Invariant subspaces for Fréchet spaces without continuous norm

Abstract

Let ( X , ( p j ) ) (X,(p_j)) be a Fréchet space with a Schauder basis and without continuous norm, where ( p j ) (p_j) is an increasing sequence of seminorms inducing the topology of X X . We show that X X satisfies the Invariant Subspace Property if and only if there exists j 0 ≥ 1 j_0\ge 1 such that ker ⁡ p j + 1 \ker p_{j+1} is of finite codimension in ker ⁡ p j \ker p_{j} for every j ≥ j 0 j\ge j_0 .