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.
Volume 131, 2021
Continuous Article Publishing mode
Click here for Editorial Note on CAP Mode