| Abstract: | We prove that in a Banach space admitting a separating polynomial, each weakly null normalized sequence has a subsequence which is equivalent to the usual basis of some , p an even integer. We show that for each even integer p, the Schatten class admits a separating polynomial. For a space with a basis admitting a 4-homogeneous separating polynomial, we relate the unconditionality of the basis with the existence of certain type of polynomials defined in terms of infinite symmetric matrices. We find relations between the properties of the entries of these matrices and the geometrical structure of the space. Finally we study the existence of convex 4-homogeneous separating polynomials. |
