Deviation from weak Banach–Saks property for countable direct sums

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 (X v ) 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(X v ) is equal to the supremum of such deviations attained on the coordinates X v . This is a quantitative version for operators of the result for the Kothe–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.