A permanence theorem for sums of sequence spaces

Abstract

Let I be the space of absolutely summable sequences. Using difficult functional analytic techniques Bennett proved that if X is a separable FK space containing 8 for all n and if 8 0 in X + 1, then I c X. Bennett also asked whether the separability assumption can be dropped. Using an elementary invertibility criterion for Banach algebras, the present note gives a self-contained proof that if z is a null sequence, X is an FK space containing 3 for all n, and X + zi = l, then X = l. This answers Bennett's question in the affirmative.