is a bounded linear operator, and if $U$ is a Dunford--Pettis operator the range of the representing measure $G(Σ) \subseteq DP(X,Y)$ 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.

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Dumitru Popa. 2007-03-21. Khinchin's inequality, Dunford--Pettis and compact operators on the space $\pmb{C([0,1],X)}$. https://arxiv.org/abs/math/0703626

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