Lifting unconditionally converging series and semigroups of operators

Abstract

We introduce and study two semigroups of operators u+ and u_, defined in terms of unconditionally converging series. We prove a lifting result for unconditionally converging series that allows us to show examples of operators in u+. We obtain perturbative characterisations for these semigroups and, as a consequence, we derive characterisations for some classes of Banach spaces in terms of the semigroups. If u+(X, Y) is non-empty and every copy of c0 in Y is complemented, then the same is true in X. We solve the perturbation class problem for the semigroup u_, and we show that a Banach space X contains no copies of ℓ∞ if and only if for every equivalent norm |·| on X, the semiembeddings of (X, |·|) belong to u+.