The structure of solid analytic FK-AK spaces

We study analytic, solid sequence spaces λ satisfying $c_{00}\subseteq \lambda \subseteq {\ell }^{\infty }$. We prove that if λ contains an element with full support, and any countable subset of the positive cone ${\lambda }^{+}$ is eventually dominated pointwise by a single element in ${\lambda }^{+}$, then λ admits an F-norm, making it into an FK space with the AK property. This characterization closely parallels S. Solecki's well known characterization of analytic P-ideals of subsets of the natural numbers. Additionally, we provide a general method to construct FK spaces with the AK property from an additive sequence of closed subsets of the positive cone of $R^N$. Our results reveal that analytic P-ideals can be viewed as a discrete analogue of FK spaces with the AK property.