We study the relation between the polynomial numerical indices of a complex vector-valued function space and the ones of its range space. It is proved that the spaces C(K, X) and L∞(μ, X) have the same polynomial numerical index as the complex Banach space X for every compact Hausdorff space K and every σ-finite measure μ, which does not hold any more in the real case. We give an example of a complex Banach space X such that, for every k ≥ 2, the polynomial numerical index of order k of X is the greatest possible, namely 1, while the one of X** is the least possible, namely kk/(1−k). We also give new examples of Banach spaces with the polynomial Daugavet property, namely L∞(μ, X) when μ is atomless, and Cw(K, X), Cw*(K, X*) when K is perfect.
Journal Article. 0 words.
Subjects: Pure Mathematics
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