2-Summing Operators on $C([0, 1], l_p)$ with Values in $l_1$
Banach spaces of continuous functions; tensor products; operator ideals; 𝑝-summing operators.
Let 𝛺 be a compact Hausdorff space, 𝑋 a Banach space, $C(𝛺,X)$ the Banach space of continuous 𝑋-valued functions on 𝛺 under the uniform norm, $U:C(𝛺,X)→ Y$ a bounded linear operator and $U^\#,U_\#$ two natural operators associated to 𝑈. For each $1≤ s <∞$, let the conditions $(𝛼)U\in 𝛱_s(C(𝛺, X), Y);(𝛽)U^\#\in 𝛱_s(C(𝛺), 𝛱_s(X, Y));(𝛾)U_\#\in 𝛱_s(X, 𝛱_s(C(𝛺), Y))$. A general result, [10,13], asserts that (𝛼) 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.