POLYNOMIAL NUMERICAL INDEX FOR SOME COMPLEX VECTOR-VALUED FUNCTION SPACES

Abstract

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 (p, X) have the same polynomial numerical index as the complex Banach space X for every compact Hausdorff space K and every sigma-finite measure mu, 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 k(k/(1, k)). We also give new examples of Banach spaces with the polynomial Daugavet property, namely L-infinity(mu, X) when mu is atomless, and C-w(K, X), C-w*(K, X*) when K is perfect.