Polynomial numerical indices of Banach spaces with 1-unconditional bases (Q448341)

This interesting paper is devoted to show that there are few Banach sequence spaces with absolute norm having polynomial numerical index of order \(2\) equal to \(1\). More precisely, it is proved that \(c_0\) is the only infinite-dimensional complex space with \(1\)-unconditional basis having a polynomial numerical index of order \(2\) equal to \(1\) and that, in the real case, there is no space of this type. An analogous result is obtained in the non-separable case, provided that the Köthe dual is norming: If \(X\) is an infinite-dimensional complex Banach sequence space with absolute norm having a polynomial numerical index of order \(2\) equal to \(1\), then \(c_0\subset X\subset \ell_\infty\). Again, in the real case there are no such spaces.NEWLINENEWLINEAs an auxiliary result, the authors give a characterization of separable lush spaces which is of independent interest.