Deviation from weak Banach–Saks property for countable direct sums

Abstract

Abstract. We introduce a seminorm for bounded linear operators between Banach spaces that shows the deviation from the weak Banach-Saks property. We prove that if (Xν ) is a sequence of Banach spaces and a Banach sequence lattice E has the Banach-Saks property, then the deviation from the weak Banach-Saks property of an operator of a certain class between direct sums E(Xν ) is equal to the supremum of such deviations attained on the coordinates Xν . This is a quantitative version for operators of the result for the Köthe-Bochner sequence spaces E(X) that if E has the Banach-Saks property, then E(X) has the weak Banach-Saks property if and only if so has X. Introduction.A Banach space X is said to have the Banach-Saks (BS) property if every bounded sequence in X contains a subsequence (x n ) whose Cesàro means