A permanence theorem for sums of sequence spaces

Abstract:

Let $l$ be the space of absolutely summable sequences. Using difficult functional analytic techniques Bennett proved that if $X$ is a separable FK space containing ${\delta ^n}$ for all $n$ and if ${\delta ^n} \to 0$ in $X + l$, then $l \subset X$. 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 ${\text {FK}}$ space containing ${\delta ^n}$ for all $n$, and $X + zl = l$, then $X = l$. This answers Bennett’s question in the affirmative.