Mean separations in Banach spaces under abstract interpolation and extrapolation

We introduce the mean separation for bounded sequences in Banach spaces and the related seminorm for bounded linear operators. The introduced quantities are closely related to the geometric characterizations of the Banach-Saks property and the alternate signs Banach-Saks property. We investigate the behavior of the mean separations for a class of operators between vector-valued Banach sequence spaces E(Xν), providing that a Banach sequence lattice E has the Banach-Saks property. We estimate the mean separations for operators under abstract interpolation and extrapolation methods. In particular, we obtain quantitative and qualitative results on the heredity of the Banach-Saks properties under these methods.