Khinchin's Inequality, Dunford-Pettis and Compact Operators on the Space $C([0, 1], X)$
We prove that if $X, Y$ are Banach spaces, 𝛺 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 $G(\Sigma)\subseteq D P(X, Y)$ 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.
Volume 129 | Issue 5
Click here for Editorial Note on CAP Mode