SOME OBSERVATIONS ON THE NUMERICAL INDEX AND THE POLYNOMIAL NUMERICAL INDEX

SOME OBSERVATIONS ON THE NUMERICAL INDEX AND THE POLYNOMIAL NUMERICAL INDEX

In this paper, we study both the numerical index and the polynomial numerical index. First, we give a sufficient condition for a Banach space X to have lushness. Second, we study the relation between the renormings of a Banach space and the k-order polynomial numerical index. This shows that every real Banach spaces of dimension greater that 1 can be renormed to have 2-order polynomial numerical index ${\alpha}$ for any ${\alpha}{\in}[0,1/18)$.

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