Arithmetic separation and Banach–Saks sets

Abstract

We introduce a measure of deviation from the Banach–Saks property for bounded subsets of Banach spaces. The measure is based on the arithmetic separation of a sequence, which is a close counterpart of James’ condition of weak noncompactness. We apply this measure to the polygon interpolation method for bounded linear operators on Banach N-tuples. In particular, we show distributions of operators with the Banach–Saks property among the polygon vertices, which imply this property for all interpolated operators. We establish similar results for a measure of deviation from the alternate signs Banach–Saks property.