A structure theorem for echelon Köthe spaces

Abstract

In [9], Lopez-Molina defined the echelon Köthe spaces $Λ^p(\Chi,?,\mu,g_k)$, which provide a suitable generalization of the echelon sequence spaces $λ^p(a_{n}^k)$ to a general measure space $\Chi, ?, \mu$ (see also [5]). In this paper, we show that the strutture of the separable echelon Köthe spaces is «nicely» close to the strutture of the echelon sequence spaces. Namely, our main result is: Let $Λ^p(\Chi,?,\mu,g_k)$ be a separable echelon Köthe space of order p, $1 ≤ p ≤ ∈fty$, with $(\Chi,?,\mu,g_k)$ purely non-atomic. Then, there is an echelon sequence space $λ^p$ so that $Λ^p$ is isomorphic to the space $λ^p(L^p)$ of all $λ^p$-summable sequences in $L^p$. An an application we show that a separable echelon Köthe space has a basis (by a basis we mean a Schauder basis), which is unconditional if $p > 1$.