Some observations on the numerical index and the polynomial numerical index

摘要

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 $lpha$ for any $lphain [0,1/18)$.