2-Summing Operators on $C([0, 1], l_p)$ with Values in $l_1$
Let 𝛺 be a compact Hausdorff space, 𝑋 a Banach space, $C(\Omega,X)$ the Banach space of continuous 𝑋-valued functions on 𝛺 under the uniform norm, $U:C(\Omega,X)\to Y$ a bounded linear operator and $U^\#,U_\#$ two natural operators associated to 𝑈. For each $1\leq s <\infty$, let the conditions $(\alpha)U\in \Pi_s(C(\Omega, X), Y);(\beta)U^\#\in \Pi_s(C(\Omega), \Pi_s(X, Y));(\gamma)U_\#\in \Pi_s(X, \Pi_s(C(\Omega), Y))$. A general result, [10,13], asserts that $(\alpha)$ implies (𝛽) and (𝛾). In this paper, in case $s=2$, we give necessary and sufficient conditions that natural operators on $C([0,1],l_p)$ with values in $l_1$ satisfies (𝛼), (𝛽) and (𝛾), which show that the above implication is the best possible result.